64 
Proceedings of the Royal Society of Edinburgh. [Sess. 
(c) We shall presently establish a general formula for the B ?l ’s, which 
will be of great help in simplifying our expression ( 5 ). 
Let us suppose that the degree of b(z)< the degree of a(z), and consider 
again equality ( 4 ). Owing to the fact that 
~B n (ui)f n '(ui) = 1 , 
it can be written 
= ^ B n (si)b(sj) ^ f n "( Ui ) 
a fn ~a( S i)fn(Si){z ~ «i) " fn\Ui){z - U t ) ' 
Let us multiply the two members by 0 and make 0 become infinite : 
observing now that 
y* fn'(Uj) _ A 
as being the sum of the residues for the rational fraction > , we obtain 
Jn \Z) 
the required formula 
/a\ xS q 
V ; ««'(*) /.(«) ' 
(d) Suppose now that we have to deal with polynomials which are 
solutions of a differential equation of hypergeometric type : this is a most 
important case. We are led to two integrals; but, using the preceding 
formula, we shall reduce them to one integral only, independent of n, and 
so the function q n will be reduced to q r 
(1 e ) It is not unnecessary to remark that the polynomial B w (0) is rather 
easy to form : it can be obtained from f n (z) by rational operations only ; 
the knowledge of the roots of f n (z) is not needed. In many cases, as we 
shall see hereafter, it is possible to form a recurrence-formula between 
the B’s and /’ s, which will be of great help in obtaining readily the values 
of B n (si). 
We shall illustrate this general theory by various examples, chosen as 
follows : 
1. a(z) is of degree 3 : Lames equation. 
2. a(z) is of degree 2, and b(z) of degree 1 : Legendre’s equation and 
its extension. 
3 . a(z) and b(z) are of degree 1 : Laguerre’s equation. 
4 . a(z) is of degree 0, 6(0) of degree 1 : Hermite’s equation. 
§ 3. Lame’s Functions. 
We shall not here give details of the treatment by the above method of 
Lame’s functions of the second kind, as this development is to appear else- 
where, together with its application to the theory of Poincare’s figures of 
