ON THE HYDRODYNAMICAL THEORY OF SEICHES. 635 
The nodes in the two parts are given by 
li(eee)—O0Lvand ihi(ep.2) 0. : , é (65). 
It may, of course, happen that one of the two equations (65) has no real root. 
Ratios OF THE SEICHE AMPLITUDES AT VARIOUS Parts or Aa BiparRaBoLtic Lake. 
§ 44. It is convenient for purposes of calculation to tabulate the following formule ; 
where ( , G, G., G denote the amplitudes of a seiche at the deepest part, the positive 
and negative end, and any point z respectively ; and a= ,/ (4c+ 1). 
GifGo= Wel, N= TEES") ya We, 4). 2s. 66). 
CfG= Nee, N= TE=* \re2\} jr= lie, 2). > er. 
Ci/Cy =a'S(c', 1)/aS(c, 1)= Cle’, 1)/C(e, 1), 
ener ia), 
=e, 1/2) Tae sl/2) . ; ; (68) . 
C/IG=Le, aIL', 3), 
=L(e,)0( aeons) | ve, 
=L(c, z)(G/G). : . (69), 
All irrespective of algebraic sign. 
Owing to the want of a simple companion fundamental integral, the Lake Function 
is not convenient when the parabolic lake is truncated. In this respect it has the same 
defect as the LecENDRE and Brssrt Functions. Its practical advantage is that it gives 
highly convergent series at points where the series for C(c, w) and S(c, w) converge 
slowly. Unhappily, the corresponding function for a convex lake has an imaginary 
argument. 
SEICHES IN RECTILINEAR LAKES. 
§ 45. If we take the origin of x at a point where the depth is h, then the law of 
depth will be /(x)=h x (1—a/a), where a is a constant, positive or negative according 
as the lake bottom slopes upwards or downwards in the direction in which increases. 
We have, therefore, with the previous notation 
Eh(1 -2/a)=u=P sin n(t-7), 
Ou ace : . . 70) , 
Gams Boos | Os 
where P is determined by 
Ciel & n?P 
ee =0 . ‘ ° 7] . 
dx* ~— yh(1 —x/a) we é ‘ 
