49 
1916-17.] Operators applied to Solution of Equations. 
noitre as to the presence of a commensurable root. If the root we are 
following be commensurable, it must be one of the series 
T 0 T It 2 . 
I + A’ I + A’ I + A’ etC ‘ 
1 + 
A- 1 
~aT 
where I is the integral part of the root, and A the coefficient of the highest 
power of x. If among these mixed numbers, when expressed as improper 
fractions, there be one whose numerator divides the absolute term, then 
we should test for a finite root by factorisation. 
85 
From the operand — in the equation 2x s — %5x 2 — S5x — 87 = 0 we have 
85 85 2 
X ~H + V X 85 
/85V 2 
\85/ x 85’ 
etc., 
so that the integral part of the root is 43. Possible commensurable roots 
are therefore 43 and 434, and the latter leads to the factorised form 
2(x — 43'5)(x 2 + x + 1) = 0. 
The operand 
121 
20 
in the equation 20^; 3 — 12 lx 2 — 12 la? — 141 =0 gives 
tr = + a fraction, so that the integral part is 7. 
20 121 43 r 
The second of the series, 7, 7 Aq, 7 t V . . . 7-4§- leads to the factorisation 
(20x — 141)(A 2 + x + l) = 0. 
It is, as will be seen from these two cases, possible to write the solution 
so as to exhibit line after line of zeros. But this would open the question 
of the simplification of operators, and in practice the above test is simply 
applied. Slightly divergent operators may also lead to a finite root. 
The equation ($ — 17)(;£— -8)(£C — 3) = # — 28^ 2 + 211x — 408 = 0 has two 
slightly diverging operators for the two larger roots. If we should pass 
a finite root over, the convergence begins to oscillate. 
Addendum. 
/ 20 V \ i 
With regard to the quintic on p. 43, we say its roots belong to the operands V, 
because, though the operators belonging to these operands act divergently, the elements 
alternate in sign, and in view of the transformation x = y-2, which decides the matter. 
This transformation gives y 5 i- 2,y i + 3y 3 + 4=y 2 + 5y — 321=0, and leads to the solution 
y= -f+14)013(^y - 00153^ -hd — '001047 (-y~ ) ? 
•000872 (~) ! < 
from which we have y = 2*638. See Burnside and Panton, chap, x, ex. 18. The other 
roots are then to be had from the 5tli roots of unity. 
XXXVII. 
(. Issued separately April S, 1917.) 
4 
