na . REPORT— 1870. 



Vli(B-C) 

 s/AJi—V 



sin am u. 



A am M, 



which express the velocities about the principal axes ia elliptic functions. 



Now, substituting these values of ^j and g- in equation (2) and integrating, 

 we have 



? ^^ r„. I (A-B)(A-C) C dn^&mudu 



VAXF-C)(A/i 



where 



{^ (A-B)(A-C) C sin"amt(c7« ^ 



'i_^^j| A(B — C) J 1— /."siu^amiasin^am uj' 



which gives 

 whore 



7. • a • « C(A-B) 



1, Q(u + ici) 



^= -nn't + - log, ; \ ; > 



• 2i • e(if — 2a) 



, _ VBC d loge etrt 



*^~ VA(B^C)(A7i-Z^)~ dn 



Now let iL'=J/4-nn'<, then »//'=—. log, — ; r- . 



Hence if we cause the axis of (.v), instead of remaining fixed, to revolve 

 with a velocity nn' round the axis of (z), we may substitute for a, a, a" the 

 following values in place of those before obtained : — 



a =cos sin (jt sin xli' -\- cos (j> cos ;p', 

 a' =cos 6 sin f cos ;p' — cos <p sin \p', 

 a"= — sin 6 sin 0, 

 /3 =cos cos (j, sin ;^' — sin <p cos ;//', 

 fi' =co9 cos ip cos )//' + sin f sin »p', 

 • /3" = — sin cos 0, 

 y =sin0sin »|/', 

 ■y' =sin cos \p', 

 y" = cos 0, 



where 



^ \Q{i(, + ia)Q{ii—ia)}y 



Q(u + ia) — 0(t( — ia) 

 {e(u + ia) 0(u— m)}^' 



2i" sin ^L'= 



* It must be remembered that since A, B, C are in order of magnitude, —A- sin- am id 

 must bo positive. 



