DYNAMICS. 29 



64. If the lines which each of two forces acting 

 singly would have caused a hody to describe in the 

 same time, make any angle whatsoever with one 

 another, the line which the body will describe in 

 that time, when both the forces act on it at the 

 same instant, is the diagonal of the parallelogram 

 under the two first-mentioned lines. 



a. This is the celebrated theorem known by the name 

 of the COMPOSITION OF FORCES. The most remark. 

 able demonstrations of it are by DAN. BERNOULLI, 

 Comment. Petrop. torn. 1.; D'ALEMBERT, Opuscules^ 

 torn. 1. cinquieme memoire ; LA PLACE, Mecanique 

 celeste, torn. 1. 1. POISSON, Mecanique, liv. i. 

 18. & 14. All these, though ingenious, are too 

 elaborate and difficult to be accounted elementa- 

 ry, and are, besides, very remote from the train of 

 reasoning by which the truth was originally disco- 

 vered 



65. If two forces be represented by the lines a and 

 6, which contain an angle = z 9 the force compound- 



ed from them will be y a 2 + 2 a b cos z + b 2 : for 



this is the diagonal which, in a parallelogram ha- 

 ving the sides a and b, and the contained angle z, 

 subtends the supplement of z. Also the sine of 

 the angle which this diagonal makes with a, is = 

 b sin z. 



66. If 



