304 ANIMAL MECHANICS. 



From the spherical triangle Sba', we find 



cos y = cos w sin a sin (<r + x) + cos o- cos a + a ; 

 or, when a> is a moderately small angle, 



cos 3/ = cos # — sm a sm (cr + a;). 



Substituting this value in the preceding equations, we find 



, o> 2 AA' . 



P - P = . sin <r sm (<j + a). 



2 p 



Now, it is easy to see, from Fig. 82, that 



A sin a = a sin (p + I sin 0 ; 

 sin (<r 4 #) = a sin 0 + sin 6 ; 



Hence we obtain 



w 2 (a sin 0 + I sin 0) (a sin 6 + V sin 0) 



p'~P = -- — Y-y — ; ( 68 > 



and, finally, for the work done, by a rotation round SZ, 

 through the small angle w — 



f» 7/1 f (a sin 0 + / sin 0) (a sin $ + /' sin 0) c?0 /V N 

 8^6 . _ j 1 . ((59) 



If we substitute, in this equation, the values of I and I' given 

 in page 266, and follow the method there employed, we obtain, 

 finally, 



. fcos *0d0 



sirSOdO 



f * m <*> 2 „ . . . ~ . 0 , fCOS 2 6d0 , x 



J = — a 2 sm 2 0 sm j3 sin /3 — . (70) 



w 2 > ' . ; ■ . r 



4— (6 sin j3 - a sin 0 cos j3) (6' sin j3' - a sin 0 cos j3') 



w 2 . ... ._ . ' . . . 0 fsin#cos&/# 

 4 — asm^{(o + 0 ) sinp sm p - asm 9 sm (p +p)j — 



