ROTATING ELLIPSOIDAL SHELL CONTAINING FLUID. 487 
Omitting for the present the terms of the second order, if we equate to zero the 
coefiicients of the 27 + 1 independent harmonics, we shall obtain a series of 27 + 1 
linear equations connecting the 27 + 1 quantities A’ which occur with Lamé products 
of order xn. These equations show that all the quantities A’ vanish except for certain 
values of \ which satisfy the determinantal equation obtained by eliminating them. 
The roots of this equation will determine the possible periods of free oscillation, and 
when the system is oscillating in the mode corresponding to any one of these roots, 
Lamé products of one order (7) only will appear in y,. 
These modes of oscillation do not involve any motion of the shell, and it is evident 
that they could not be generated or destroyed by any disturbance communicated to 
the shell, if the fluid be free from viscosity. 
We proceed now to examine more closely the modes which depend on terms of the 
second order. As we have seen above, terms of different orders correspond to different 
fundamental modes; and therefore we may for the future suppose that the second 
order terms alone exist in Wj. 
§ 5. Lamé Functions of Second Order. 
A Lamé function of order 7 is a function R of one of the four forms 

R = Es R = V/ p* Zee b? 0 P20; R = Vp" —¢ 0 lee R = J (p° a b°) (p> Ta, c’) . Psy 
where P,, denotes a rational, integral, algebraic function of p of degree n, and R 
satisfies the differential equation 
PR 
9 9 9 dh E 9 
ap? see os ie) re el ep 1)p?>—B]R. (20), 

(2° — *) (pe) 
where B is a suitably chosen constant. 
The Lamé functions of the second order are, therefore, the three functions 

Ee PEC 5 (FO) Boe 2 (eA) 
together with two functions of the form p? +8... ... .. +--+ + (22). 
To find these latter functions, substitute in equation (20) with n = 2; we obtain 
2 (p? — B) (p? — 2) + 2p? . (2p? — B — 2) = (6p? — B) (p? + A). 
Equating coefficients of p®, and the terms independent of p, we get 
—4(2+¢)=68—B, 26%? = — BB. 
