V 



346 Paralldogram of Forces. 



i 



x^^ (2) ; then, since the directions of R 

 and S, make equal angles with that of z\ each of which is express- 



a 

 ed by ^ which equals the angle made by the direction of z^^ with 



that of x^ we shall evidently have n^'R and n^'S, for the values of 

 R and S, when resolved in the direction of z^^ whose sum must 

 equals', .\n''{R-\-S)=z' \ hence (2) becomes n''\R'\-^)=x', 

 (3). 



Again; by resolving S in the direction of i:, we have by what has 

 been proved, S cos. g — the value of S when resolved in the direction 

 of a:-, and by adding R, which acts in the direction of x, we shall 

 have S COS. a-j-R =0:^, since by the nature of the components and 

 their resultant, the resultant Vihen resolved in any direction, must 

 equal the sum of the components when resolved in the same direc- . 

 tion ; hence by substituting the value of x' in (3) we get S cos. a4- 

 R = n'^^(R+S); or since R=^S, we have 2n^^^ =:!+ cos. a, (4), 



a a 



but by trig. 1+cos. a=2cos.3 ^^ •"• (4) becomes n^'^ cos. 5' (5). 



By (5), we have n^V=z'cos. 5 =z^ resolved'in the direction o(xi 

 and in the same way for a force z^^, whose direction makes the an- 



^ 2 ~ 2^ with that of oc, we shall have z^' cos. ^ = z^' resolved 



2 



2 



in the direction of x, and soon ; hence, if Z denotes any force whose 



direction makes the angle ^^ with the direction of x, where m rep- 

 resents any positive integer, and if X denotes the value of Z, when 



resolved in the direction of x, we shall get Z cos. ^ = ^> °^' ^^ 



restoring the value of a, we have Z cos. \ •^— / = X, (6). 



Hence, the force z, whose direction makes the angle ^ with that of 

 ar, is easily resolved In the direction of x ; for since the integers m 

 and n, are arbitrary, they may evidently be taken such, that 



4+2nP 



g, shall differ from fi by an angle less than any given angle, 



