DETERMINATION OF THE LONGITUDINAL SEICHES OE LAKE GENEVA. 631 
Thus the problem resolves itself into finding the functions I ra (0, x), I' ra (0, x), which 
are determined from (5) as 
l'*(0, 4- (11) 
Jo p(x) 
l»(0,x)= [\(0,s)ds ( 12 ) 
Jo 
with the initial form 
l 0 (0,x) = x ...... (13) 
3. Arrangements for Computation. 
It is clear that if p(a) is zero the integrands in (5) and (6) become infinite 
for s = a. These infinities, however, can be avoided in a very elegant manner as 
follows. 
Equation (35) of the paper by J. Proudman, already quoted, gives the relation 
R(x, v ,\)^,x,X)-n(^,x,\)l-R(x, v ,X) = R^, v ,X) . . . (14) 
which is a direct consequence of the fact that R(£, x, X) and R(a:, n, are two 
solutions of the fundamental differential equation. If these functions be expressed 
as power series in X by (3), and the coefficients of equated, we have 
2, | b(f, *)“ L(£ x )^fn-r( x , ??)] = hz(£ v) ■ ■ (15) 
Hence, if the functions I„(0, x) and L(x, a) are known, together with their first 
differential coefficients with respect to x, the value of I„(0, a) can be at once 
determined, and no infinities occur in any of the integrands. 
We have already pointed out that the variable x may be measured from either 
end of the lake, but we have assumed that £ is less than »?. It is now convenient to 
split the lake into two portions at some arbitrary point x = X. Let measurements 
from the west end of the lake yield the functions l n (0, X) and I' n (0, X), and let 
measurements from the east end yield the functions I n (X, a), I , n (X, a). Wherever 
necessary, the system will be denoted by w or e to indicate the manner of measuring 
the variable, so that we have 
I»(£’7) = I n(a-v> a ~£) ■ (16) 
and, if x = a— x, 
w e e 
I n (x, rj) = I n (a- r], a- x) = l n (a — 7j, x) .... ( 17 ) 
0 w 0 e 0 e 
^fn( x >v) ~ — V’ a ~ *)' ~ — dx' L( a — V> x ) • • • (1®) 
In particular, our problem requires the functions for which £=0 and *1 = a. We 
shall choose X, the arbitrary value of x, to be such that the corresponding section 
divides the lake into two equal surface areas ; that is, X = \a. Let dashes denote 
differentiation with respect, to x or to x\ according to the system ; then we shall 
