404] 
the second term is 
ON THE ROTATION OF A SOLID BODY. 
143 
cos A cot l (q cos n — r cos on) 
q 
+ — cos p cot m (r cos l — p cos n ) 
r 
+ — cos v cot n (p cos on —q cos l ), 
and the third term is 
p • 
+ ~ sin A cosec l ( q cos m + r cos n ) 
q . 
+ ^sm/i cosec on (r cos n +p cos l) 
r . 
H— sin v cosec n (p cos l + q cos m). 
Hence the second and third terms together are 
pq ( „ cos l cos n cos m cos n . cos m . cos l 
= - cos A :—: cos a —7 1- sm A —— r + sin u, 
(o \ sm l sm m sm l sin m 
pq j— cos A sin n cos (v — A) + sin A sin n sin (v — A) 
w }+ cos p sin n cos (p — v) + sin p sin n sin (p, — v ) 
pq . (— cos A cos (v — A) + sin A sin (v — A)| „ 
(o (+ cos p cos (p — v) 4- sm p sm (p — v)J 
pq . 
— sm n 
w 
cos {A + (v — A)}) 
[+ COs {p-(p- !/)}) 
+ &c., 
= — sin n (— cos v + cos v) + &c., = 0 ; 
w 
we have therefore 
cos j dj cos (f) — sin j sin (p dcf) 
= — sin a da sin l cos A — sin /3 d/3 sin m cos p — sin 7 dy sin n cos v, 
= d — . sin l cos A + d - . sin m cos p + d — . sin n sin v 
(O O) co 
= + — (sin l cos A dp + sin on cos pdq + sin n cos v do') 
CO 
- (sin l cos A p + sin m cos pq + sin n cos v r) 
co~ 
= — cot j cos <f> d - — sin j sin <jf> dcf). 
+ &c.,