73 
1914-15.] Integral-Equations and Lame’s Functions. 
The integral I, therefore, satisfies the differential equation ( 1 ): and it is 
a doubly-periodic function of x. But any douhly-periodic function of x 
satisfying equation ( 1 ) must be a constant multiple of the Lame’s function 
y{x), which corresponds to the particular value of A concerned : and thus 
we have the required result 
. 4 K 
ll{x) = A. I TJ'^y{s) ds . 
•'0 
§ 3. A Definite Case worked out directly . — It is somewhat surprising 
to find how much difficulty is attached to the verification of even the 
simplest particular cases of this theorem by direct integration. As an 
example, let us take n — 2, in which case one of Lame’s functions is 
y{x) = srd x — t 
where 
t = 71 - ^2 + ^4} . 
This function is the doubly-periodic solution of the equation 
where 
A = -2{l-hA:2- + . 
In this case we have to consider the integral 
,- 4 K 
/ {dn X dns + k cosh y cn x cns + kk' sinh rj snx sn sY{k^ sn^ s -t) ds . 
Jo 
We can omit terms of the (expanded) integrand which are odd functions of s, 
since these give zero when integrated between 0 and 4K : and we can also 
omit the term which contains dn s cns multiplied by a function of sn s, as 
this is the perfect differential of a doubly-periodic function and therefore 
vanishes when integrated over this range. Thus the integral may be 
written (arranging the integrand according to powers of sn s) 
riK 
I { (d7i^ X + Id cosh2 y crd x) + sn'^ s{ - k‘^ drd x — k^ cosh^ rj crd x 
+ Idk'd sinh2 rj srd x) } {Id srd s-t) ds . 
Now we have 
—{sn s cn s dn 
ds 
l-2(l+A;2)s7^2s+3A;2s7^4s, 
and therefore if the integrals 
yK .4K 
I ds and / sn‘^s ds 
'0 Jo 
are denoted by I and J respectively, we have 
•4K 
1 
