The periodic solutions of the linear differential equation
which reduce to Mathieu functions when v = 0 or 1, will be known as the associated Mathieu functions. The significance of this terminology will appear in the following section.
(Received February 17 1922)
(Accepted May 11 1922)
‡ Proceedings, 40 (1922) pp. 28–29.
* I.e. a regular singularity with exponent-difference ½. It is to be remembered that the coalescence of two elementary singularities produces in general a regular singularity with arbitrary exponent-difference; the coalescence of three elementary singularities generates an irregular singularity of the first speoies, and so on.
† When v = ½ the equation bears the same relation to Legendre's equation as its general form bears to the associated Legendre equation.
‡ It has bean proved by the present writer, Proc. Camb. Phil. Soc., 21 (1922) p. 117 [OpenURL Query Data] [Google Scholar], that when v=0 or 1 the equation cannot have two periodic solutions except for θ=0. It is shown in the present section that this is true for all values of v.
* Hermite, Crelle's Journal, 89 (1891) p. 18 [OpenURL Query Data] [Google Scholar], Œuvres, 4 p. 18. A still further generalisation of Lamp's equation is given by Darboux, Comptes rendus 1882, and de Sparre, Acta Math. 4, (1883) pp. 105–140, 289–321. [OpenURL Query Data] [Google Scholar]