75 
1914-15.] Integral-Equations and Lame’s Functions. 
§ 4. Extension of the Theorem. — It will be seen that the proof in § 2 
makes no use of the fact that the range of integration is from 0 to 4K 
beyond the inference that the values of U and y{s) are the same at the end 
of the range of integration as at the beginning. The theorem may there- 
fore be stated in the following extended form : 
The functions of Lame are the solutions of the homogeneous integral- 
equation 
yix) = A I {dn x dns -{■ h cosh rj cnx cns + kk' sinh rj sn x sn sf y{s) ds , 
Jc 
where C denotes a path of integration beginning at any point in the plane 
of the complex variable s, and ending at the same point or any other point 
of the plane which is congruent with it. Of course, the path must not pass 
through any of the poles of the functions sn, cn, dn, which are at the points 
congruent with s = iK'. 
In particular, we can take the range to be the straight line from s = K 
to s = K -h 4iK'. Thus 
rVi+UK' 
y{x) = A / {dn x dns + k cosh t] cnx cns + k k' sinh t] snx sn s)^ y{s) ds . 
Jk 
Writing s = K-f-f, x = K-\-z, and making the corresponding change in 
the differential equation, we have the result that 
(k' + kk' cosh 7] snz snt -l- k sinh rj cn z cn ty 
dn^ z dn^ t 
u{t) d,t 
is an integral-equation satisfied by the doubly -periodic solutions of the 
equation 
d‘^u 
dz^ 
= I n{n -f- 
cn^z 
dn^ z 
It might be expected that an interesting particular case of the general 
theorem would be obtained by supposing the path of integration to be a 
simple closed contour enclosing one of the poles of the integrand, in which 
case the Lame’s function of x would be expressed as a constant multiple of 
the residue of the integrand at this pole. A closer examination shows, 
however, that the residues at the poles are zero, so that this proposal leads 
to no result. 
§ 5. Derivation of the Integral-Equation whose Sohitions are the 
Functions of Mathieu. — We shall now show how the integral-equation for 
the Mathieu or elliptic-cylinder functions, which was obtained by the 
present writer in 1903, can be obtained as a special case of the integral- 
equation whose solutions are the functions of Lame. 
