110 
CHRISTIE. 
or, as a special case, when only the -i-tides are sought, 
2 c qi V qi sin \>p qi — [A — 2 c T V r sin v /?,.] = O') 
q r C 5 
2* s qi V qi cos v/9 qi — [B — 2 s r V r cos v ft] = 0 ) ’ 
q r 
where 
^ = . 2 T k cosp( 2 &-f- 1 )^ 
(7) 
B = v cosec p — . 2 2! sm p (2 & -f 1) — 
r m k k ' m 
V x = 
s^n v /3 r sin- m v p r 
v*- 
sin ft sin (v ft — p sin (v ft + p 
sm v p . sm m v /? . 
sm /5 ql sm (v /J qt - P sin (v /S ql + p 
and i r includes neither y nor i. The summations with re¬ 
spect to k are from & = 0 to k — m — 1. Equations (7), in 
which each unknown is affected by a factor of diminution, 
take the place of the last two series of equations given by 
( 3 ) and afford by their solution the c’s and s’s, and hence 
the a’s and e’s, for substitution in the forms (4). 
In like manner we have from ( 6 ) 
0 =F Yvy + Tv" 
0 = tf p / + 
2 c qi W qi sin v p qi — \E — 2 c r W r sin v /? r ] = 0 
q r 
2 s qi W qi cos v /S qi — [F — 2 s r TT r cos v /5 r ] = 0 
or 
(8) 
where 
E=Nv seep ^ . 2 Q k cosp (2 & + 1)~ 
m k 
F= Nv cosec p ^ . 2 Q k sin p (2 £ + 1) ^ 
