144 Statics. 



Now regarding FA as radius, the perpendiculars FG, FC are 

 the sines of the angles FAG, FAC, or of a, 6 ; we have, therefore, 

 by calling FG, D, and FC, D f , 



D : U : : sin a : sin 6, 



whence 



sin a D 



~^T'' = 77 

 and 



q = J?JL, OTq :p::D:iy, 



178. which agrees with what has already been proved. 



If the directions pA,qA, are parallel, the angles GAF, FAE, 

 FAC, are considered as infinitely small, and consequently as 

 Trig 17 nav i n g tne same ra tio as their sines. We may accordingly sub- 

 stitute sin a -f- sin e for sin (a -j- e), and sin b sin e for 



sin (b e), 

 and we shall have 



p (sin a -f- sin e) 

 % sin 6 sin e 



But we have just seen, that 



D : D 1 : : sin a : sin 6, 

 and for the same reason, 



sin a : sine :: FG : FE, 



:: D : * cos/j 



whence 



D sin 6 

 sin a = , 



and 



<? cos /"sin a dcos /*sin 6 

 sm e = = fc- - ; 



substituting in the value of </, for sin a and sin e the values above 

 found, we have 



