200 PROCEEDINGS OF SECTION A. 
In order that we may have % = - ¥,,_ g, we must have 
eo. ee 1+ 4G) 
ha'+K YG (22) 
We have now 3 arbitrary constants, viz., A, B, E. To deter- 
mine these, we have when 9 = o (vo i. = —L ) 
’ ae - 
a aie ee 
v =|0 3 a — 0 i; = : W 
: l : 
From (21) and (22), remembering that 7 = 0, and restoring 
the unaccented letters, 
na? E A a? MO ] . eS oa 
parr ke (1° ve ) meen * (ree Ly 
PHP (8 
PAIGE 
[ (yA - wB) (1-e cos 2u7)—-(uA+vB)sin2u7 xe ] 
(23) 
From (16) 
ZT 2vVT 
(vy A +B) (1-e cos 2u7m) + (uw A-—vBoe sin 2 LT = O (24) 
From (9) by means of (16) and (18), and remembering - =o 
: a 
- y(1+2yu?2)A-p(2v?)B i +[ Jo(r +22) Atm(x—202) Bt cos 2.7 
f aA Ae: Bl aS BP eee DA) 
= gui t2 75) V(r Qa) 3 sin zum |e 2K (25) 
Tw 
2v 
If, as is generally the case, e 
last three equations become 
ma? Res 2 oe OLY | Se oe 
hat+K YG we tv? Y Ga? * 
2vT a 
[WA +»B)sinzur+ (vA - 4B) cos2um fe Jae (26) 
(u A —vB)sin2u -(vA+7yB) cos 247 =0 
is large compared with 1, the 
_ {o(r+2n%)A +m (1-202) Bh cos 2mm 
SSPE Oe 
+ ja (t+ 202) A —v(I +202) Bh sin aun =e e 
( 
ic (28) 
From (27) and (28) 
2 —2v 0 
(« A-v B)cos2urt(yA + 4B) sin2zuar= - Ok e (29) 
From (26) and (27) 
Eee By Y 1c yeer Yk 2» K 
rare k ( ee == mre * (eee 
207 
(A cos 2p + Bsin2p7) e (30) 
