62 Proceedings of the Royal Society of Edinburgh. [Sess. 
We shall assume that the two following conditions are satisfied : 
1. All the roots of f n {f) are simple. 
2. None of them is a root of a(z). 
We may remark that these conditions are satisfied for nearly all the 
polynomials of mathematical physics. 
We shall write 
K z )_ yjA. 
a(z) g(z) ' 
A second solution of equation (E), known as the function of the second 
kind connected with/ w , can be immediately written under Euler’s form : 
( 1 ) 
qjf) =/»(z) f t 
- [fn(t)Y 
the lower limit of the integral being conveniently chosen. 
We shall now introduce two polynomials A n (z) and of degrees 
respectively lower than these of ff and f n , and such that 
(2) A n (z)f n (z) + B n (z)f n '(z) = 1, 
a property sometimes described as Bezout’s formula. 
We can then write 
fnf) 
A, 
Bn - fn )g dt. 
fnif) J \fn ' fn 
Integrating by parts, and supposing that vanishes at the lower 
J n 
limit of the integral, we obtain 
qfz) B n (z)g(z) , /7A„ + B„: , B n g 
+ 
g dt. 
fnif) fnif) ' J V fn ‘ fnff 
We shall write the quantity between brackets 
A„ + 6,/ ^ B n b 
f n frfl* 
and try to find a new expression for it. 
From identity (2) we have 
(3) A n ff + A ff n + B fff + B n ff = 0, 
whence 
A n + B n _ ^ _ B n f n 
fn fn f nfn 
Denoting now by u x , u. 2 , . . . u n the roots of f n , and by v lt v 2 , . 
the roots of f', we have 
A n '(vi) 
f'~Zu 
. V 
n - 1 
fa irtfa'(Vi)(z-n) 
B A B >,•)/„>,) ^ B„('i>j) 
fafa - •«,) fJVijf -Vi) ' 
