ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 653. 
Seeing that it is possible to refer the four differential equations of the Seiche-problem to 
one and the same general equation, viz., that given under (3), it may be of interest to dis- 
euss here the properties of this in many ways remarkable general type, and to derive its 
_ particular solutions. The results obtained seem, on the one hand, to be of practical value 
in regard to the computations involved in the evolution of the periods and nodes of the 
seiches, while, on the other hand, the mathematical investigation here given establishes 
an important relation between the Seiche-functions discovered by Professor CHRYSTAL 
and other functions frequently used in mathematical and physical problems, notably 
those of LEGENDRE, a relation which increases the importance of the Seiche-functions 
from the mathematical point of view. It will be recognised at once that equation (3) 
is a special case of the hypergeometric differential equation 
p I 
ol — 0) 55 +[y- (a+ B+1)o] S4- aby =0, (11) 
whose solution is represented by the contour-integral 
a- -B-1 —a 
y=const x [ U "ew: (w—v) du (12) 
Cc 
C 0 i 
Indeed, if we substitute v= == and a=n+2a,8=—1, y=a+4 we have 
Belen dy 
(_- wo?) 4 —(2a+ Lae + n(n + 2a)y=0 (3) 
which, by introducing «= ou under the sign of integration in (12), is satisfied by the 
integral 
(ue are -4 
= const x bai Ae 
Jo (t we Wyre 
Gin (13) 
If we write 
w= (il SUP) BeNe 
we find easily that Y is a solution of the equation : 
ON, aY 
(1 — Ww) Fp t 2a - 3) Fp t (m+ 1) (n+ 2a-1)Y=0, 
and hence, substituting in (11) and (12) a=n+1, B=1—n-2a, y=3-a: 
(1 = Na | 
Yeconstx {Gaye t, 
‘so that we find as another solution of (3) : 
ee (1 — f?)\nt+a-% 
y = const x (1 — w?)? Ww o G-upn (14) 
For a=4, the integrals (13) and (14) become identical, viz., 
(1 -#)" 
y= const x bs (t zs wrth 
