1916-17.] Operators applied to Solution of Equations. 
43 
v - 
the second operand, and so on. Whenever our way is blocked we 
Pn- 2 
pick out the term giving rise to the most divergently acting operator, and 
combine this in a binomial form with that coefficient farthest to the right 
which has not yet been made the numerator for a solution, and there 
will be roots belonging to the operand got from this binomial form by 
root extraction, unless this is frustrated by some term farther to the left, 
which is then taken as a new basis for roots by root extraction. Clearly, 
the farther we proceed to the left, or the greater r 2 — r 1 is in the form 
/R \ 1 
R 2 af j — R yf 1 , giving the operand yg]y 2 A the greater number of roots we 
have from this one operand, and in this regard roots belonging to integral- 
indexed operands (r 2 — r l = l) are the least advantageous of all to calculate. 
This is to some extent counteracted by the fact that when we are evolving 
3, 4, or more roots together the convergence is apt to be slow, and we 
shall often find it preferable to isolate any particular root we are following, 
as the smallest root in an equation reduced by Horner’s process or a slight 
modification of this we shall give presently. 
Examples : — 
The equation x Sj rX 2 — 2x — 1 has one operand, — ^ and 2 belonging to 
x 2 = 2. 
/ 8 Y 
The equation 7afi + 2(bc 3 + 3£c 2 — 16^ — 8 has three operands, ( — J and 
i__?° 
i- 7 
The equation tc 5 + 12x 4 + 59cc 3 -f-15Cf» 2 + 201x — 207 has five operands, 
'207Y 
* 
I go no further into the general theory at present, however, as the 
consideration of the exceptional cases would lead us too far. Another 
paper is necessary to deal with this point and the simplifications that 
may be introduced into these expressions for the roots as first thrown off 
by the operators. 
A new basis is also afforded for the occurrence of algebraic solutions, 
this basis being the simplicity or complexity of the p th roots of unity, p 
being the quantity (r 2 — rq) above, indicating the root of the operand taken. 
The sixth roots of unity being comparatively simple, so that they may 
be grouped in twos and threes, a sextic with six roots derived from 
can have these roots expressed algebraically. 
* See note at end. 
Not so the corre- 
