ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 485 
§ 4. Transformation of Boundary Equations. 
Let us now transform our boundary conditions to the surface of the sphere (B). 
We can at once express the right-hand member of equation (10) in terms of 
3 YQ CARS GI 


, Ria: PX 
costa — = ===, 
p p 
; Py Py 
Coe a eP—F — /(—— #8) 
Yai PZ 
cae — /(p? — &) 
where P’ has the same signification with reference to the ellipsoid (A’) as P has with 
reference to the ellipsoid (A). 
The right-hand member therefore becomes 

= 8 r Ge YZ Po 0’ b? | 
we. P | Se Se Wea "| C8) 
Consider next a single term of order n in yy, say f= R’M’N’. We have seen 
that M’N’ is expressible in the form ¢S,, where € is some quantity which does not 
vary over the surface of the sphere (B) and §, is a spherical harmonic function of 
degree n in X, Y, Z. 
If dv’ denote an element of the normal to the surface (A’), we have P’ dn’ = p’ dp’, 
and therefore, 
a Ov OW Ov __ PY oy, 
COS @ eee cos 8’ + -7 cos y' = Bie amen Oa: 


Now M’, N’ are independent of p’. Therefore, when f, = R’M'N’, 
a Sabiaill Wy ROR, / pa APs 
cos a + = cos B’ “ee Ay cos y' = ry Ay ee mee Gp’: os 1) (iG): 
Next let ds be an element of a line through (a, y, z’) whose direction-cosines are 
( meg COS ces 0) , and which, therefore, lies in the surface (A’). 
sin 7” sin y'’ 
Then 


ae Ov, ae 
Ox 



and when y, = R’M’N’ = R’S,, 
a : 
é 
aii 8 
i — — (Rd .8, + Be (17). 
