1903 - 4 .] Prof. Chrystal on Mathematical Theory of Seiches. 329 
/ ^ 
where u = A(x) £ , v = J dx b(x ) , cr(v) = A(a) 6(a;) ; and g and t have 
the usual meanings. 
A natural* seiche of frequency n is therefore determined by the 
equations 
A(x) £ = u = P sin nt 4- Q cos nl , .... (3); 
where P and Q are solutions of 
cP P 
dv 2 
+ 
0 
0 ) 
Since cr(y) is a slowly varying function of v, we might take it to be 
either a linear or a quadratic integral function of v. On the former 
assumption the solution of (4) is found to depend on Bessel’s Func- 
tions. It is found, however, that the assumption o-(v) = h( 1 - v 2 /a 2 ) 
is more convenient for obtaining approximate representations of 
the cases that occur in nature. The solution in this case is found 
to depend on certain functions which we may call the Seiche 
Functions , defined, for — 1 ■<«;<: + 1, by the following convergent 
series : — 
C('-, to) = 1 - — w" + 
i'(c -1.2) 
1.2 x 3.4*' 
;(c-1.2)(c-3.4) 0 
1.2 x 3.4 x 5.6 W 
c/ s c c(c — 2.3) 
S(c, w>) = w - y^'w 3 + 0 3 x 4.5 ^ 
c ( c ~ 2-3) (c - 4,5) 
2.3 x 4.5 x 6.7 + 
S(<v «0 = 1 - 
1.2 
c(c + 1.2) 
1.2 x 3.4 
c(c+1.2)(c + 3.4) , 
- 4 - 
1.2 x 3.4 x 5.6 + 
;(c, w) = W - ^w 3 + 
c(c* + 2.3) 
2.3 x 4.5* 
c(c + 2.3) (c + 4.5) 7 
2.3 x 4.5V 6.7 w + 
The functions C and S are synectic integrals of the differential 
equation 
+ ^ = 0 (5); 
* As opposed to a forced seiche, whose period depends partly on the period of 
the disturbing agency. Some of the seiches on Lake Erie are. I believe, of this 
nature. 
