
61 F.2d 58 (1932)
DON LEE, Inc.,
v.
WALKER.
No. 6700.
Circuit Court of Appeals, Ninth Circuit.
August 16, 1932.
Frank Parker Davis, of Chicago, Ill., William H. Hunt, of San Francisco, Cal., Harry W. Lindsey, Jr., of Chicago, Ill., and Charles M. Fryer, of San Francisco, Cal., for appellant.
Chas. E. Townsend, Wm. A. Loftus, and Felix T. Smith, all of San Francisco, Cal., for appellee.
Before WILBUR and SAWTELLE, Circuit Judges.
WILBUR, Circuit Judge.
This action was brought by Clinton L. Walker, appellee, to recover damages for the infringement of a patent, No. 1,575,239, issued to him March 2, 1926, by the United States Patent Office for an improvement in method of counterbalancing engine main shafts, and to enjoin further infringement. The court entered an interlocutory decree affirming the validity of the patent and finding that the patent had been infringed by the defendant, Don Lee, Incorporated, and by General Motors Corporation, a Delaware corporation, manufacturer of the Cadillac automobile which was in privity with the defendant and has assumed the defense of the action, enjoining further infringement by the defendant, Don Lee, Incorporated, and all those in privity with the defendant, and ordered an accounting to ascertain the profits realized by such infringement. From this interlocutory decree the defendant appeals, claiming that the patent is invalid and that there was no infringement thereof, even if valid.
The patent, as indicated by its title, is one for a "method" of counterbalancing main engine "shafts," not engines, and, as the usual and obvious method of counterbalancing eccentric weights is by other weights, the patent counts upon the method of ascertaining such weights and their points of application to the main shaft, that is, their number and their longitudinal positions on the main shaft, and the distance of the center of gravity thereof from the center of the shaft. The patent does not specify the weights nor the longitudinal positions thereof, except as hereinafter *59 noted in claim 3, nor their distance from the center of the shaft. The counterweights necessarily depend upon the weights to be balanced, and these depend upon the design of the engine which is not covered by the patent. The patent, therefore, purports to cover a method of arriving at these factors of weight and position by a computation described in the patent. In short, it is a patent of a mathematical formula for the solution of a problem in dynamics. There are five claims in the patent. Claims 1, 2, 3, and 5 relate to the balancing of the main shaft of a particular type known in the engineering field as the "Sharp type." The first claim of the patent is as follows:
"Having thus described my invention, what I claim and desire to secure by Letters Patent is:
"1. A method of counterbalancing an engine main shaft having four throws, with the throws set as follows: Throws 1 and 4 set at 180° apart; throws 2 and 3 set at 180° to each other but at 90° to throws 1 and 4; which consists in treating each half of the shaft as a cantilever supported at the central transverse plane of the shaft, determining the bending moment of the cantilever at the point of support produced by the centrifugal force of each throw, multiplied by the distance from the central transverse plane of the shaft, and counteracting this bending moment by weights of such mass and radius of mass center and distance from the central transverse plane of the shaft that their bending moment will be equal and opposite to that of the throws to be balanced."
The patent (page 2, lines 2 to 5, inclusive) discloses the conventional formula for determining the centrifugal force: "* * * Centrifugal force equals .00034 WRN2, where W is the weight, R its radius of gyration, and N the number of revolutions."
Claim No. 1 amounts to this, given the centrifugal force of each throw ascertained by the conventional formulæ, and for convenience indicated by the letters F, F2, F3, and F4, for throws 1, 2, 3, and 4, respectively, and representing the distance from the central transverse plane as L1, L2, L3, and L4, respectively, we have the bending moment on one side of the transverse plane which we will indicate by B, as L1 F1 plus L2 F2=B, and on the other side, which we will indicate as B2, as follows: L3 F3 plus L4 F4=B2. Having thus ascertained B, we are to counterbalance, or counteract, it by "weights of such mass (x) and radius of mass center (y) and distance from the transverse plane (z) that their bending moment will equal B, and on the other side of the reference plane similarly B2."
Let x equal the weight and y the radius and z equal its distance from the transverse plane on one side, we have (.00034 xyn2) z=B. Thus, we have three unknown quantities in the equation. The claimant does not undertake in this equation to specify any one of the unknown quantities, but simply claims that whenever the three, x, y, and z, are so combined in the above formulæ that they equal B, his patent is infringed. In this connection the following criticism by claimant's solicitors in their argument to the Patent Office on Walker Amendment C, of Sharp's solution of the problem in his book published in 1907, p. 117, is interesting: "No data is given as to how to compute the required masses, their axial angles, nor their position longitudinally on the shaft"; they claim Sharp "does not determine the mass, the radius of mass center, the longitudinal position nor the axial angle of the required counterbalance weights." (See Dr. Durand's testimony in footnote on this subject.)
Upon this question, Dr. Durand, one of the defendant's experts, also testified upon cross-examination as follows: "Q. Having once determined a moment such as Walker explains in his patent and deciding to compensate for deflection by placing a relatively large weight between throws one and two and three and four, would there be any particular problem in putting part of each weight at some other point on the shaft? A. No problem at all, simply a matter of mechanics as in all these cases of substitution and equivalents."
It should be added here, however, as will appear subsequently herein, that Dr. Durand regarded the whole matter of compensating both the dynamic forces and the bending moment as a purely engineering problem to be solved by mathematics and not one requiring inventive genius.
Claim 2 is as follows: "2. A counterbalanced engine main shaft having four throws, with throws 1 and 4 set 180° apart and throws 2 and 3 set 180° apart but at 90° from throws 1 and 4, and counterbalancing weights for each half of the shaft, said weights being positioned with respect to the throws to be balanced so that their center of gravity falls within the angle produced by extending the center line of the outer throw and a line bisecting the angle of the two throws."
As the angle between throws 1 and 2 is 90°, the bisector of this angle is 45° from *60 both, and the counterweight is to be placed somewhere within the 45° angle formed by this bisector and the center line of the outer throw extended.
Claim 3 places the counterweights as follows: "* * * One of such weights on each half of the shaft being positioned midway between the end throw and the next adjacent throw, whereby to prevent deflection of the shaft between the center and end thereof."
It will be observed that this claim fixes the unknown quantity (z) in the formulæ above stated.
Claim 5 is as follows: "5. A counterbalanced engine main shaft having four throws, with throws 1 and 4 set 180° apart, and throws 2 and 3 set 180° apart but at 90° from throws 1 and 4, and a single counterbalancing weight positioned between the two throws on each half of the shaft and having its center of gravity within the angle produced by extending the center line of the outer throw and a line bisecting the angle of the two throws, such weight being effective to assist in counterbalancing the two adjacent throws."
This claim does not give the method of ascertaining the counterbalancing weights as in claims 1 and 3, but places them anywhere between the two throws (not midway as in claim 2).
Claim 5 is not involved here. That the patent emphasizes the bending effect on the shaft of all off center, or eccentric masses on the shaft moving at high velocity, is evident. That the claims are based upon a method of meeting this difficulty is apparent.
With these preliminary observations we proceed to a consideration of the evidence. There does not seem to be any great conflict, if there is any, among the experts who testified, although the terminology and emphasis varies somewhat. Dr. Lydik S. Jacobsen, testifying in rebuttal as an expert (note 1, see end of opinion), stated that in the main he agreed with Dr. Durand.
Dr. Wm. F. Durand, who testified as an expert (note 2, see end of opinion) for the defendant, after quoting from the patent the object thereof as stated page 1, lines 40 to 49, and p. 1, lines 26 to 32, inclusive, he says: "There is certainly nothing contained in the disclosure of the patent which is not contained in the text-books, and in so far as the application of the method outlined in the patent is narrowed to the specific investigation of one-half of the shaft at a time, it would seem to narrow the application of this particular method to a shaft of the Sharp type, whereas, considering the whole shaft at once as a complex system of masses and forces, we arrive at a general result applicable in any and all cases." In other words, as we understand it, general and accepted formulæ would give a correct result in all cases, but the method of the patent only gives a correct result in one particular case included in the general formula. If this were true, no one would contend that a special application of a general formula known to the engineering world could be patented, for the greater includes the lesser. Again he says:
"I have failed to find anything which to me is distinctly novel in the disclosure of the patent. The disclosure seems to emphasize certain elements of procedure in principle, the selection of a central reference plane is certainly not novel, because, from the various early days of the science of mechanics, we have found that we could take our reference plane anywhere. The whole world is before us, so far as this one goes, for the selection of that reference plane, as I have attempted to demonstrate a few minutes ago.
"In so far as the matter of reducing the ultimate balance to expression in terms of two forces or two masses is concerned, that certainly is not novel, for, from the beginning of time, with reference to the science of mechanics, we have known that a number of forces may be represented by one, or again, one may be represented by many; it is one of those basic elements of equivalents, if many may be represented by one, one may be represented by many, and the many may be in numbers 2 or more. So that those two basic features, at least, are entirely well known and grounded in the very fundamentals of the science of mechanics."
When the witness' attention was drawn to the following argument of the patentee's solicitors, in amendment C, dated July 7, 1925, marked, "Paper No. 8": "What applicant has done is to provide a simple and accurate method whereby such a shaft may be balanced against the centrifugal forces of the oppositely disposed cranks," he said, "He has, of course, shown a method which gives the correct result in the case of the Sharp shaft. It gives the same result as that which would be arrived at by following any of the other methods to which reference has been made or more fundamentally by applying the well-known elementary principles of mechanics of this particular problem."
In support of the contention of Dr. Durand and of the appellant that the problem *61 of balancing the moving parts of an internal combustion engine is one familiar to the engineering profession, and treated extensively in the text-books and papers of that profession, several such articles were introduced in evidence, including one published in 1902 by W. E. Dalby, Professor of Engineering and Applied Mathematics in the City and Guilds of London Institute, Technical College, Finsbury; also a text-book on High Speeds Internal Combustion Engines, by Arthur W. Judge, Diplomate of Imperial College of Science and Technology, Associate Member of the Institution of Automobile Engineers, and particularly chapter VII dealing with "Engine Balance"; also a text-book, published in 1907 by Archibald Sharp, B. S., member of the Institution of Automobile Engineers, etc.; a text-book on the "Theory of Machines," by Robt. W. Angus, member of the American Society of Engineers, Professor of Mechanical Engineering, University of Toronto, etc., and particularly section 245 thereof dealing with "Balancing any number of rotating masses located on different planes normal to a shaft revolving at uniform speed." These articles were also referred to by the patentees in their paper No. 8, above mentioned, in connection with their amended application C (July 7, 1925), wherein they attempted to distinguish the claimant's method from those theretofore known. Dr. Durand was examined extensively upon this attempted distinction between the claimant's method and those used by these authors, and his testimony is so clear, definite, and convincing to the effect that these authors not only recognized the problem dealt with by the claimant, but also disclosed methods of solving the same which in the particular form of engine and crankshaft dealt with by claimant produced the same result, that we append extensive quotations from his testimony on the subject (note 3, see end of opinion), and give his definitions of the mechanical terms used in subnotes thereto.
It will assist in determining the validity of the patent in question to consider the claim made herein of infringement, for that claim will exemplify the character of monopoly claimed by the patentee, although later we will return to the question of the prior art as it affects the claimed monopoly. The appellee answers the question in his brief as follows:
"Was the Cadillac shaft counterbalanced according to Walker's method?
"The Walker method of course is the method of moments, using the central transverse plane as the reference plane and placing the weights on the shaft in the most convenient and desirable position, whereby to bring about static balance, dynamic balance, and deflection balance, while employing fewer weights than were required under the old system of equal and opposite weights. The shaft itself, Defendant's Exhibit 10, and the admissions of Chase heretofore noted, are amply sufficient to prove the presence of the Walker method in the Cadillac shaft."

"Defendant's Infringing Device.
"The device complained of is a fourweighted shaft as now used in the Cadillac and LaSalle automobiles, and is represented diagramatically in the sketch below.

"It will be noted that such a shaft contains eight crank cheeks, representing off-center masses, each of which would have to be balanced by an equal and diametrically opposite weight, amounting to eight weights (or six under the modified method first tried out by the engineers for the Cadillac Company and which was admitted to be unsuccessful). However, the infringing shaft is a fourweighted *62 shaft balanced according to Plaintiff Walker's methods, and so successful in practice that the General Motors Corporation advertised it as `an epochal achievement,' and included it in all LaSalle automobiles when that type of car was later introduced to the public as a supplement to the Cadillac line.
"As Walker was the first to work out this three-bearing crankshaft of the type in question, with a lesser number of weights than the off-center masses to be balanced, making due provision for deflection balance, he is entitled to rank as the inventor; and it is not seriously contended by defendant that such an arrangement and method are not to be found in the Cadillac and LaSalle shafts. As a matter of fact, the defendants did not even contend that the four claims in suit were not readable upon the Cadillac shaft, except to argue that there were methods other than the method of moments calculated from the central, transverse plane, as a reference plane, whereby a similar result might be accomplished.
"However, General Motors' star witness, Mr. Chase, admits that he used the Walker method of moments when it became necessary to reduce the number of counterweights from six to four."
The testimony of Mr. Chase thus referred to is as follows:
"XQ. What method did you follow in those computations? A. Early in the work I used the methods exhibited by or disclosed in Exhibit 3. Later on when we began using less than six weights, it became necessary to use other methods.
"XQ. What was that other method? A. Moments of weights around the center of the shaft.
"XQ. Using the central transverse plane as the reference plane? A. Yes, sir. I believe there is something in this Dalby book regarding the use of the central transverse plane as a reference plane. There is an example on page 150 of a symmetrical engine. There is a reference plane in the center of the shaft there. Here is another illustration of a six throw shaft, using a central reference plane. I also mentioned Judge in this first example on page 150 that I referred to. The cranks are on either side of the reference plane, a similar distance from the reference plane; that is, cranks 2 and 3 are both the same distance from the reference plane. Cranks 1 and 4 are the same distance from the reference plane. The angular position between cranks 2 and 3 is shown as 113 degrees and 8 minutes. The angle between cranks 3 and 4 is 148 degrees and 13 minutes. The angle between cranks 2 and 1 is 148 degrees and 13 minutes. I suppose the method which Professor Dalby uses would produce a separate weight for each crank-pin, but I have not examined it carefully enough.
"Referring to the book by Judge I find the reference plane at the central transverse point on page 325. That is a case of a two-cylinder engine with cranks at 90 degrees. The reference plane is taken midway between the two cranks. I do not find in either of those books any shaft like Exhibit 10, showing an example worked out like this shaft, Exhibit 10. This example given on page 150 of Dalby's book hasn't anything to do with counterbalancing weights. It merely suggests the possibility of solving a problem of that kind with a central reference plane, but not for the purpose of computing counterbalance weights. I merely noted that it showed the reference plane in the center of the shaft. This example is not a parallel case. It is simply a suggestion for the use of the central reference plane, and for no other purpose. In this second example in Professor Dalby's book, page 158, the purpose of selecting the central reference plane hasn't anything to do with computing counterbalance weights. This example in Judge's work, page 325, has to do with the force produced by the inertia forces. There is no mention made there of any counterbalance weights. The example is not cited for the purpose of showing the use of counterbalance weights."
If, as the appellee contends, Chase's testimony is an admission of infringement, it is because the patentee has a monopoly upon a system of calculation of forces to be balanced upon a revolving shaft which takes account of the forces to be balanced, that is, centrifugal force of eccentric weights "around the center of the shaft," and the use of the central transverse plane as a reference plane, that is, the calculation of the bending, or elastic, or deflection force, whichever it may be called, with reference to that plane, or, perhaps more properly, with reference to the point of intersection of that plane with the center line of the shaft.
It is clear then that the patentee seeks a monopoly of a formula for determining dynamic forces, and this although those forces were fully recognized and considered by engineers in published text-books long before the appellee applied for his patent. Dr. Jacobsen, testifying for patentee, stated in part as follows: "I have examined and believe I understand the prior patents, the publications and the prior uses referred to herein. *63 I have been present here in the courtroom during the trial. I find that the combination disclosed in the Walker patent differs from all the other references given. It differs especially in regard to the attention paid to the question of elastic balance or deflection balance. In explaining how that is taken care of in the Walker patent, I find in the preamble or introduction of the Walker patent, No. 1,575,239, on page 1, line 55, a sentence beginning like this: `In other words, the shaft may be designed to give a better balance of the inertia forces of the reciprocal masses and to more nearly neutralize the bearing pressures caused by the centrifugal forces of the rotating masses.'"
But on cross-examination the witness testified as indicated in the footnote that "the method of arriving at the magnitude of the balance weight, or at the magnitude of the balancing couples, does not affect the quantity of the balance at all." Brooks Walker, a mechanical engineer graduating from the University of California in 1925 and since devoting his time to automotive and combustion engines, was also relied upon by the appellee to establish infringement.
In dealing with the question of priority, the appellee's contention as to the scope and nature of his patent and to the question of infringement is again manifested. He quotes from a letter written by him in October, 1918, to Pierson. After stating therein the nature of the Sharp shaft, he says: "I then counterbalance it with a pair of weights, one between cranks one and two, and the other between three and four." The brief then states: "The foregoing presents a complete disclosure and description of the exact method involved in this controversy and is comprehensive enough to embrace the Chase-Cadillac shaft of four-weighted variety." It will be observed, if this statement is to be taken seriously, that the appellee assumes therein that, given the statement that the counterweights were between cranks as indicated, a mechanical engineer could compute the mass, radius, and angular position thereof, and this is exactly in accord with the testimony of Dr. Durand. Appellee also contends that four weights, as used in the "Chase-Cadillac" shaft, is the mechanical equivalent thereof, and consequently covered by the patent. That they (the four weights) are the mechanical equivalent of two weights, if the respective weights are properly placed in each case, is freely conceded by the experts, in fact, Dr. Durand fully explains that fact. If then it is true that four weights, or any other number of weights properly placed, are the mechanical equivalent of two weights placed between the cranks, and therefore covered by the patent, the appellee's claim resolves itself into a claim of a monopoly on the idea of counterbalancing a crankshaft at all, because the only method of doing so is by counterweights properly placed. Of course, in view of the fact that this idea is old and is discussed and solved in the text-books long before the appellee claims to have conceived the idea, the claim of invention and of monopoly thus broadly stated must fall. In justice to the appellee and his expert witnesses, it is proper to say that it is made clear by such witnesses that the appellee's only claim to novelty in the placing of the two weights is that they are so placed that they not only balance the opposing off-center masses dynamically, but also balance the bending, turning, deflection, or elastic moment about the point in the center line of the shaft intersected by "the central transverse plane." This is made clear by the appellee himself in his testimony as to his disclosure to Mr. Chase of the General Research Corporation, in part, as follows:
"And the interest of the entire discussion, which occupied the whole forenoon, was practically all in regard to the crank arrangement and the method of counterbalancing, and I explained in detail to them how I had proceeded to balance the shaft, using the central transverse plane and computed the centrifugal forces of the off-center masses and composing them into a single component which I balanced with a single weight in this case, providing an equal and opposite bending moment to the bending moment of the off-center masses. I then explained to him how, the two ends of the motor being alike, one computation sufficed for both ends of the motor."
"I explained to them that where I had balanced the shaft with one weight on each half, that I could divide those weights into two or more weights, providing I preserved the same plane of action for the resultant of those weights as I would have for the single weight; that in my computation for the off-center masses and the weights, it resolved itself down, after eliminating certain constants that are there for various speeds, into a product of w., which would represent the weight, r., which would represent its radius of mass center, and l, which is the distance from the central transverse plane; that when that w. r. l. of the off-center mass is once determined, a weight or more weights could be applied to *64 the shaft to put it in balance, so long as you stayed in the same plane of action, and so long as the combined w. r. l. of the weight is equal to that of the single weight. I went into this very fully with them and I pointed out to Mr. Chase and to the others how I balanced my weights and why I had placed them so."
In short, the plaintiff claims a monopoly of any method of balancing the bending moment of the Sharp shaft about its central point by weights which also balance the eccentric masses "dynamically." Appellee in his brief in effect so contends. We quote to that effect as follows:
"Walker's use of four weights.
"It ought to be sufficient answer to the argument of the appellant that there is no corroborated testimony which would indicate that Walker had any conception of a 4weighted shaft prior to the time he saw the Cadillac shaft (which was late in 1923), to refer to the obvious fact that the invention is of such a nature as to make it impossible for one to have a conception of one form without having a conception of the other. Dr. Durand concedes this point, where he testified:
"'Q. Isn't it a fact that Walker employs a balancing means located off the 45-degree line, that is where he uses the single weight on each half  does the 45-degree line indicate to you that he knew how to balance that shaft with more than one weight on each half? A. It would.
"`Q. In other words, he has combined two vectors into one? A. Yes.
"`Q. And it would follow from that that he could resolve one vector into those two components? A. Undoubtedly.'
"The Walker notebook (Exhibit 8) kept in regular course of his experiments and seen in daily use by him by his son Brooks and by his mechanic Hoffman during the period prior to the completion of the Walker engine, contains the following entry:
"`I could put part of the weight in the flywheel and part on the shaft between crank 1 and 2 on one end and 3 and 4 on the other, or perhaps put it all there. To put the weight intermediate, I would have to separate the blocks and do away with the two bearings. It would be best to put just enough intermediate to oppose the tendency of the shaft to deflect and put the balance required at the end of the shaft where space permitted.'
"This is a complete disclosure of the fourweighted Cadillac shaft."
Before closing the discussion upon the nature and scope of the appellee's patent claims, we must consider the prior patents introduced in evidence by appellant. The one involving disclosures most nearly like the appellee's claim (No. 2) is the German patent issued to Klose in January, 1911. This patent shows one or two counterbalances placed on each half of a four-throw shaft similar to the Sharp shaft. The center of gravity of these weights falls within the angles produced by extending the center line of the outer throw and a line bisecting the angle of the two throws. It should also be noted that the Klose patent deals with the question of balancing out of the tilting moments in such a shaft. There is an obvious mistake in the drawing accompanying the Klose patent, in that at one end the weights g g0 are interchanged; however, as pointed out by Dr. Durand, the text deals with couples having equal weights, and the drawing could not mislead a person sufficiently intelligent to understand the text of the specifications of the patent. The method of balancing the "tilting moment" is stated in the Klose patent with reference to the diagram which follows the description. We quote as follows:
"The primary inertia forces, which operate in the various crank planes, produce, as stated in the beginning, resultants mR of uniform magnitude directed radially to the circumference of the crankcircle. As those for each two opposed cranks are directed oppositely at the distances u and v (Fig. 3) they form two couples with the moments R0 x u and R x v; these same couples may be annulled by couples acting in the opposite direction which can be constituted by rotating weights; in the foregoing instance the couple R0 x u by a couple g0 x w and the couple R x v by a couple g x w. The weights g0 and g may also be replaced by a weight G, so that the balancing is effected by two counterweights G G, which are opposed to each other.
"From the above discussion it follows that the arrangement shown and described of such a four crank engine operates in such manner that free inertia forces and tilting moments cannot occur therein.
"In order for the impulses to occur at equal periods it follows from the discussion that both the one series of pistons I, III, II, IV, as well as the other series, I, III, II, IV, come into action upon cranks displaced successively at 90° whereby the torque impulses appear to be uniformly distributed.


*65 "Patent Claim.
"System of balancing for eight cylinder reciprocating engines with a four crankshaft, upon the cranks of which equal crank-connecting rod pairs operate in planes at 90° to each other, distinguished by the fact that two opposed cranks are positioned at right angles to the other two opposed cranks whereby the crank circles of each pair of opposed cranks are positioned symmetrically with reference to one and the same median plane and the tilting moments are counterbalanced by rotating counterweights located opposite to each other."
The drawings accompanying the Klose patent are as follows (note that the patent drawings shows g0 and g erroneously interchanged in the upper left end of the upper left half of the drawing).
Dr. Durand, after quoting the foregoing excerpt from the Klose patent with reference to couples, states: "The Klose patent discloses a shaft of the type under discussion in the present case applied to a V-8 engine in which the primary and secondary forces are themselves automatically balanced, leaving only the moment to be balanced, and discloses furthermore specific directions for applying a system of four counterbalance weights which again in accordance with the statements of the patent may be reduced to their equivalent resultant as represented by the large weights G and G. The line of action, or what is referred to in the patent as the center of gravity of these counterbalance weights of Klose falls within the angle that is defined in the Walker patent, that is, off the so-called bisector plane."

*66 It should be noted that the Klose plan for balancing the turning, deflection, distortion, bending, or elastic moment, as it is variously called, by two couples, each balancing two throws, one couple balancing the end throws and the other the other two throws, not only shows a clear conception of the importance of balancing these forces where the primary and secondary forces are automatically balanced (as in the eight cylinder V-type engine with the Sharp shaft), but discloses a method of computing such counterweights or couples. No suggestion is advanced by either party or by the experts to the effect that his method of computing the turning forces to be balanced is not correct, or that his method of determining the balancing couples is erroneous, and we see no error therein.
"Once we have the balancing couple determined in magnitude and direction," says Dr. Durand, "it then becomes a matter of indefinite choice as to location, radius of action and mass, only so long as the definite and fundamental conditions of the couple are fulfilled."
The "Dalby method" of balance has already been discussed herein and the testimony of Dr. Durand in reference thereto set out in the notes. The testimony of the patentee, Clinton L. Walker, seems to admit the fact that the "Dalby method" is accurate. He testifies as follows: "You cannot do the mechanical work of physically distributing the metal weight by my method, but you can figure the amounts of the weight and their radius on the mass center and where they should be, very readily. I never have tried it with the other methods. My method was so much easier for me to handle that I always employed that in my balancing work, figuring from the center of the transverse plane both ways. I saw the demonstration of the Dalby method as explained by Dr. Durand. I presume that method gave the same resultant moment. I would not say that it is the same, because the w. r. l. considering the method of taking each half might be different from the w. r. l. considering the shaft as a whole. By pursuing the Dalby method, with that type of shaft, you get the same result as with my method taking each half of the shaft alone; by following the Dalby method it could be solved. Personally I have never used the Dalby method. I have read his books and also read the book by Sharp, and followed it, but I have never personally used the method myself."
In addition to the text-books covering the balancing of the Sharp shaft and the Klose patent, on the same subject there was placed in actual use in the Marmon car a balanced crankshaft of that type with counterbalancing weights midway between the throws used in 1904 and 1905 and described in the "Horseless Age" issued September 27, 1905. We reproduce cut thereof from appellant's brief, Exhibit 86. The throws are marked 1, 2, 3, 4, the bearings are indicated by letter A, and the counterweights by shaded portions.

*67 Dr. Durand testified that the Cadillac crankshaft claimed to be an infringement and the Marmon balance crankshaft as to the arrangement of counterweights for deflection balance were on the same basic idea. In both it should be stated the center of gravity of the counterweights is on the bisector plane, and not off it as in the Walker patent.
The Duesenberg shaft (patent No. 1,584,279) shows an eight-throw, three-bearing shaft with counterweights midway between the bearings.
We conclude that appellee's patent claim No. 1 for a method of computation for determining the position and weights necessary to counterbalance distortion in a Sharp shaft with an eight-cylinder engine V-type motor is not patentable. We agree with appellant's contention that such a computation is not "a new and useful art, machine, manufacture or composition of matter" within the meaning of section 4886, Rev. Stat. (35 USCA § 31). Neither is it novel, being merely a special case already covered by a general case, or formula, and also by the method of graphic statics which gives a solution without formula or computation, forces being represented by the length of lines and their direction being parallel to the action of the force, the solution being indicated by the length and direction of the line indicating the resultant force.
As to angular position of counterweights, claim No. 2 is anticipated by the Sharp text-book and Klose's German patent.
Claim No. 3 is in effect a restatement of claim No. 1 with the addition that the counterweight is to be "positioned midway between the end throws and the next adjacent throw." The latter idea does not constitute invention, and was anticipated by Klose and by Duesenberg and by the Marmon shaft, where weights were placed midway between the throws as claimed by Walker.
Claim No. 4 merely makes the same claims generally for any shaft "having a plurality of throws on each side of a central transverse plane"; that is, two or more throws on each side. This claim is anticipated by the Marmon and Duesenberg shafts, and involves no invention over prior art and mechanical engineering skill.
Claim No. 5 is not involved in this litigation.
In arriving at the conclusion that the patent sued upon is void as to claims 1, 2, 3, and 4, we have not overlooked the rule which is applied by the appellate court in equity cases where the witnesses appear before the trial court and a question of veracity arises among them, but in the case at bar there is no substantial disagreement and no question of veracity is involved upon the matters we have dealt with.
Having concluded that the patent sued upon is invalid as to claims 1, 2, 3, and 4, it is unnecessary to consider the other questions involved.
The interlocutory decree is reversed, and the trial court is directed to enter a decree of dismissal upon the ground that the patent in suit is void as to the claims involved, namely, 1, 2, 3, and 4.
NOTE 1.  I am 32 years of age and live at Mayfield. My profession is Assistant Professor of Mechanical Engineering at Stanford University. I graduated in mechanical engineering at Stanford University, in 1921. Then I went to the Westinghouse Company in East Pittsburg, Pennsylvania, and did engineering research with that organization until 1924. During that time the majority of my work dealt with the balancing problems of rotating bodies, vibration problems, and problems in strength of material or elasticity. In 1924 Dr. Durand asked me to come to Stanford University as an instructor in mechanical engineering, and I have been there since. In 1927 I was naturalized and became an American citizen, before the Honorable Judge here. In 1927 also I got my doctor's degree in physics, Stanford University. My courses at Stanford University at present are dynamics of machinery and strength of materials. * * *
Mr. Lindsey. Q. Now, the method of counterbalancing as shown in the Walker patent is to take each half of the shaft and compute the moment, is it not? A. I presume that is the method by which he arrived at the magnitude of his weights.
Q. Does that method in itself afford any better balance, elastic, dynamic or static balance, than the Dalby method or the couple method or the Sharp method or the Klose method or any other method that has been discussed during the course of this trial? A. The method of arriving at the magnitude of the balance weight or at the magnitude of the balancing couples does not affect the quantity of the balance at all.
Q. It does not matter what method you follow as long as you get your resulting moment, and then you can split that up in any way you care to and distribute your weight longitudinally along the shaft wherever you may have space: is that correct? A. Yes.
Q. Then you agree with Professor Durand? A. I agree with Professor Durand.
Q. And you agree with Professor Durand concerning the disclosure of these text-books with respect to these different methods? A. I agree with Professor Durand in regard to the text-books giving an elaborate and fully described method for arriving at the unbalance. * * *
"Q. How old would you say the problem of elastic balance is? A. It is difficult to answer that question. I have not seen a great deal of emphasis placed on that problem until about XXXX-XXXX. But the reason for that may be that before that time I was not interested in the balancing problem any more than generally interested.
*68 Q. But there was some literature on the subject prior to 1920  weren't the general principles understood, the basic principles of elastic balance? A. The basic principles, as mentioned by Professor Durand, involving the problem of counterbalancing, have been known and understood for many years.
NOTE 2.  I am a resident of Palo Alto, California, 70 years old, a professor emeritus in engineering, Stanford University. I graduated from the United States Naval Academy in 1880 and then was a member of the Engineering Corps of the United States Navy from 1880 to 1887 and then professor of mechanical engineering at Michigan Agricultural and Mechanical College from 1887 to 1891, and then professor of Marine Engineering at Cornell University, 1891 to 1904, and then professor of mechanical engineering at Stanford University 1904 to 1924, when I became Professor Emeritus. I am a member of a considerable number of societies, that is, engineering and scientific societies and other bodies of that character and have contributed articles on the subject of engineering in various phases to engineering periodicals. I am the author of a number of books dealing with engineering subjects. Before the time when I became Professor Emeritus I practiced as a consulting engineer and am now consulting engineer to the Government and municipalities, including the United States Government, and the cities of San Francisco and Los Angeles.
In the course of my long experience as a teacher of engineering and as a practiser in this art of engineering, I have had occasion frequently to deal with the subject of the balancing of engines and the balancing of engine crankshafts. I have been familiar for something more than 50 years, with text-books dealing with the general subject of counterbalancing of engines and counterbalancing of crank-shafts of engines, beginning with the text-book of Todhunter and Rankine, text-books in mechanics which were used in the Naval Academy and through which I first became acquainted with the subject of mechanics, and then on down through a considerable number of authors such as Dalby, whom I have known personally well, Angus, whom I have known personally, and Sharp, and numerous magazine articles.
NOTE 3.  Mr. Davis. Q. I will ask your attention to certain representations by Mr. Walker's solicitors in the prosecution of the Walker application in reference to what is revealed or taught by certain text-books that were discussed between the Patent Office Examiner and those solicitors and what, according to the solicitors, they failed to reveal, and I first invite your attention to pages 5 and 60 of paper No. 6, Walker amendment, dated February 25, 1925, where a quotation is made from Angus: "The theory of machines." Will you please state what significance is in the statement of the quotation which appears in capital letters, reading: "The Reference [Plane] Must Always Contain one of the Unknown Masses?" A. Turning to the book in question, page 310, I first desire to read the title of Paragraph 245 in order to make clear the particular problem with which the author is here dealing. The title is, "Balancing any number of rotating masses located on different planes normal to the shaft revolving at the uniform speed." From this statement of the problem it is clear that the author is here dealing with the entire problem of engine balance as I outlined it this morning, in which the forces (subnote 1) are not necessarily the same in amount, and the angular disposition about the circumference of the circle may all be by way of irregular angles. He therefore has to take account of the general case (subnote 2) and in particular of the resultant under the two counts which I mentioned this morning, namely, the resultant force of translation and the resultant moment. (Subnote 3.) In the second paragraph of this article I find these words: "As before, this may be done by the use of two additional weights revolving with the shaft and located in two planes of revolution which may be arbitrarily selected."
Subnote 1. The forces [in the problem of counterbalancing an engine shaft] are due to the centrifugal action, and, as such, must necessarily act at right angles to the shaft.
Subnote 2. General case. In the general case of the balancing of an independent crank-shaft, for example, however, the case will include the possibility of any number of off-center masses of varying magnitude located at varying or irregular distances, at irregular spacing on the shaft, and with an irregular distribution of angular relation around the circumference.
Subnote 3. Force of translation and the resultant [bending] moment. * * * In such a [general] case it is necessary to take account of the resultant [force] under two heads  first, the resultant force of translation, which tends to carry the shaft away bodily  pull it out of its bearings and carry it away to some other point, and, second, a resultant moment which tends to tip or bend the shaft into a different angle or position but without reference to translation.
I point to the words "arbitrarily selected" as indicating the entire freedom of choice with reference to these two planes. Then following comes the quotation referred to: "It is convenient to use the lefthand plane, or that through O, as the plane of reference and, in fact, the reference plane must always contain one of the unknown masses." I would point out in the first place that in this statement there is no restriction whatsoever with reference to the location of these planes and particularly refers to the unknown masses. Then he states that the reference plane must always contain one of the unknown masses. The same basic fact in mechanics might have been expressed perhaps more happily in these words: "The reference plane may be taken anywhere at will and when once taken one of the balancing forces must be located in that plane. In other words, there is an entire freedom of choice so far as the selection of plane is concerned, and in this particular case where he is dealing with a residual force of translation as well as with a residual moment, it so develops from the principle of mechanics that according to his formula of solution one of his balance weights will find its place in this particular plane; but in the case of ordinary engine balance of internal combustion engines such as these engines we are concerned with in this case, and as I pointed out this morning, there is no residual force of translation and this particular instruction therefore disappears entirely from the problem, it is not pertinent, and the resultant couple which will secure adequate balance may be located anywhere.
Immediately following the quotation made from Angus by the solicitors they say, "From *69 careful study this seems to be the rule of Dalby, Professor Sharp and others," which I take it to refer to the emphasized statement as to the location of the reference plane. Turning to the text-book by Dalby, page 33, article 28, I again wish to read the title of the article: "Balancing any number of given masses by means of masses placed in two given planes." Reading further in the discussion of the article, but without quoting the entire paragraph, I would state that it is very clearly evident that he also is here dealing with the general case; he is dealing with any number of forces located anywhere and distributed at any angle about the circumference. He is therefore dealing with precisely the same problem as the author Angus and it is therefore not unnatural that he should follow the same indicated rule. I note, however, this language a little further down in the article: "The artifice used to obtain a solution of the problem consists in taking the reference plane coincident with the plane of revolution of one of the balancing masses." That is in essence the same rule as given by Angus. But I call attention to the fact that this author calls it an artifice, implying that it is something which is done by way of convenience, or that it might have some special advantage, and he goes on to point out that it has this advantage that, with such a particular location, this particular balancing mass has no moment (subnote 4) and therefore disappears from the problem.
Subnote 4. Moment. The term "moment" is intended to measure the turning, either produced by a force with reference to turning about some specified point, taken for convenience or otherwise, as a point of turning. * * * It all goes back to the principle of the lever, which again goes back to Archimedes or to the Egyptians, or presumably still earlier, the basic reason why we must multiply the force by the arm. * * *
Turning over the page at the end of this article, I find a subheading, "Checking the Accuracy of the Work," which reads in part as follows: "Having found the balancing masses, add them to the drawing in proper positions relatively to the given masses; choose a new reference plane anywhere." Again the "anywhere" implies entire freedom of choice with reference to the location of the reference plane.
With reference to the author Sharp, I should say that this author deals with this entire problem in a somewhat different way. There are various routes by which we may start from given initial data and arrive at the terminus of a definite solution of a problem of mechanics. Author Sharp adopts a different mode of treatment, which is fundamentally correct, but which follows a somewhat different route. It is known as the method of graphical statics. He does not take reference planes as such; he does not deal with couples and moments in the same sense as Dalby and Angus, but he arrives at the same result by a perfectly logical and correct route, but he does not follow the same route in detail, and therefore it is not with him a question of the location of the reference plane.
The solicitors in connection with this quotation made from Angus and Dalby, to which I have been referring, stated that, applying what they considered the rule of these authors, brings the counterbalance weights or the lines of action on the bisector plane. This is what the solicitors say: "Following this rule it brings the counterweights as above stated on the bisector of the angle of cranks 1 and 2 for one-half shaft, and on the bisector of the angle of 3 and 4 for the other end, and in both cases on the opposite side of the shaft to their respective crank arms to be counterbalanced." I do not consider that to be a proper conclusion. If the plain directions of Angus, Dalby and Sharp on that matter are followed, the correct solution will result. In fact, as I said this morning, by the aid of the diagram, the solution develops not on the bisector plane but at the proper angle.
Walker's solicitors also make the statement: "Applicant's weights do not come out on the bisector of the angle above referred to, but their centers of gravity are situated in an axial plane between this bisector and the axial plane passing through the centers of cranks number 1 and number 4 and at an angle of between 18½° and 27½° from the axial plane of cranks 1 and 4 on the side of each end crank toward its next adjacent crank." I would say that the result is the same whether it is arrived at by the particular method applicable to this shaft and proposed by Mr. Walker or used by Professors Angus and Dalby, and I should like to interject just at this moment this thought, that the so-called Dalby method or the Angus method is not a Dalby method or an Angus method in any sense of the possessive, because they did not originate these methods. These methods run back to the origin of time, as I said this morning, in the science of the mechanics; they adopted a certain routine for solving different problems, or, in other words, for applying the fundamental principles of mechanics to the particular problem of balancing engine shafts. I have personally known Mr. Dalby and I know they would be the very first to discard any claim to originality, in the basic features of their methods. They are methods followed by these authors, and not these authors' methods.
My attention is next invited to page 6 of Walker Amendment C (dated July 7, 1925, and marked Paper No. 8), where Mr. Walker's solicitors set forth "What applicant has done is to provide a simple and accurate method whereby such a shaft may be balanced against the centrifugal forces of the oppositely disposed cranks." He has, of course, shown a method which gives the correct result in the case of the Sharp shaft. It gives the same result as that which would be arrived at by following any of the other methods to which reference has been made or more fundamentally by applying the well-known elementary principles of mechanics of this particular problem.
Now, on page 8 of this same paper, "No. 8," there is this said: "The books herein cited, being the best known works on the subject known to this applicant, accept the 4-throw in one plane shaft as the accepted type and make no mention of any other, Professor Sharp and Arthur W. Judge being the only ones to suggest a shaft similar in crank arrangements to this applicant's, but they stopped with a mere suggestion and did not even work out the necessary system of counterbalancing." That is the system of counterbalancing the type shown in the Walker patent. On page 117 of the text-book by the author Sharp is a paragraph headed, "8-cylinder 4-crank Engine," and on page 118, the following page, Figure 11, is an outline sketch of the shaft which we have here called the Sharp shaft, *70 evidently intended as a part of the 8-cylinder, 90-degree V engine. Turning back to page 117 I note these words:
"If the 4 cranks are set at 90 degrees, Figure 11, the angle between the imaginary secondary cranks is 180 degrees. Thus the secondary balance is similar to the primary balance of a 4-cylinder, 4 crank engine (Figure 7, Chapter 3) running at twice the speed. Therefore in this case the secondary balance is perfect. To maintain the perfect primary balance the 4 counterbalancing masses, or an equivalent system represented by BB, Figure 11, must be fastened to the crank shaft."
With reference to this quotation I would point out that the reference to Figure 7 is to a previous figure which is found on page 112 and which illustrates the simple pair of cylinders at 90 degrees. There is in this diagram plainly illustrated a single counterbalance indicated by a large black dot, evidently placed and intended to counterbalance properly the moving parts of this particular combination of cylinders and crank-shaft; in this particular figure there are only two cylinders. The heading at the paragraph referred to is, "8-cylinder, 4-crank action." We here find, then, the same sort of a reference, "the four counterbalance masses"; he evidently refers back to Figure 7 where he has pictured one such mass for a single crank and two cylinders and naturally there will be four such masses with 4 cranks and 8 cylinders. And he states that they must be applied in manner as indicated in Figure 3, which is perfectly correct for a distributed system of weights. Then he goes on to say: "Or an equivalent system represented by BB, Figure 11," which is exactly the counterbalance of the so-called Sharp shaft reduced to two counterbalance weights, with the indication of the proper method not on the bisector plane. It would appear, therefore, that the author has really given definite indications of the character of the balance for this particular shaft. I should like also merely to read into the record the contents of two or three sections earlier in the book in order to show that he has once for all dealt with the basic problem, and that he does not therefore need to repeat the particular details of procedure in connection with the discussion of a series of different types of engine arrangement. Turning to page 41 of this same author we have a paragraph headed: "Masses in different longitudinal and transverse planes," the obvious connection being the problem of balancing of the masses in different longitudinal and transverse planes. This paragraph is divided up into a series of subparagraphs of which I will not read the individual headings, but I should like to refer more especially to a subheading 16, which reads: "Recapitulation of general graphical method." In this paragraph I find a series of directions, a series of guide-posts, so to speak, numbered 1, 2, 3, 4, 5 and 6, and containing explicit and complete directions for procedure in this case. Therefore, having dealt with the general case once and for all, he does not, as I said a moment ago, find it necessary to repeat in connection with each individual type of engine dealt with.
On this page and on the following page of this Walker amendment C, where it discusses a book published by Sharp, page 117, Sharp's book of 1907 is discussed by the solicitors and the assertion is made by them that, "no data is given as to how to compute the required masses, their axial angles, nor their position longitudinally on the shaft, nor is any proof offered that such weight would effect primary balance," and then a little later refer to a graphical method of laying out a 4-throw shaft with masses in different longitudinal and transverse planes, as appearing at page 41 of Professor Sharp's book, and they assert that the author "does not determine the mass, the radius of mass center, the longitudinal position, nor the axial angle of the required counterbalance weights." The diagram in question forms a part of the article to which I have previously referred, and constitutes a part of the illustrative material relating to his general method of dealing with this problem. The determination at this point is simply a determination of the balancing couple (subnote 5) required in magnitude and direction; that is plainly indicated, and that is as far as he chooses to go and as far as he needs to go. Once we have the balancing couple determined in magnitude and direction it then becomes a matter of indefinite choice as to location, radius of action and mass, only so long as the definite and fundamental conditions of the couple are fulfilled.
Subnote 5. A couple, Dr. Durand defines as follows: "* * * Consisting of two equal and opposite forces conspiring together in the same sense as the illustration I have given a moment ago of the automobile wheel or the auger."
Arthur W. Judge, in his text-book on "High Speed Internal Combustion Engines," supra, thus defines a centrifugal couple:
"If a shaft AB of length a be provided with arms of equal length r at its ends, and equal weights M be attached to these arms (Fig. 181).

then the centrifugal force upon each arm when the shaft is rotating will be M w2r, and the shaft will experience a couple of moment M w2r a tending to rotate or twist it in the manner indicated by the arrow, that is, about an axis perpendicular to the direction of rotation and of the centrifugal forces themselves. It is important to notice that this couple will be of constant magnitude but of varying direction  the axis of the couple rotating in a plane perpendicular to the axis of the shaft and at the same angular velocity.
"Such a couple would occur in the case of a two-cylinder engine with cranks at 180°, and would remain unbalanced. In several other types of engines also unbalanced couples occur and are resisted by the reaction of the supports of the engine."
Following, this statement is made by the solicitors at page 9 of Walker's amendment C, "In the synopsis of results in this book, page 104, arrangement No. 35, he gives the mathematical details pertaining to a shaft for an `8-cylinder motor,' with 4 cranks at 90 degrees set at 90 degrees. The data given does not give the information *71 pertaining to the required counterbalance mass." Turning to the table in question, which is labeled here "table 13, Synopsis of Results," I find under No. 35 the reference referred to. It is to be noted that this particular table gives a synopsis of results of a very large number of engines of different arrangement, different crank angles, different detailed relationships of one kind or another, and under varying conditions. Then I find under Column 1 the heading, "Type of Engine." Then there are four subheadings under that general heading, "No balance masses, half balance, over balance, connecting rod balance," and, following parallel across the page are the various results which will follow from these various operating conditions. In other words, I would point out that this particular table makes no pretense of giving directions how to counterbalance or as to whether or not a counterbalance is required; it assumes that a counterbalance is required; it assumes that various kinds of counterbalances have been applied, partially or complete. Then it purports to give the various results under such headings, "Largest unbalanced forces"  meaning force of translation,  "the largest unbalanced longitudinal couple, the largest unbalanced transverse couple," and so on, in a series of other headings. The earlier pages of the book, as I have already noted, contain full directions for adequately and properly balancing any and all of these various types of engines.
The solicitors thereupon refer to the work of Arthur W. Judge, saying, "In a book by Arthur W. Judge appears a copy of the diagram and text found on pages 117 and 118 of Professor Sharp's book, with no further information on the subject." I believe the page referred to is 340 of Judge. It does not give it here. Turning to Judge, page 340, Figure 207, and entitled: "Alternative arrangement for 8-cylinder V-type Engine," I find the same type of shaft, the so-called Sharp Crank Shaft, indicated as one of the possible and one of the especially serviceable types of crank-shaft available for 4-crank engines, with special reference to the fact that the unbalanced secondary forces in this case are automatically balanced out, that is to say, the primary advantage of this particular disposition of the cranks is definitely pointed out by this author on this point.
Referring to the earlier pages in the book, namely 312 and 313, on page 312 I find a general heading: "General Case of Engine Balance," and divided under two subheadings: "Graphical method," and "The Analytical Method." It is clear the author first treats this general problem by a graphical method and secondly by a matter of algebraic analysis, in either case arriving at the same result. He does show, therefore, in this paragraph, especially taken in connection with other teachings in the book, adequate and complete directions for counterbalancing this or any other arrangement of the engine cranks.
On page 12 of this paper, Walker Amendment C, it has this to say: "It was considered necessary by all of these writers to use as the reference plane the plane of revolution of one of the counterweights. It has been suggested by the Examiner that it is immaterial where the reference plane is taken. This is only true where the two halves of the shaft are identical in form and weight, so that the tendency to gyrate about the central point can be represented by a couple. By taking the central transverse axis as the reference plane, each half of the shaft can be computed independently of the other, and the shaft put in perfect balance even though the two halves of the shaft are dissimilar." The statement seems to cover the broad claim that any shaft may be properly counterbalanced by considering either half separately, and by taking moments about the central transverse plane. If I accurately and correctly interpret the reading of the specification at this particular point, it does not seem to be correct, nor does it seem to give the proper result.
Referring then to the specification of the Walker patent in suit, the following statement occurs at line 69, page 2: "This rule can be applied to any shaft with the cranks or eccentrics set at any angle, and the same rule is followed for each half of the shaft." I assume that the rule referred to is the one set out in the preceding paragraph in Column 1 of that page 2. As I said a moment ago, if I correctly interpret the direction of the rule, it does not seem to give the correct result in cases other than when applied to the so-called Sharp Shaft, and this may perhaps be pointed out by means of certain exhibits which were introduced yesterday,  for example, Defendant's Exhibit No. 68, which is a diagram headed: "90 degrees spiral shaft, Walker Method." Following the general practice which I indicated this morning with reference to the measurement of the moment and the laying off of the moment, there would result for one-half of the shaft, apparently, Figure 4 is a resultant from Q to the other end of the line, and with an indicated balance couple measured by the same length in an inverse direction; if then we pass to the other side of the reference plane and consider the other half of the shaft separately and independently, following the same or a like rule, we should obtain Figure 5, in which the resultant moment is again indicated in magnitude and direction by the distance from Q to the other end of the line, and the indicated balance moment by the same line in an inverse direction. If we are to realize the balance couple indicated on Figure 4 by a single weight  I should have said if we were to realize the moment by the amount in Figure 4, by a single moment, we must take it from such a position as that indicated in E in Figure 2; if we are to realize the balance moment of Figure 5 we must take some such position as that indicated by Figure F in Figure 2. This would then give the two weights not directly opposite. It is to be noted that these two weights located in this manner and determined in magnitude in this fashion, would give a correct balance moment so far as moment is concerned, as far as the tendency toward tipping the shaft is concerned, but the indication of two weights not directly opposite for purposes of balancing will of course introduce an unbalanced force of translation and will destroy the automatic balance for force of translation which otherwise would exist. It is to be noted that whether these moments are taken in these figures 4 and 5 in one way or the other the result is the same. As I pointed out this morning the result of a problem in mechanics does not depend on arbitrary choice of one direction or the other; it runs to deeper and more fundamental features, and it will be seen that, as outlined in Figure 5, for example, I have followed the direction of the cranks themselves, *72 3 and 4 would go inverse to the direction of 3 and 4, but the ultimate result will be the same, in any case, indicating a moment inclined at the angle as shown in Figure 5. The same result seems to be indicated again in Exhibit 71, which relates to the 90-degree shaft, with the end pairs at the 180 degrees counterbalanced by a method of considering each half independently. In this particular case the moments for cranks 1 and 2 are to be laid out vertically as indicated in Figure 4 indicating a resulting moment as shown in the diagram, while for the other two cranks they are to be laid off as indicated in Figure 5, indicating a resulting moment as they are there shown. This again would indicate one resulting moment vertically and the other horizontally. If those are to be realized as one weight each, then it would seem that it would require weights as indicated at A and B in Figure 2. This again would give a correct balance of the moment, but, as in the previous case, would interpose an unbalanced force of translation requiring further treatment in order to bring a shaft into complete balance relative to both force of translation and rocking moment.
Mr. Davis. Will you look, please, at the sketches which were introduced on cross-examination of the plaintiff's witness, Mr. Walker, and to Defendant's Exhibits 69, 72 and 74, which were submitted as representing the following of Dalby's instruction in the matter of counterbalancing, both the type of shaft that is illustrated on the sketches 68 and 71, that you have been considering, and also the type of shaft that you have referred to as the Sharp shaft, and state whether or not you find they do correctly demonstrate the applicability of the Dalby method or the method of which Dalby treats.
A. On Defendant's Exhibit No. 69, the problem dealt with is the so-called spiral shaft with a reference plane through one of the weights. Checking from the diagram I find that it does correctly represent the method followed by Dalby, or again the application of the fundamental method or principle which I outlined this morning and that it, in a single vector polygon in Figure 4, gives the final resultant properly indicated as to direction and magnitude. They are laid out under the same fundamental rule as I stated this morning, and I need not repeat it.
Exhibit 72 is a shaft with the end pairs at 180 degrees with a reference plane through one of the weights. Here again the moments are computed, exactly, estimated exactly in the same fashion as those in the previous cases. The picture which is shown in Figure 4 is laid off exactly in accordance with the same rule as that which I introduced this morning, and, in other words, in accordance, exactly, with the same rule as that laid down by Dalby, and it correctly indicates in direction and magnitude by the dotted line the resultant balancing moment.
Exhibit No. 74, which is the Sharp shaft, is a reference plane through one weight. This really is a repetition of what I gave in substance this morning, because that was the Sharp shaft and I said at that time that the method which I then followed gave the correct result and that it gave that result independent of the plane or reference. * * *
My attention has been invited to the introductory part of the specifications of this Walker patent in suit where the following reference is made  where it refers to the usual method employed when applying and positioning counterweights being "fairly satisfactory where the cranks at one side of the central transverse axis of the shaft are symmetrically arranged," etc., and it is then said: "Where the cranks or throws of a shaft are not so arranged, the application of counterweights presents a different problem, and this problem is more complicated where space is not available for a counterweight opposite each crank or throw to be balanced." I do not think that problem with which the patentee Walker dealt was any different from or any more complicated than that which the text-book writers dealt with in their books and with which I myself have been familiar in the practice of my profession. Referring to some problems that I have had to meet that would be at least as complicated; my own experience in balancing problems were especially in the line of balancing of marine engines which presents a far more generalized case than this problem of the automobile engine. In the typical marine engine the moving parts are different in weight, and it results that the forces applied are different in magnitude. Likewise, for various reasons, the crank angles are sometimes irregular, rather than being uniformly distributed around the circumference; furthermore, in order to reduce to the farthest possible degree the unbalanced forces in that form of engine, in order to minimize vibration which is transmitted to the hull of the ship, it becomes necessary, it becomes specifically demanded in the sense that no expense shall be spared to realize the most perfect balance achievable, and for those reasons the reciprocating parts are considered with the greatest care; not only the reciprocating parts of the main engine, but the reciprocating parts of the valve gear and various other moving parts of the engine, which have an irregular motion, acceleration and deceleration, and which by said motion would tend to transmit the vibration tendencies to the hull of the ship; the problem really therefore is very much more complicated than that of the simple balancing of an engine crankshaft consisting of 4 cranks at 90 degrees.
Then in the next paragraph of the introductory part of the specifications, which paragraph begins at line 33, the object of the invention is set out as follows: "The object of the invention is to generally improve and simplify the counterbalancing of a crank-shaft of the type mentioned. The further object is to provide a method of counterbalancing crank-shafts which will permit the use of a single weight to counterbalance a number of cranks or throws, or which permits division of a single counterweight into a number of smaller weights placed at convenient points longitudinally of the shaft." Those objects directly follow from the plain teachings of these various writers.
Now following this last quoted statement of the object of the Walker patent, he says, "This system of counterbalancing permits a wider range of design than where both halves of the shaft are symmetrically arranged, and in the same angular phase on opposite sides of the central transverse plane. In other words, the shaft may be designed to give a better balance of the inertia forces of the reciprocating masses and to more nearly neutralize the bearing pressures caused by the centrifugal forces of the rotating masses." There is certainly nothing *73 contained in the disclosure of the patent which is not contained in the text-books, and in so far as the application of the method outlined in the patent is narrower to the specific investigation of one-half of the shaft at a time, it would seem to narrow the application of this particular method to a shaft of the Sharp type, whereas, considering the whole shaft at once as a complex system of masses and forces, we arrive at a general result applicable in any and all cases.
