DR WALLACE ON A FUNCTIONAL EQUATION. 



631 



the first volume of the Petersburgh Commentaries : his demonstration was after- 

 wards improved by D'Alembert in the first and sixth volumes of his Opuscules. 

 It has also been adopted by late writers, and, in particular, by Poisson, in his 

 Traite de Mecanique, and by Whewell, in his Analytical Statics. It is the third 

 method which I mean to follow ; and, excepting the particular mode of establish- 

 ing the analytical principle, my demonstration will differ but little from Poisson's : 

 he has resolved the functional equation in two different ways in the two editions 

 of his book, but my solution is different from both. 



15. The axioms of statics, on which the following investigation is to rest, are 

 these: 



(1.) The direction of the resultant of any two forces is in the plane of the 

 forces ; and when they are equal, it bisects the angle made by the straight lines, 

 which indicate their direction. 



(2.) When the directions of the constituent and resultant forces coincide, this 

 last is equal to both the others. And if the angles which the constituents make 

 with the resultant be supposed to increase, the resultant will decrease continual- 

 ly, until it become =0. The directions of the constituents will then be perpen- 

 dicular to that of the resultant. 



(3.) If each of the constituent forces be increased or diminished in any ratio, 

 the same for both, the resultant will be increased or diminished in the same ratio : 

 that is, if the forces P, Q, and their resultant R, change their values, and become 



P', Q', R', and if pr = Qr; then shall ^= pr= qt-; also p-= pT, and q =-q7 • 

 The general problem now to be resolved is this. 



Problem. — To find R, the resultant of any two given forces P and Q, which 



act at a point ; also^ its direction. 



We shall begin with the case in which the 

 given forces are equal. 



16. Case I. Let P and P be two equal forces, 

 which act in the directions AB, AB', and R their 

 resultant, which acts in the direction AC, a line bi- 

 secting the angle BAB' : It is required to find the 

 magnitude of the force R. 



It is evident from our third axiom, that, while 



the angles BAC, B'AC, continue the same, p must 

 be a constant quantity, but this quantity will change 



■R 



if the angle change. Therefore, -p- must be some 

 function of the angle BAC ; and, denoting the angle 



Fis. 1. 



