72 BEVIEWS — ANALYTICAL STATICS. 



acted upon may form a part. But to return to M. Poisson's proof, to 

 ■which our attention was directed by finding it in Mr. Todhuuter's 

 hook. It may sound a bold assertion to make concerning a proof 

 published by such a man as Poisson, but we cannot help coming to 

 the conclusion that it is a complete fallacy. We cannot give the 

 proof at length, but the following general description of it will 

 enable us to point out where the fallacy lies. Assuming that the 

 direction of the resultant of two equal forces will bisect the angle 

 between the directions of the two forces themselves, he takes two 

 equal forces, P, inclined at an angle 2x, whose resultant is R, and 

 assumes B=P f (x) ; his object being to determine the form of the 

 function f. By resolving each of the forces P into two equal forces, 

 Q, inclined at an angle 2 z ; he arrives at the equation 



/ (*)• /(*; = /(*+*) + / (*-e) tl) 



This functional equation he has to solve, i.e., he has to find the 

 most general solut on, and to limit it by consideration* derived from 

 the special problem before him. This he proceeds to do as follows r 

 " We see at once that f(x) = 2 cos c x is a solution, c being any 

 constant quantity. We proceed to shew that this is the only solution, 

 and that c=l." Mr. Todhunter, perhaps, scarcely conveys Poisson' a- 

 meaning here. His words are : " Or je dis que cette expression de 

 la fonction f (x) est la seule qui satisfasse a l'equation (1), et que de 

 plus dans la question qui nous occupe la constante c est l'unite." 



As far as we can make out, the reasoning which follows is not in- 

 tended to shew that the equation (1) admits of no other solution, 

 (which we are required to take upon M. Poisson's assertion) but only 

 that in the particular case before us c = 1. The steps by which it 

 is endeavored to prove this are as follows. First, it is asserted that 

 it is evidently true that c = 1, or that f (x) = 2 cos x y when a? is 

 zero, for then the directions of the two forces P would coincide, and 

 the resultant P would be 2P, and we must therefore have 

 f (0) = 2. Again he shews that the conditions of the problem are 

 satisfied by assuming f{x) = 2 cos x in another particular case, viz., 

 when x = 60° in which case the resultant S = P, which involves 

 the assertion f (60°) = 1 which as cos 60 Q = | is satisfied by writ- 

 ino f (x) = 2 cos x. A most ingenious proof is then inserted to 

 shew that if the relation f (x) = 2 cos x is satisfied for x = and 

 for any other value of x, it must be satisfied for all values of x. The 

 proof of this assertion is derived entirely from the equation (1) itself, 

 and inasmuch as the object in view is altogether to choose from the 

 different solutions of the equation that one which suits the physical 



