1916-17.] Operators applied to Solution of Equations. 41 
From the second of these three tests, 
n 
n - m 
n n—m 
m n _ m 0 m _ A 
111 
n 
B m 
<h 
we deduce the test for the convergent action of the operator 
DI 1 . D c T (n - 1) . D b " 
which is the general type of operator for operands with integral indices. 
This formula, with m = 1, gives 
AC ,i_1 
B n 
71 
n —1 
C 
as the test for the convergency of the root suggested by the operand ^ 
in Ax n — B,r + C = 0. By putting n = 2, 3, 4, etc., we have the tests for 
quadratic, cubic, quartic, etc., operators, viz. 
?! AC <1 3* AC= J Acy ! , etCi> respectively. 
I 1 B 2 2 2 B 3 3 3 B 4 
Suppose now that (2) above be true, then by raising 
n n—m 
n \ rn n - m C m . A 
71 - in 
in 
n 
B m 
to the power of 
m 
n — m 
, we have 
n 
n - m 
n m m 
n—m _ 77 i\n—m Q _ J^n—m 
m ) 
n 
■jD n—m 
<1 
(4) 
The numerical factor here is 
n 
i n—m 
7l J 
in' 
n V l_m m 
— (n - m) \m) n — m 
—m \ 7 
Comparing this now with the numerical factor in 3, we see that 
mV- 
) m 
U' 
is a proper fraction raised to a power greater than 1, whereas 
/ r fi\ 71 
\ m T~ m a m4xec ^ number or quantity greater than 1 raised to a power, 
and is therefore >1. The whole expression in (4) is greater than the 
similar expression in (3), and therefore if (2) hold, so does (3). 
In other words, if there be m roots of the equation Ax v — B^c m H-C = 0 
belonging to the operand (g) ] there are also n — m roots belonging to the 
operand 
