76 
Proceedings of the Eoyal Society of Edinburgh. [Sess. 
The differential equation for the Mathieu functions may be written 
+ (9) 
and may be derived as a limiting form of Lame’s equation (1), by making 
k tend to zero while at the same time n tends to infinity in such a way 
that the product nk is equal to the finite quantity /x. 
The integral-equation which we have obtained for the Lame functions 
now becomes 
I jtx , . I ^ 
y(x) = XJ j dnx dns + ~ (cosh y cn x cns + k! siiih y sn x sn s) y(s) ds , 
and when k tends to zero dn s tends to unity, while sn s tends to sin s and 
cn s tends to cos s. Thus 
y{x) = A|'|l + ^(co.h y COS X cos s + sinh y sin x sin s) | y(s) ds . 
When n tends to infinity this becomes 
y{x)==\re>^^^oshr,coszcoss + smhr,smxsms)^^^^ _ 
Jo 
and this is the most general form of the integral-equation satisfied by the 
function of Mathieu. 
As was remarked in § 1, the well-known trigonometrical definite-integral 
for the Bessel functions is a limiting case of this latter integral-equation. 
In order to obtain it, we shall first take a new independent variable 
^ = ZjLl COS X 
in Mathieu’s differential equation (9), when the equation (9) becomes 
d^y 
d.y 
dl 
+ 1 + 
9 , y 
IX- + 
r).y = o 
( 11 ) 
and the integral-equation (10) becomes (taking y to be zero) 
,-2t7 
y(x) = X . 
. ( 12 ) 
We can now convert the differential equation (11) into Bessel’s 
differential equation by making ju tend to zero while keeping ^ finite : in 
order to make the resemblance to the ordinary form of Bessel’s equation 
complete, we write —n^ in place of the constant A. Thus y(os) becomes 
the Bessel function Jn(0- 2/(®) becomes the solution of the equation 
