630 MESSRS A. T. DOODSON, R. M. CAREY, AND R. BALDWIN, THEORETICAL 
applied. A noteworthy feature of Proudman’s method is that it can take account 
of all the natural irregularities of the body of water, and its application has been 
found to give very satisfactory results. In §§ 9-11 will be found a comparison 
between observations and the results of the theoretical determination of the longi- 
tudinal seiches of Geneva ; better agreement could hardly be expected. 
General Account of the Method. 
2. Mathematical Basis. 
The solution we are using involves the function 
where 
and 
for rC> 0. 
R(4 
A) = 2(-A) h *G> v ) 
I 0 (£ v) = y-£ 
u(, V )-r r 
J = s = ( p(s) 
= r r 
J S'=( J s=s' p{s) 
dsds' 
-dsds 
(3) 
(4) 
(5) 
( 6 ) 
The functions I n (£, y) depend only on the transverse sections corresponding to 
x = £, x = y, and the variable x may be measured from either end of the lake, provided 
that £ always refers to that section which is the nearer to that end ; that is, £ is less 
than y. This is obviously true for n = Q, and its general truth can then be readily 
seen from (5) and (6). 
The conditions of the problem are such that the solution is given by 
R(0, a , A) = 0 ..... (7) 
A value of X- which satisfies the above equation will then determine the period of 
a free oscillation; if possible values of X are denoted by X 1; X 2) . . . . . ., then 
the periods are given by 
T = 2 ir /( g\ s )i (8) 
and the corresponding mode of motion is given by 
V = R(0, x, \ s ) . . . . (9) 
on omitting a factor which is a simple harmonic function of the time. W e shall then 
have 
£= • . • • • ( 10 ) 
r n (o, x) = ^-i„(o, x). 
where 
