Parallelogram of Forces. 345 



Art. XXII. — Of the Parallelogram of Forces; by Prof. 

 Theodore Strong. 



Suppose that two forces denoted by x and y, whose directions 

 form a right angle, are simultaneously applied to a particle of mat- 

 ter M ; to find the direction and quantity of their resultant. 



Put P=3.14159, &£C. = the semicircumference of a circle whose 

 radius =1, z = the resultant, and & = the angle which its direction 



P 



makes with that of a:, then "9 — ^ equals the angle which its direction 



makes with that of y. 



Since z, is the resultant of x and y, it is manifest that x is that 

 part of z which acts in the direction of x, or that x denotes the value 

 of 2;, when resolved in the direction of a;, also y represents the value 

 of 2, when resolved in the direction ofy; and z equals the sum of 

 the values of j; and y, when resolved in the direction of 2;. 



If X equals y, it is evident that the direction of z bisects the an- 



P . . . P 



gle q5 which the directions of x and y make with each other, •'• t 



equals the angle which the direction of z makes with that of a; or y 

 in this case; put x=y=n'z=^ the value of z when resolved in the 

 direction of a; or y, then we shall manifestly have n'x, n'y, for x and 

 y, when resolved in the direction of z, whose sum must equal z, 

 hence n'x-\-n'y=2n'x=z, or since x=n'z, we get 2w'-s =z, or 



/T P . 



n'=\/ - =cos. —, by trigonometry; also by trigonometry, if w 

 .* 4 



represents any integral number, (which we shall suppose to be pos- 



P /P \ /P ^\ P 



itive,) cos.^" =cos. 1 ^4-2wP ), ••. ??/ = cos. ( j 4-2wP I . Put v 



-l-2wP==a, (1), then suppose the direction of a force represented 



'. ^ . . . 



by z' bisects the arc a, or makes the angle ^ with the direction of 



x; it is evident that z' may be considered as the resultant of two 

 equal forces R and S, of which R acts in the direction of a:, and S in 



the direction of z, which makes the angle a (or t-j | with that of a?, 



supposing R and S, to be resolved in the direction of z'. Let x' 

 equal the value of z' when resolved in the direction of a:, and sup- 

 VoL. XXIX.—No. 2. 44 



