two Rectangular Forces acting on a Point. 1 07 



The latter system of forces may be split into two, namely* 

 first X and Y, which give a resultant R inclined to the axis 

 of a; at an angle d, and, secondly, — Y and X, the former in 

 the direction of the negative axis of x, and the latter in the 

 direction of the positive axis of ^; this system, being perfectly 

 similar to the first, will give the same resultant in magnitude 

 inclined to the axis of y at an angle d, and therefore to the 



positive axis of x at an angle -^ + 5. 



These two equal resultants, therefore, include a right angle, 

 which their resultant, that is, the resultant of the proposed 

 system, bisects ; hence 



'^ X b 1 . ^ 1 + tan 



<^ = -— + fl and tan A = , -4- 



4? ^ 1 — tan 



and putting for tan tan 5 their values deduced from the 

 equations (2.), (3.)> we have, whatever may be X or Y, 



X + Y\" X" + ¥» 



/X+_Yy _ X" + Y" 



which evidently requires that n •=. \\ thus the direction of the 

 resultant is completely known. 



With respect to the magnitude, we have seen that -^ re- 

 mains invariable at the same time with 5, which determines 

 the direction of the resultant. 



Now X may be regarded as the resultant of a force II' in 

 the tlirection of R, and p iu a direction perpendicular to R. 



R' and X have the same inclination as X and R, therefore 



R'_ X 

 X ■" R • 



p and X have the same inclination as Y and R, therefore 



A- Y 

 X"" R* 



The com]X)nents of X are therefore R' = -rr 



X Y 



p = -ir- 



Similarly Y may be decomposed into R" in the same direction 

 as R' and R, and p' in a direction perpendicular to R and op- 

 posite to the direction of p. 



P2 



