208 a smith’s prize dissertation. [587 
But to obtain this more directly, take A, B, C for the angles between the forces 
Q and R, R and P, P and Q respectively, then A + B + C = 2-7T, and thence 
ol = a, 
¡3 = a+C, 
ry = a + C + A — a + 27r — B, 
whence writing a = \ir, or taking the line of displacement at right angles to the 
force P, we have 
a = ^7r, /3 = + G, 7 = 2tt + r — B, 
and the equation becomes OP — Q sin G + R sin B = 0, that is, Q : R = sin B : sin C; and 
similarly R : P = sin G : sinZl, that is, 
P : Q : R = sin A : sin B : sin C, 
equations which in fact express that each force is equal and opposite to the diagonal 
of the parallelogram formed by the other two forces.