31 
1916-17.] Operators applied to Solution of Equations. 
difference between the two least roots, there remains now a linear form 
+ 2A(a 2 — a x ) 4* 3Aa 1 + B = 0, giving 
^2 = — — — 2(a 2 — cq) — 3a x = eq + a 2 + a 3 — 2(a„ 2 — cq ) — 3cq — a.. 
a. 
And x = £ + «2 = « 3 > a root shown retracted into a single term, its initial 
operand. On referring to the solutions (a), ( b ), and (c) in § III, it will 
be seen that (a) vanishes completely with E, and reduces the other two 
to single rays ; the vanishing of C next causes (c) to vanish, and reduces (b) 
to a single term. 
The proof here outlined is easily seen to be of general application, and 
may be extended to include equal and imaginary roots. A pair of equal 
roots will fall out in their turn when the indicator passes through their 
value, taking with them two coefficients, and when the real part of two 
imaginary roots a±i/3 is reached, these will also disappear, along with 
two. coefficients, if the origin be shifted so as to make the radius vector 
(x — a) 2 + /3 2 disappear. 
This proposition is of importance practically in enabling us to see, often 
by mere inspection, that the roots cannot all belong to operands with integral 
indices. Such equations are in fact the exceptional ones, as will be seen 
from the following definition of an operand. As operand of an equation 
may, or must, be taken a value of x derived from any two terms in the 
equation, such as R,r r — R^ 1 — 0, or x — 
V R 
If r — r 1 = 1 , we have an 
integral indexed operand, otherwise a fractional-indexed one. 
To introduce the roots derived from fractional-indexed operands, we 
begin with the trinomial form Ax n — Bx m + C = 0. Lagrange’s Theorem 
now failing, we employ common Reversion of Series. 
Writing first 
we have first 
C A , 
- + ^ x 
B E 
or x 
C A 
X ~ l T > + 
,n | to 
X — 
Next 
x = 
giving 
x 
n—m+l 
1 1 A C m 
m _j_ 
m 
Tl- f-i 
B ^ 
