1917-18.] Reduction-formula for Functions of Second Kind. 
But from (3), 
63 
A n(Vi) B n (Vi) 
and so 
fn'iPi) fn(Vi) 
= 0, 
An + B„' A B n(Ui)f n "(Ui) 
Zj f '2 
fa 
ttfn\Ui)(z - Ui) 
Now from (E) we have 
whence 
fn'(u;) = _ b(Ui) 
fa (Ui) Cl(Ui) 
We can also write 
(R 
fn t~la(Ui)fn'(Ui) Z-Ui 
B a b = A B>^W , V - 
7/0 f..'(Ui)(z — Ui) Zj a '( 
a fn i-ia(ui)ff{ui){z - Ui) £fa'(Si)f n (si)(z - «*) 
where s ± , s 2 , . . . s rn are the roots of a(z). 
[We suppose here that the degree of b(z) is inferior or equal to the 
degree of a(z), which is generally the case in the equations of applied 
mathematics; we shall, however, see afterwards an example of the con- 
trary, introducing into this equality a constant term.] 
We thus obtain 
K + B/ _ BJ> = _ y B n (Sj)b(sj) , 
fn afn S a' (Si)f n (Si)(z - Si) ’ 
and, using this formula, we obtain for q n the required expression : 
(5) 
<ln(z) = - B n (z)g(z) —f n (z 
B n(si)b(sj) j ~g {t)dt 
f°>'(Si)fn(Si)J t~ S i 
We shall now discuss this result: 
(a) Comparing this value of q n with the expression given by (1), we 
see that it contains m integrals instead of one ; but, firstly, these m 
integrals are similar in form; and secondly, not containing they are 
far more simple than the integral in (1). 
(b) The polynomials of mathematical physics are generally solutions 
of differential equations where a(z) and b(z) are independent of the degree 
n, which appears only in c(z). The m integrals in (5) are therefore inde- 
pendent of n, and appear in the expression of the function q of any order. 
So, if m<n, we can reduce q n to the m functions q 1 , q 2 , . . . q m , and our 
formula (5) will be a general reduction-formula for q n . We must note 
that the function q 0 cannot be introduced there, for,/ 0 being a constant, 
the polynomial B 0 is nugatory. This reduction -formula seems to be 
specially suitable for the numerical computations which often occur in 
harmonic analysis and allied questions. 
