646 
Proceedings of Poyal Society of Edinburgh. [sess. 
both vanish. Under ordinary circumstances there is therefore no 
seiche corresponding to this solution. 
It should be observed, however, that a solution 
- x 2 ) 2 = ( a 2 - < A + B log — + -- 1 sin nt , . (49) 
( a - x ) 
where n 2 = a 2 gh, and T == r/a J(gh) = 7 rl/ J(gd ) , . (50) 
exists, which satisfies .the conditions 
£ = ,, 9 X sin ^> when x = p ; . . (51) 
and 
i = hf^)i sinnt ’ when *=g> • • ( 52 > 
where A, and /x are not both zero. This solution is, in fact, 
£h(a 2 - x 2 ) 2 = ^ X ( ~ <j a/x - + (A - /x) log a 1 sin nt . (53) 
66 — l) [ CL — X ) 
In particular, if q= -p, so that /3= — a, and if also A = /x, this 
solution takes the simple form 
£ = 
A 
h(a 2 - x 2 )f 
sin nt . 
■ ( 54 ) 
§17. Although this ‘anomalous seiche,’ as we may call it, 
cannot in general exist, it might happen that the lakes were 
connected at the ends P and Q with canals or other lakes, in such 
a manner that the boundary conditions (51) and (52) were fulfilled. 
We should then have a seiche of longer period than the uninodal 
of the normal series of seiches. In fact it is easy to see, by 
making q= -p in (16) and then causing p to approach the value 
a, that for any finite value of v we have 
L T„ = 2 tt/cl J(gh ) .... (55) 
q=—p=a 
The anomalous seiche is therefore the limit toward which any or 
all of the normal seiches approach when we extend our practical 
lake P 0 Q more and more nearly to the theoretical limit A 0 A'. 
§ 18. Apart from the anomalous seiche, the possibility of which 
arises from the fact that a concave quartic lake approaches the 
shores of its theoretic end with an infinitely small gradient, the 
results obtained above repeat the main features of the seiche 
