482 MR. 8. 8S. HOUGH ON THE OSCILLATIONS. OF A 
Corresponding to any series of points whose coordinates are denoted by (a, y, 2), 
we shall obtain a new series whose coordinates are (a, y, 2’), which will be real or 
imaginary according as )? is greater or less than 4w°. We will take as the standard 
case that in which \?> 407. 
If the point (a, y, z) trace out any surface whose equation is f(x, y, z) = 0, ee 
corresponding point(«,¥,2’) will trace out a new surface whose equation is f(«,y,72') = 
The part of this latter surface which corresponds to the real part of the nae 
SF (&, y; 2) = 0 will, however, be purely imaginary if )? < 4?. 
If (cos a, cos B, cos y), (cos @’, cos 8’, cos y’) be the direction-cosines of the normals 
to the two surfaces, we have 
cos « : cos B’ : cosy’ = of/ ox : of / dy : of / oz’ 
= of / dx : of / dy : z (of / dz) 
= cos a : cos 8 : 7 cos y. 
Substituting in equations (7), (8), the differential equation for yp, takes the form 
O°, Oa? 4 O's, Oy? + Oru Oz> = Oke 1. 4e) seu: 
while if f (a, y ,z) = 0 be the equation to the undisturbed boundary of the fluid, y, 
must satisfy the equation 
11 cos a! ae cos 8’ + cos 7’ | — 20i {F* c0s a! — WF cos 6} 
he Gs cos y’ — 72’ cos B') 
a cee + 6, (72! cos a! — = cos ¥) | 
_+ 6,' (w cos B’ — y cos @’) 
(10) 
at the surface f(a, y, z’) = 0. 
The problem of finding the motion of the fluid is thus reduced to that of obtaining 
solutions of equation (9) consistent with the boundary condition (10) at the surface 
J Gh Oh C2) = © 
§ 3. Case of Ellipsoidal Surface. 
Hitherto, no assumption has been made as to the form of the surface of the fluid. 
Let us now suppose that it is given by the equation 
a oP ee eee 
