

A NEW PEOOF OP THE PAEALLELOGEAM OP FOECES. 357 



A XEW PEOOF OF THE PAEALLELOGEAM OF FOECES. 



BY THE EET. GEOEGE PAXTON TOUNG, M. A., 



PEOFESSOE OF LOGIC AND METAPHYSICS, ENOX'S COLLEGE, TOEOXTO. 



Bead before the Canadian Institute, March 15th, 1856. 



In the following proof it is merely 

 assumed that the direction of the 

 resultant of two equal forces bisects 

 the angle (being less than two right 

 angles) between the directions of 

 the forces. 



If AOa = m6, BOb = (m-l)6, 



COc =r O-2)0 DOd = 26, 



BOe=0, e being < a right angle 



and m being a whole number ; and if 



^ OV bisect the several angles AOa> 



BOb, &c; the resultant of two forces, each equal to unity, in OA 



and 0«,_lies in the direction OV. Call it B m ; and let i? m _„ 



B m _ 2 , B 2 , B r , represent the resultants, likewise in OV, of 



the same forces, when they act in OB, Ob, in OC, Oc, &c. Then a 

 unit of force in OA, and another in 00, are together equivalent to a 

 single force B x in OB. Also a unit of force in Oa, and one in Oc 

 are together equivalent to a single force R 1 in Ob. Therefore a unit 

 of force in each of the lines OA, Oa, 00, Oc, may be replaced by a 

 force H 1 in each of the lines OB, Ob. But the resultant of a unit 

 of force in each of the directions OA and Oa, is R„„ in the line V; 

 the resultant of a unit of force in each of the directions OC and Oc, 

 is B m _. 2 in V ; and the resultant of a uuit in each of the direc- 

 tions OB [and Ob is B 1 B m -i in OV. Therefore, 



R m + R m -2 = B x Rm-i • And similarly, 



R,a-\ + J5V-3 = B Bm-2 



B 3 + B L = B 1 B 2 

 K + • 2 = B/ 

 If B x may be assumed equal to 2 cos 0, the above equations be- 

 come B, 2 = 2 cos 20, B s = 2 cos 30, R m = 2 cos w<f>. Now, 



to shew that the assumption B x = 2 cos is legitimate, it might 

 perhaps be enough to observe that the resultant of two forces cannot 

 be > than the sum of the forces, so that B x must be intermediate be- 



