BEYIEWS — ANALYTICAL STATICS. 73 



problem which led to it, we might a priori doubt the usefulness of 

 such a course. In effect the reasoning is worth nothing. In the 

 first place that /(0) = 2 may be deduced at once from equation 

 (1) by putting z — o, and the succeeding reasoning literally gives us 

 no information whatever. If, indeed, it could have been said that 

 unity was the only value of e which would satisfy the conditions of 

 the problem when a; = 60°, the proposition would be established, but, 

 unfortunately, this is nob the case, for an infinite number of values 

 might be given to c such that the conditions of the problem might 

 be satisfied in this particular instance. For example we might put 

 c = 5, for then we should have 



/(60 Q ) = 2 cos (5 x 60°) = 2 cos 300« = 1, 

 as it should be : and the fallaciousness of Poisson's reasoning is at 

 once apparent from this, that the very same words which he employs 

 to shew that f (x) = 2 cos x is the proper solution of (1) might 

 be employed to shew that f (x) = cos 5 x ought to be selected. 



We may notice that a very simple mechanical consideration will 

 suffice for the selection of the true value of c, if it be granted that 

 the solution of (1) is necessarily of the form f (x) = 2 cos ex. 

 "When x = 0, the equal forces P act in the same direction and the 

 resultant is the greatest possible ; when x = 90°, the angle between 

 the forces is 180°, and the resultant is zero, and it does not seem 

 too much to assume that as x increases from 0°, to 90 c , the resul- 

 tant will diminish continuously. This being granted it is at once 

 evident that c must be unity, for cos c x must vary from unity to 

 zero continuously, as x varies from 0° to 90°. We are by no means 

 prepared to say that this form of proof of the parallelogram of forces 

 can be made perfect. The solution of functional equations always 

 involves more or less of doubt and obscurity, and what is called the 

 the " general solution" of such an equation is by no means necessarily 

 most general that can be conceived. Certainly Mr. Todhunter deserves 

 our thanks for giving vis the classical proposition of Poisson instead 

 of the method which had been substituted by Mr. Pratt, which is 

 just as unsatisfactory as Poisson's and much more clumsy. We could 

 have wished, however, that Mr. Todhunter had called attention to 

 this singular fallacy. It seems scarcely fair to the student to put a 

 proof in. his hands, especially with such a name attached to it, with- 

 out giving him so much as a hint that it contains anything unsatis- 

 factory. 



G. C. L 



