196 
MR. Ct. H. BRYAN ON THE WAVES ON A ROTATING 
Hence, we find 
^ J 
(1 - /c") Kh = ^0 (j^o) + b(^T_ (^^o) + 1 {vq) 
X (V — 1)'^^ (/x) 
= K% 
f/c- - 1 
by (15), (16). 
Whence 
X {vq^ — 1)"'^ {[Ji) 
h = C,/ CT T,/"^ (ix) 
where 
■ (38), 
I; ■'« (-'«> + ^ G) \ W - 1)*" • (39); 
moreover, tlie equation of the disturbed surface of the liquid is 
C — ^0 + ^^0 :.(4^)5 
where 
S^Q z= h~~ = C/ CT" tan a/c^ (/x) .(41), 
The Boundary-Conditions. 
7. Let Vq be the potential of a mass of the liquid filling the spheroid 
.. (18), 
and let v be the potential of a distribution of the liquid of thickness everywhere 
equal to h over the surface of the spheroid. The combination of these two distribu¬ 
tions is equivalent to the liquid mass as disturbed by the waves, so that 
Vi=Vo-fr’ .(42). 
For the free waves, the boundary-equation (10) requires that 
\p = Yq-\- V \(jr' {x^ y^) + const.(43), 
all over the surface (40). 
Now xjj, V, 8^q are small quantities of the first order. Hence, expanding by 
Taylor’s theorem, we have to first order 
