* 
Vou 7, 1921 MATHEMATICS: J. H. McDONALD 67 
formula g„ + x = (n + v + l)g„ + sg„- i with initial values g_j = 0, 
go = 1 and are connected with f n by the relation f n = g„f n + „ + 2g„ _ r 
i. Putting A t , = g„g/ + i — g„ + i g/ where differentiation is with 
respect to js it is known (Hurwitz, Math. Ann., 33, 1889, p. 246) 
that A„ + 2 = (n + v + 2)g 2 „ + ! + £ 2 A„, Aj = * + 1, A 2 = (n+ l) 2 (n + 2) 
and since -( ^±2 J = — and Ai = n + 1, it follows that ^±- ] is an in- 
dz\ g v J g v 2 g v 
creasing function of z if n + 1 > 0. 
If differentiation is taken with respect to n and D v = g„ 
dg v + 1 
ft, + 1 — it is found that D v = g v 2 - zB v _ u D 0 = 1, A = (n + l) 2 - z. 
dn 
For, from ^t- 1 = g„ + (» + v + 1) ^ + 0 < ^- 1 with use of the 
equation g„ + L = + v + 1) g„ + jsg„ _ 1, it follows that g v 
^ + 1 
dn 
K 1 ^ = 
Sv — 1 , 
dn 
— z ( g v _ 1 — — g„ — — - J. If a negative value is as- 
\ dn dn f 
signed to z D v >0 and + 1 is an increasing function of n, with the con- 
gv 
dition as before n -f 1 > 0. 
From these properties of + 1 it follows if Z\ is a root of g„ = 0 that 
gv 
g v _ !(0i)>O or <0 according as g v changes from negative to positive or 
the reverse when x increases through Z\. Denoting the dependence of 
g v on n by g* it follows if n' is slightly greater than n that g?'(2i)>0 
n' / \ 
in the first case and <0 in the second because in both cases , > 
f - — = 0. If Zi denotes the root of gf(z) = 0 which differs slightly 
gp- lfe) 
from 21 it follows in both cases that |si'|> The roots of f n = 0 
are known to be the limits of the roots of g* (z) = 0, v = 1,2. . . ., hence, 
if p k denote the k th root of f n = 0 and p k > of / M / = 0, |p*'| .> |pjfe|- 
The equality can easily be excluded. This is the theorem stated at the 
beginning. It has been assumed that z<0 and if n + 1>0, all the roots 
Z\ are <0 in fact. 
A formula of another kind may be obtained as follows : Differentiating 
the equation f n = g v j n + v + zg v _ x f n + v + , and equating the result 
to /„ + 1 expressed in terms of f n+l> + i,f n+v + 2 , and comparing co- 
efficients it is found that & + 1 = g n v + (n + v + l)(g v )' + {zg n v _ 0' 
and g? + l - g : = ( g : +l y. 
An application of the second equation may be made to the calculation 
