1917-18.] Reduction-formula for Functions of Second Kind, 61 
IX. — A Reduction-formula for the Functions of the Second Kind 
connected with the Polynomials of Applied Mathematics. 
By Pierre Humbert. Communicated by Professor E. T. Whit- 
taker, F.R.S. 
(MS. received February 4, 1918. Read March 4, 1918.) 
§ 1. Introductory. 
In the course of an investigation regarding rotating fluids, I have had 
occasion to make considerable use of Legendre’s function of the second 
kind, which, as is well known, may be expressed in the form 
Q n(z) = £P n (z) kg - + ] 
where P w is Legendre’s polynomial, and f n _ x a polynomial of degree n— 1. 
Various expressions of f n _ x have been given, particularly in terms of 
Legendre’s polynomials of lower degrees ; but I found that none was 
satisfactory in regard to the practical computations which I had to 
perform. I tried, therefore, to obtain a suitable form ; and, introducing 
a polynomial B n (z), of degree n— 1, such that 
A n {z)R n (z) + B n (z)R n ' (z) = l, 
where A n is a polynomial of degree n— 2, I found the following remark- 
able expression : * 
f n _^ Z ) = ~ B „(S) . 
Z 2 — 1 
The same process allowed me in a similar way to obtain a simple 
reduction-formula for Lame’s function of the second kind.]- I then formed 
the idea of extending this method to the treatment of other functions, 
and the result of this investigation is the theme of the present paper. 
§ 2. The General Method. 
Let us consider a polynomial f n (z), of degree n, satisfying the differential 
equation of the second order 
(E) + c{ - z)fni?) = 0 
where a , b, c are polynomials. 
* Comptes Rendus , t. 165, p. 759. t Ibid ., p. 699. 
