70 Proceedings of the Eoyal Society of Edinburgh. [Sess. 
VII.— On an Integral-Equation whose Solutions are the Functions 
of Lam4. By Professor E. T. Whittaker, F.R.S. 
(MS. received December 7, 1914. Read December 7, 1914.) 
§ I. Object of Paper.— TY iq chief object of the present paper is to establish 
the followino^ theorem : 
The functions of Lame [that is to say, the doubly -periodic solutions of 
the differential equation 
sn^ x + K)y'\ . . . • (1) 
are the solutions of the homogeneous integral-equation 
y{x) = \ I idn X dn s + k cosh rj cnx C7is + kk! sinh t] snx sn s)”^(s) ds , (2) 
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where rj denotes an arbitrary constant. 
It will be found that this result plays much the same part in the theory 
of Lame’s functions as the theorem 
P = constant X j - 1 coss}”cos wsJs . . • (3) 
does in the theory of Legendre’s functions, or the theorem 
1 r 
— — / cos {ns - X sin s)ds . . . . (4) 
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does in the theory of Bessel’s functions : or (to take a case in which the 
analogy is still more marked) as the integral-equation 
y(x)=^XrV^^^^^^^^y(s)ds ..... (5) 
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does in the theory of the Mathieu or elliptic-cylinder functions. It 
will be shown, in fact (§ 5 infra), that (5) is a limiting case of (2), 
while (4) may be regarded as a limiting case of (5) and also as a limiting 
case of (3). 
It will be noticed that the integrals occurring in (3) and (4) are ordinary 
definite integrals which do not involve the Pi” or the under the integral 
sign : whereas (2) and (5) are integral-equations, that is to say, the function 
y occurs both on the left-hand side and also under the integral sign. This 
