1904-5.] Prof. Chrystal on Mathematical Theory of Seiches. 637 
Some further Results in the Mathematical Theory of 
Seiches. By Professor Chrystal. 
(MS. received March 20. 1905. Read same date.) 
§ 1. In the practical calculation of the periods and nodes of the 
lakes we have hitherto examined it has been found that the use 
of the Seiche Functions gives the best results. Indeed, as will 
appear from details presently to he submitted to the Society by 
Mr E. M. Wedderburn and myself, the agreement between theory 
and observation, so far as we have gone, is beyond what might 
reasonably have been expected. Also the phenomena of concave 
lakes, i.e. such as have no shallows or points of minimum depth, 
are easily deducihle from the formulae given in an abstract ( Proc . 
R.S.E. , vol. xxv. p. 328, 6th Oct. 1904) which I communicated to 
the Society on 18th July 1904. On the other hand, the theory of 
convex lakes is less easy of manipulation, chiefly owing to the 
difficulty in calculating the roots of the equations (£(c , 1 ) = 0 , 
1) = 0. 
S 2. It seems, therefore, to he worth while to work out the 
theory for another class of cases, where all the solutions can be 
expressed by means of elementary transcendents. The normal or 
o- - v - curve in these cases is a simple quartic curve ; viz. : — 
o- = 7?-(l - v 2 /a 2 ) 2 for concave lakes; o- = h{\ + v 2 /a 2 ) 2 for convex lakes. 
§ 3. My starting-point was a slightly generalised form of an 
equation used by Stokes in 1849, in his well-known paper on the 
Breaking of Railway Bridges (Collected Works, vol. ii. § 7, p. 186), 
viz. 
(fty 
(z - a) 2 (z - b ) 2 -~ + cy = 0 , . . . ( 1 ) 
the general solution of which is 
y = A (z - a) m (z - b) n + B (z - a) n {z - b) m , . . ( 2 ) 
where m and n are the roots of the quadratic 
p 2 - p + c/(a- b) 2 = 0. 
( 3 ) 
