504 A MEMOIR ON THE PROBLEM OF THE ROTATION OF A SOLID BODY. [217 
where, on the right hand, < ~- and ^ may be considered as standing for given 
functions of for the present purpose, however, multiplying by dt, and 
dil di1 
substituting for 2n 7 " and j— the values dh and dk, we have 
etc/ dz? 
- 2n dt—mh + m/c+- d -^ dt, 
dli (B-A)pq dS 
which agrees with the foregoing value of dg, or we have 
dg = — 2)i dt. 
J dh 
The comparison of the terms involving dk gives 
cZH ( 5» dkl _ ^ v t k sin F a cos F, ir . (h - Cr 2 ) k v 
dk + * ~dg~^ + k?-C 2 r 2 * + (k 2 -C 2 r 2 )(B-A)pq Z ’ 
or, substituting and reducing, 
- f=- 2 ” s f - ® s " s cot * f os - ■ s > § +sin ^ - s > * §+«* F w)} 
_ (h — Cr 2 ) k dfl 
(k 2 — C 2 r 2 ) (B — A) pq dS ’ 
where, on the right-hand side, ~ and d ~ may be considered as standing for given 
functions of 
dfl dfl dfl 
dT ’ dS’ dF’ fol the P resent purpose, however, multiplying by dt and 
putting for 2n -j-g dt and dt the values dh and dk, we have 
dfl 
d£l 
~lF dt = ~ ® dh ~ ® dk ~ k cot F * j cos O 8 * ~ S ) dt + sin ( 8 * ~ 8 )( cosec F + cot F 
(h — Cr 2 ) k 
dF 
~ dt 
J Cl U'Vy 
dT 
,dfl 
dS 
(k 2 — C 2 r 2 ) (B — A) pq dS 
which agrees with the foregoing value of cZct, or we have 
7 dfl , 
d ™=-dk dt ’ 
and we ha\ e thus the system of formulae for the variation of the elements in the 
problem of rotation, given in Art. I.