Parallelogrcnn of Forces. 345 



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



Theodore Strong. 



J 



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 ^t, then "o — ^ 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 z, w^hcn resolved in the direction of t, also y represents the value 

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

 the values of a; and y, when resolved in the direction of z. 



an 



P 



P 



gle gj which the directions of x and y make with each other, .*• -^ 



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

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

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

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

 hence n'x^n'y=2n'x=z, or since x=7i'z, we get 2n'=z =z, or 



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



2 4 



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



p /p \ /P \ P 



itive,) cos. J =cos. K-+2»PJ, .•.n'=cos. ^^ +2nPj. Put ^- 



+2«P=a, (1), then suppose the direction of a force represented 

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



^ ; 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 r, and S in 



the direction of z, which makes the angle a (or 4'] ^^"i^h that of x, 



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 x, and sup- 



VoL. XXIX.— No. 2. 44 



