38 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
The point about the non-reduplication of elements may be established 
by reversing for a few steps any tetranomial or pentenomial form. It is 
proper perhaps to point out further that, so far as the trinomial form 
A 
Ax n — B^ + C = 0 is concerned, there is another operator, n . ^ . D c _(n_1> . D B n_1 , 
equally effective in giving the solution. In the solution (a) of the cubic 
Acc 3 -f Ba? 2 + Gr + E = 0 in § III , e.g., the operator — 2? . D E . D c will give 
the left-hand ray, which is the solution of B^ 2 + Cf +E = 0. It will not, 
however, develop any ray parallel to this one, and therefore operators of 
this type must be rejected. 
V. CoNVERGENCY FOR A TYPICAL OPERATOR. 
We proceed now to deduce a test for the convergency of the develop- 
ment arising from the operators on any operand. 
Let us examine the convergency of the n roots belonging to the 
operand 
in the trinomial Ax n ± B&” 1 ± C = 0 * 
(n > m). 
We have 
3 
giving 
s V A/ 
first, and then 
This gives 
X 
1 
m—n+1 
ra-j-1 5 
A ~n~ 
_ -1 
so that the operator is D B . D c v?i . D ‘ 
and the n roots, if they belong here, will be given by the formula 
x == 
\A J n 1 
C' 
m—n+l 
m+1 
A~ 
n 
2m — n + 1 
71 
B 2 
21 
2m—2n+\ 
c ’ r ” 
2m +1 
1 3 7n -7i+ 1 3m - 271 + 1 
_l — • • 
7i n 7i 
3m— 3n+l 
B 3 C M 
O I 3m +1 
A n 
+ etc. 
* We take here such a combination of signs as excludes negative operands. We are 
considering merely arithmetical ratios, and for an imaginary root its modulus may be 
substituted. 
