330 Proceedings of Royal Society of Edinburgh. [sess. 
and are connected by the relation 
C (c, id) S'(c, w) - C \c, w) S (c, w) = 1 . . . (6),* 
where the dashes denote differentiation with respect to w. On 
account of the fact that C and S have certain of the properties of 
cos w and sin w, and in a certain limiting case reduce to these 
functions, we may call them the seiche-cosine and the seiche-sine 
respectively. From another point of view they are limiting cases 
of the hypergeometric function ; but from this fact no practical 
advantage has been found hitherto. 
In like manner (£(c, w ) and @(c, w), which we may call the 
liypey'bolic seiche-cosine and hyperbolic seiche-sine , are integrals of 
(1 + <^-T + cP = 0, (7) 
and 
(S(c, w) <S'(c-, w) - &(c, w ) (5(c, w)= 1 . . . (8) 
For the particular values w = 1 and w = i (where i is the imaginary 
unit) we have 
C(c, 1) = I 
V 1 1 . 2 ) 
l 1 3.4) 
S(<U) = ( 
f C \ 
/ c \ 
( 43 ) 
t 1 -kr) • ■ ■ 
II 
K + U) 
( 1 + 33) 
( 1 + 5.e) •' 
,11 
[ 1+ M 
( 1 + 43) 
( 1 + 6v)--- 
It follows from Sturm’s Oscillation Theorem regarding the solu- 
tions of a linear differential equation, such as (5), that, for any 
given real value of v <: l , there are an infinite number of positive 
real values of c which satisfy the equations 
C(c, v) = 0 , S (c, v) = 0 ; 
(K(c,v) = 0 , <S(c,v) = 0; 
and that the roots of either of the equations of one of these pairs 
separate the roots of the other. 
The analogue of the relation cos 2 x + sin = 1 for the circular functions. 
