1917-18.] Reduction-formula for Functions of Second Kind. 65 
equilibrium for a rotating mass of fluid. We shall only state the following 
results : 
When Lame’s function of the first kind is a polynomial, i.e. when n is 
even, there is no difficulty at all, and the function of the second kind S n can 
readily be reduced to functions S of the first and second order. 
When n is odd, Lame’s function being no longer a polynomial, but the 
product of a polynomial T n by an irrational factor, the method fails. 
However, it can be applied with certain modifications : taking for A n and 
the polynomials connected with T ?l by Bezout’s formula, and introducing, 
in the course of the work, double roots for the decomposition of rational 
fractions, a formula analogous to the preceding, though a little more com- 
plicated, can be obtained, which allows the reduction of S n to the two 
functions S of order 1 and 2. 
§ 4. The Extended Legendre’s Polynomials, C n v (z). 
These functions, studied by Gegenbauer, furnish us a good and com- 
plete example of our method. Let us first briefly recall some of their 
properties.* 
The polynomial C n v (z) is defined by the expansion 
(1 — 2 aZ + a 2 ) - " = ^7 a n Cn v (z). 
n=0 
It satisfies the differential equation 
(1 — z 2 )y" - ('2v + 1 )zy + n( 2v + n)y = 0. 
It can be expressed in terms of Gauss’s hypergeometric function by 
the formula : 
n\ / \ r(?i ~b 2r) 0 i 1 — z\ 
c ” (z)= ix»Ww ; ,,+i; ~r)- 
Amongst the recurrence-formulae of this function we shall use the two 
following : 
(I) nC/ = (n + 2v - l)^C n _/ + {f - l)V n _; v 
(II) nzCn - (n +2r- l)C n _/ + (z 2 - l)C n \ 
With C n v is associated a function of the second kind, which we shall write 
H/(z) 
C/(2) f 
di 
j„(l-^)>-+4[C/(0] 2 
* For the properties, expansions, recurrence-formulae, etc., of the function C?F, the 
reader is referred to Whittaker and Watson, Modern Analysis , p. 323. See also Appell and 
Lambert, Generalisations des fonctions splieriqucs (Edition Frangaise de l’Encyclopedie, ii, 5), 
p. 237. 
VOL. XXXVIII. 5 
