224 Mr. Thomas Weddle on a New Analytical 



the angle contained by their directions, and is never greater 

 than their sum. 



Let the forces P, Q, II, acting upon the point O in the 

 directions OP, OQ, OR, be in equi- 

 librium, so that R, acting in the oppo- 

 site direction R O, is the resultant of 

 P and Q. Denote the angles QOR, 

 R OP, and P O Q by «, /3, and v. 

 Make < ROQ'= <ROQ = «, and 

 <ROP' = <ROP = /3; and to the 

 point O apply in the direction O P', 

 a force P' = P ; in O Q', a force 

 Q'=Q; and in OR, a force R'=R; 



hence the forces P', Q', R' are lespectively equal to P, Q, R, 

 and the angles contained by the directions of the former forces 

 are equal to those contained by the directions of the latter ; 

 now P, Q, R are in equilibrium, hence P', Q', R' are also in 

 equilibrium; wherefore the forces P, P', Q, Q', and R + R', 

 whose directions are O P, O P', O Q, O Q', and O R, must 

 balance each other, and consequently the resultant of the first 

 four forces must be equal and opposite to 11+ R'. 



Moreover, since <ROP = <ROP' = |3, RO (produced, 

 if necessary) will bisect the angle POP'; hence the resultant 

 R x of the equal forces P and P' will act in R O, and R x will 

 be a function of P and (3 ; now, since the numerical values of 

 Rj and P vary with the unit of force, and that of /3 is inde- 

 pendent of it, the ratio y> must be independent of this unit, 



and consequently a function of /3 only; denote this function 

 by 2 <p /3, .-. R t = 2 P . 4> |3. 



Now Rj cannot, for any value of .3, exceed P + P' or 2 P, 

 wherefore if \J/ (3 = cos -1 <p /3, the value of ^ /3 will always be 

 real, and we may therefore assume 



Ri = 2P.cosvI//3. 

 Hence also, if R 2 be the resultant of the equal forces Q and 

 Q', we must have 



R 2 = 2 Q . cos rj/ a. 



Moreover, since Ri and R 2 act in the same straight line, 

 R, + R 2 is the resultant of P, P', Q, Q' ; and consequently 

 R, + R 2 =R+R' = 2R, 



.-. R = Pcos\{//3 + Qcos\J/« (1.) 



Similarly, Q = Pcos\J/y + RcosrJ/a, . . . . (2.) 



and P= Qcos^y + Rcos\I//3, . . . . (3.) 



