354 Proceedings of the Royal Society of Edinburgh. 
Also, from Lagrange’s equations 
[Sess. 
i 
dt 
Mr 2 } — Mrf(8 2 + sin 2 6<j> 2 ) 
.dV 
% 
Therefore ~ 2 {hr 2 ) 
= rr(0 2 + sin 2 (9</> 2 ) + r 2 (0 2 + sin 2 04 > 2 ) + 2 rr(00 + sin 2 + sin 0 cos 66(f> 2 ) 
10V.. f( 0 0V .02V . S 2 V . 0 2 V 
M 0r JV1\0£ 0?’ 0r 2 r000r rsm Odcfidr 
. 0 2 V . 0 2 V .0 2 Y 
+ x + y + 1 
dxdr ' dydr 0z0r 
= { r(^' 2 + sin'- } ^ - 3r 2 ( ft + sin 2 6ft) - £ + 
2 . A 0V 2 . . 1 0V 
- —r u- — - —r smM — . . 
M r 06* M sin 6* d</> 
Taking mean values, we have 
v/2 
M 2 
+ [r 2 (0 2 -1- sin 2 $cjj> 2 ) 2 ] - 3[r 2 (d 2 + sin 2 0<j> 2 )] - ^ 
•>■2 
02 V 
dr 2 
_2 
M 
r(6 2 + sin 2 $</> 2 )— 
dr 
The Lagrangian equations referring to the angular co-ordinates will 
now be considered. 
We have 
M-(l sin 2 6 ft) = - 2rf.P - M<£ 2 — (r 2 sin 2 0). 
dr . d<f> dr 
Therefore 
M Y (r 2 sin 2 <9<^ 2 ) = - 2<i- f ^ - 2M<£-(r 2 sin 2 6)A - 2A 9 ^ - M<£ 2 4k 2 sin 2 6). 
dt 2 dt\d<f>/ dt drf> dt 2 
Substituting in Lagrange’s equations for f, 0, and and simplifying, 
(d/ 2 . 
we obtain for M -^(r 2 sin 2 d<^ 2 ) the expression 
CLv 
d / 9-0 /1\0V 0 ; . ^ d( 
— — (r 2 sim 6 1 ) -- - 2<f>r sm 6* — — - 
2 9 dt' dcf> r cftVrsi 
0V 
r- 1 site 
sin Odcf) 
9 ( . .7 12 
+ A < — + M</>— (r 2 sin 2 0) > - 2r 2 sin 2 0M</> 4 - 2r 2 cos 2 0M$ 2 </> 2 
Mr 2 sm 2 6 I 0<£ dt ) 
— 2Mr<j£> 2 — (r sin 2 0) + 2 r</» 2 sin 2 + 2</> 2 sin 0 cos 0^ . 
dt dr dO 
