42 
Proceedings of the Royal Society of Edinburgh. 
Suppose now that (2) does not hold, or that 
[Sess. 
n \m n - m C m . A 
Therefore 
n — m 
n — m 
n 
m 
m 
B“ 
n 
tP 
<1. 
n — m 
C m . A 
(The case when this expression = 1 we exclude for the present.) 
m 
Raising the expression last written to the power of — , we have 
'll/ 
n — m 
n 
m 
n - m 
m—n 
B . 0""“ 
m 
A n 
< 1 , 
and comparing this arithmetically with (1), we see that if (2) does not hold, 
(1) must hold, and there must be n roots of the trinomial Ax' 1 — Bcr m + C = 0 
to be had from 
If the above expression, 
n - m 
n 
m 
n — m 
hi 
b ,Ti 
n—m 
C"^“. A 
or its reciprocal = 1, then either m and n — m both must be submultiples of 
n, or some one of the coefficients A, B, and C must be incommensurable. 
Excluding the latter, the only possible case when m and n — m both are 
submultiples of n is when ?i = 2m, so that the equation is really a quad- 
ratic in x m . 
§ VI. CONVERGENCY IN THE GENERAL CASE. 
We have ascertained thus the existence in every case of n converging 
series belonging to the trinomial Ax 11 d=BaJ m zbC = 0, the simplest form from 
which an operator can be formed. If another term -J-RaT be included in 
this form, then either the new operator arising from this term will 
co-operate with the operator of the simpler form in the formation of a 
doubly infinite series which still converges, or this term ±Raf' itself 
becomes the ridge of demarcation for solutions lying on either side of it, 
as the term — Bx m did in the trinomial form. In the general case, we 
start with the two coefficients on the right, testing for an integral-indexed 
solution, and if the different operators act convergently, we then turn to 
