27 
1916-17.] Operators applied to Solution of Equations. 
F HF 4 A 2 F 7 
The elements — jp-, etc< ’ cons tituting the solution of the 
quartic trinomial Aa? 4 +E^ + F = 0, we may suppose ranged along the 
2 -axis, the other parallel lines being disposed parallel to the x and y axes 
respectively. The solution is thus triply infinite. 
(/) In the case of an equation employing four or more operators, we 
simply develop one horizontal line after another. 
Thus for the solution of the quintic Ax 5 -fB^ 4 — C^ 3 + E£ 2 +Fx-f G = 0, 
E 
belonging to the operand we have 
E F 1 G 1 B 
A 
leading to 
x ~ C + O' x + O' x- + <f + of 
E F C G C 3 BE 3 AE ! 
c + c , e + g'E 5 + c'c 3 + oc 5 ’ 
and therefore to the operators 
- Dp 1 . Do 1 . D e 2 , D^ 1 . Dc 2 . D b 3 , II b 1 . Di 1 . D 0 2 , - D.P . D E 2 . D c 3 . 
These four, acting on the four elements of the second line, yield 
F 2 C 0 FGC 2 FB o FAE 3 2 C 3 
E 3 ~ 3 E 4 + C 2 + 2 C 3 
ns ~ 3 ^4 + + 2 + 0 + + 2 / +5 + 3 
GA 3 2 E 3 _ABE 4 _A 2 E 5 
G 
G 5 
C 6 
C 7 
E.g., let A = B = F = G = 1, and C = E = 10, then these elements give x = 1*48, 
which will be correct to two decimal places, though, for the closer approxi- 
mation, the equation ought to be transformed. 
The number of elements (including the zero) in this third line is 10, 
being the homogeneous products of four things two at a time ; and generally 
the number of elements in the second, third, fourth, etc., horizontal lines 
representing a solution of an equation of the n th degree (or an equation 
with n -\- 1 terms) is ^EL, ^H.,, etc. 
(g) When the above series converge, then they converge to a root of the 
equation. When they first converge and ultimately diverge, then in spite 
of such divergence among the ultimately infinitesimal elements the root 
may be approximated to by following the series in its more important 
elements, the opening ones, though the calculation of the root may be 
facilitated by conducting the investigation into the proximity of the 
suggested root. We must, in fact, change the origin. 
In the analogous case in Differential Equations, Poincare has shown 
