486 MR. S. S. HOUGH ON THE OSCILLATIONS OF A 
Now, the element ds lies entirely in the surface (A’), and corresponding to every 
point of it there will be a point X, Y, Z which lies on the sphere (B). Hence, as 
R’, « do not vary over the surface of the sphere, they will remain constant as we 
pass along the element ds. Thus we have 


0 , 
ay (R’e) = 0 
and, therefore, from (17) 
Ovi. Sh ey (eS OX) OS el aseneZ 
oe oe Sle ae aU plas ae 
But from (13) we see that 





Cg ( COSIE\ = = Ee cosec vy’ on ) Gases 
as Oi Tf any) p (p> — b) 1 pJ/ PB) uy 
ONY 1 Yi A UTR (= =)= Pz , cosee PX , cosee ; 
8 S(p—B) Os — /(p? 8) \sin 9] — p?/(p? — eae NES) % 
OZ il Oza IE 
SP (Cane?) mae 
Therefore, 
dy AY SEAR alll GiB ates 1 seein 
0s pr/(p? — 2°) {x ay mek peosee Me 
and 
= — ae — ! oa — life Re | OS, | 
cos a’ cos B' = sin y’ ~. = GaP) | aw Yay - (18). 
From (15), (16), (18) we see that the boundary equation, on the assumption of the 
form (14) for w,, takes the form 

” 124’ 5, cS. }- Bui ZA’ = u waz ws ye) 

oY ox 
2 — 2) ee) ze 
= = 946 Gp V0 + Oo ga IK + 0, } 
VEGA AT Tg aA Ts em *Yy 0) 
Now the function Xa a —Y = is itself a spherical harmonic function of order x, 
OSn OSn . . 
and both 8, and X = a7 Yi ax may be expressed as linear functions of the 2n + 1 
independent harmonics of the nth order. 
