71 
1914-15.] Integral-Equations and Lame’s Functions. 
is an instance of a general theorem, which I propose to establish in another 
paper, namely, that integral-equations play the same part in relation to 
differential equations with four regular singularities that ordinary definite 
integrals do in relation to differential equations with three regular 
singularities.* 
It may be remarked that the differential equation (1) has Lame’s 
functions for its solutions when the parameter A has certain special values : 
but unless A has one of these special values, the solution of the differential 
equation (1) is not doubly-periodic, and consequently is not one of the 
Lame’s functions required in Applied Mathematics. The integral-equation 
(2), on the other hand, does not involve the parameter A at all : and its 
solutions are Lame’s functions and no others. 
§ 2. Proof of the Theorem. — In order to establish the result (2), we shall 
require two curious properties of the expression 
dnx dns + k cosh r] cnx cns + kk' sinh t] snx sns . 
This expression we shall denote by U. 
In the first place, we have 
dHJ_dfV 
dx'^ ds'^ 
= - k'^ dn s dn X (I - 2 sn- x) - k cosh y) cn s cnx {\ -2 k^ sn^ x) 
- kk' sinh yj sn s sn x {1 - 2k^^ sn^ x) - [ - k^ dn s dn x {\ - 2 sn^ s) 
- k cosh -q cns cnx - 21^^ sn‘^ x) - kk' sinh -q sn s sn x — 2k‘^ sn‘^ s)] 
= 2{sn‘^ X - sn^ s) {k^ dn s dnx -k- k^ cosh 'q cns cnx + k^k' sinh y] sns sn x) 
— 2k!^{sn'^x-sn‘^s)\] ........... (6) 
In the second place, we have 
/^Y _ /auy 
\^X ) \06* / 
— iy — W" snx cnx dn s — k cosh q dnx snx cns + kk' sinh q cn x dn x sn sf 
— { — k^ sn s cn s dnx - k cosh q dns sns cnx + kk' sinh q cns dns sn xf. 
* Lame’s differential equation (1) when expressed in algebraic form has four regular 
singularities. Mathieu’s differential equation 
^ + {a + cos2 = 0 , 
dx^ 
when expressed in algebraic form has two regular singularities and one irregular singularity, 
which latter may be regarded as formed by the confluence of two other regular singularities, 
making four in all. Legendre’s differential equation has three regular singularities : and 
Bessel’s equation is a confluent form of Legendre’s. 
