484 MR. S. 8. HOUGH ON THE OSCILLATIONS OF A 
To find the value of y, at the surface (A), we may transform the expression (14) for 
ws, first to the surface (B) and then from (B) to (A). 
Now, by a known property of Lamé products (vide Heme, ‘ Kugelfunctionen,’ vol. 1, 
§ 89), if M, N be two conjugate functions of order 7, the product MN at the surface 
(A) will transform into a surface harmonic of order » at the surface (B); and, con- 
versely, any surface harmonic of order 7 at the surface (B), when transformed to the 
surface (A), can be expanded in a series consisting of Lamé products with constant 
coefficients, each of which products will be of the n™ order. 
The same conclusions will hold for the surface (A’) and the sphere (B). 
We can thus express the value of , at the surface (A) in terms of a series of 
Lamé products, in which each term will be of the same order as that from which it 
arises in (14). 
The couples on the shell due to fluid pressure are 
[[p do (y cosy — zcos £), {|p do (z cos y — « cos f), [|p do(w cos B — y cos y), 
where do is an element of the surface and the integrals are taken over the whole 
surface. 
If P denote the perpendicular from the centre on the tangent plane to the ellipsoid 
P 
A) Cc d 1 = = = 
( pan pr/ (p? = b?) (p? = c’) 


~ G2) =P) 


3 
y cos y — 2008 8 = Py: | 1 1 | Pyz (c? — 02) 
pP—e p? ap 
and Pyz is proportional to 

l/ (yw? — BP) (2 — 2) / (2? — *) (2 — ”) = IMN,, 
where M,, N, are two conjugate Lamé functions of the second order. But, if MN be 
any two conjugate Lamé functions different from M,N,, //MNM,N,do=0. For, if 
we transform to the surface of the sphere (B), /do is equal to the corresponding 
element of the spherical surface, and MN, M,N, transform into two different surface 
harmonies. 
Thus the only term in ys, which can give rise to any couple about the axis of « will 
be the term involving the Lamé product M,N,. 
Similarly the terms which can give rise to couples about the other axes will be of 
the second order. These, as we have seen above, all arise from terms of the second 
order in (14), and, in order to evaluate these couples, it will be unnecessary for us to 
calculate any coefficients in y, other than those of terms of the second order. 
