248 Dr. Stevelly on the Composition of Forces. 



the squares of P andQ, or R 2 =P 2 + Q 2 ; and therefore its magni- 

 tude must he represented by the length of B C, the line joining 

 the extremities of the lines representing the two component forces. 

 For if, to fix our ideas, we suppose A R to be the direction of 

 the resultant of P and Q represented by A B and A C, and at A 

 erect xAx' both ways perpendicularly to AR,then, because B AC 

 is by hypothesis a right angle, <2?AB= RAC, and #'AC = BAR. 

 Hence the direction of P divides the right angle x A R into the 

 same angles that A R divides the right angle CAB, and the di- 

 rection of Q divides the right angle a?'AR into angles equal to 

 those into which A R divides the angle B A C. Hence, if we 

 suppose P to be replaced by its two equivalent components in 

 the directions A R and A x, and Q by its two equivalent com- 

 ponents in the directions A R and A x r } and call these respectively 

 V, ?", 0!, Q", we shall have (by II.) 



and therefore 



P" Q A Q"-T 

 P = R and Q--R' 



P"=?Q=Q". 



Hence the resultants of P and Q along A x and A x 1 are equal 

 and opposite ; and therefore, of the four which are together 

 equivalent to P and Q and therefore to R, the components P' 

 and Q', which act both in the direction A R, must be together 

 equal to R. But again (by II.), 



and 

 hence 



P' 

 P" 



P 

 = R' 



therefore P'= 



p 2 



R ; 



Q'. 

 Q" 



Q 

 R' 



therefore Q' = 



Q 2 



R' 



P' + Q'= 



r ps ,Q 2 





R 2 = 



=P 2 + Q 2 . 





and 



Hence, whatever be the direction of R (and that has yet to be 

 determined) , its magnitude is represented by the length of B C, 

 which joins the extremities of A B and A C, which represent 

 P and Q both in magnitude and direction ; and this is also 

 equal to the diagonal of the rectangle whose sides are AB 

 and AC*. 



* This, with a portion of the preceding, constitutes what is familiarly 

 known as " Laplace's Principle." 



