1916 - 17 .] Operators applied to Solution of Equations. 
35 
accurate account of the whole residue of the process to its most distant 
utterance. 
The solutions so developed are 
(operators D^b, . D E (S_1) . Eq § , D ( d C) . . D^). 
(S' 
IB 1 CEy 
3 A - 3 T A* 
1 2 B 2 Ei 1 I R C Ei 1 Q (PE^ 1 
+ 3 ‘ 3 ’ 2! At + 3 ’ 3 ‘ 1' 1' Af + 3 ' ' 2! A 1 
(0 
1 4 1 B 3 E-t_l j Q B 2 CE 
3 3 3 3! At 
1 B 4 
+ ±.2.1.0.—, etc. 
3 4!’ 
1 2 1 B C 2 E-t 112 C 3 E-t 
Q 
2! 1 A 2 + 3 3 3 1 2 ! At + 
3 3 3 3! At 
+ 
1 5 2 1 B :1 C E 
f -4 
1 4 1 2B 2 C 2 E-t 1 
+ O • o • o • « ZW - 7T7 
3 3 3 3 3! 1 At 3 3 3 3 2! 2! At 
1 2 1.4 C 4 E-t , 
— . — , etc., etc. 
+ 3 3 3 3 4! At 
-.1.0, etc. 
Before remarking on this development, I shall give an example. 
Rearranging, as we must always do, in powers of the surd, we have 
x = \- 
3 A 
/E N 
\ s 
ri+ iBC 
2 B 3 ( 1 
"8l EA 2 + 81 
C 3 
A J 
2 B 2 C 2 
7 B 4 C . 
pfP 
\A, 
I 
9 EA 
AE 2 
81 A 2 E 2 
729 A 3 E 2 ’ J 
e\* 
a; 
1 C 1 B- 
1 BC 2 
B 3 C 1 
+ 
C 4 
B 5 
.3 E 9 AE 27 AE 2 243 A 2 E 2 243 AE 3 729 A 3 E 2 
This gives for the equation + + — 100 = 0 
, etc. 
-i+100- 
o 
1 
1 + 
1 
1 
1 
900 4050 810000 
etc. 
100 * 
1 
1 
1 
300 900 270000 
, etc. 
4+ (4*6415)(1*000865) - (21*544)(*0022185) or 4*2644 (approx.). 
O 
And to get the other two roots we must affect 4*6415 and 21*544, first by 
co and os 2 respectively, and then by or and os, where w is a special cube 
root of 1. 
It is advisable to give still another example before discussing the 
coefficients in (7). The reader will easily find the remaining (■ n — m ) roots 
of the equation Ax n — Bx m + C = 0 given from the operand by the 
operator D ( _ C) . D 
They are 
m 
n—m 
A 
D n—m 
B 
1 
n — m 
+ 
1 
(-C) 
m — 1 
j^n—m 
n - m 
n — 1 
B n 
+ 
n — m 
2m— 1 
n + m - 1 ( - C) 2 A n ~ m 
Zn—l 
p>?i — to 
n — m 
2 ! 
1 2 n + m - 1 n + 2 m - 1 ( - C) 3 A 
3m -1 
n—m 
3! 
Sn — l 
7)1 
, etc. 
B n - 
n - m 
n — m 
n - m 
