ANIMAL MECHANICS. 265 

 which reduces the preceding equation to 



cos y = (sin a sin (<r - x) + cos <r cos (cr - x)} 



O) 2 . . , >. 



-. sin a sin (<r - x) ; 



2 



or, finally, 



w 2 . . , x 

 cos y = cos surer sin (<r - x). 



Substituting this value of cos y in the preceding equations, 

 we find x 



p* = p % + w 2 .AA' sin <t sin (er - a?) (46) 



In Fig. (69), the angle a is iTSa ; and if we draw a perpen- 

 dicular aX from a, upon the bisector FX, it is easy to see 

 that 



YX YO + OX 



sin a = — 77- = -5 — • 



ao ao 



Let 



OS = a, YOS = 0, aOZ = 0, 



and we obtain 



a cos d> + / cos 0 

 sm a = 2 -j ; 



and, in like manner, since <r - x = KSa 



. , a cos (h + 1' cos 0 

 sm (er - 4?) = -j, . 



Hence, equation (46) becomes 



p' 2 = p 2 + w 2 (a cos $ + I cos 0) (a cos (p + /'cos 0), 



or, 



g (a cos 0 + I cos 0) (a cos $ + I' cos 0) 



j? 3 : 



