618 PROFESSOR CHRYSTAL 
From EvLer’s identity, viz. 
(T2501 Sti oe eee (1 — u,) 
=l-u =f enna o-l)+.- 
es. (—)Pu,(u,-1)(-1)...... Gee 
we see that 
-(1_A\(y_ o/s 
le; X51} vy=(1 Hel = Lae ad © ; 
Je(y c/x _¢/x 
SiGe ye (1 Se =) ae ee ME ad w. 
In our applications we shall for the most part put \=1, or \=-—1. Then we may 
omit the argument A and write 
ye oe ce le 
Cle, o)=1- Fert et . Cr 
Ne ee a ee 
S(r,v)=v- yr eee Bay ba ; : : (16). 
We shall call C(c, v) and S(c,v) the Seiche-cosine and the Seiche-sine respectively. 
Also | 
SA eu? ce+l.2) 4 ) 
= CR UICKCct co) ae ee 
S(e,v)= et po aE : : : (18). 
These may be called the hyperbolic seiche-cosine and the hyperbolic seiche-sine. 
When c has one of the integral values 1.2,3.4,...., C(e,v) reduces) tome 
rational integral function of v; and when c has one of the integral values 2.3, 
Ae . . , the like happens to S(c, v). 
The same holds for &(c, v) and S(c, v) with regard to the negative integral values 
—1.2,-3.4,....,and —2.3, —4.5,.... ; but this is of little interest mua 
seiche problem, for which the values of c must be positive. 
By a well-known property of the solutions of a linear equation of the second order, 
we must have C(c,v) S(c,v) — C(c, v) S(c, v) = constant, where the dash denotes 
differentiation with respect to v. In the present case this constant is easily seen to 
be unity ; and we have 
C(c, v) S(c, v) —Ce, v) S(c, v)=1 a F ; (1O}§ 
Cc, 1)S'(e, ») —Ci(e, v)G(e,v)=1 . : : : (20). 
These are the analogues of the relation cos *@+sin °6=1 for the circular functions ; and 
they are very useful in seiche calculations. We might also define a seiche-tangent, 
seiche-cotangent, seiche-secant, and seiche-cosecant. We shall only have occasion to 
use the seiche-cotangent, viz., C(c,v)/S(c,v), which we shall denote by K¢é, »). 
These functions have many curious properties more or less analogous to those of the 
circular functions: e.g. K’/(c,v)=—1/S%(c,v), but it is needless to encumber the 
present paper with details of this description. 
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