21 
1916-17.] Operators applied to Solution of Equations. 
In Physics, when a new theory or hypothesis is broached, it is con- 
sidered sufficient if such theory be shown to involve no contradiction 
with other theories already established, and to be itself supported by 
incontrovertible facts. Further, the consideration of the latter, the facts 
underlying the hypothesis, must precede any attempt to explain the 
hypothesis or to fathom its significance. This is the position we now 
adopt here. Having squared the matter superficially with our formulae, 
and shown that there is no antagonism between the new evaluation of 
D -1 - and the former one, we shall have our hands quite full in considering 
OC 
some of the concomitants of this fact. 
§ III. Solutions of the Cubic, Quartic, Quintic, etc. 
We pass, therefore, to the application of the above mode of solution to 
the higher equations, beginning with the cubic A.X 2, + B^c 2 + Cx -j- E = 0. 
(a) The first operand is got by writing 
E B 
C c ; 
/ V* — — rv*2* ai 3 
A 
d 
and then, by substitution, 
x= - — 
E B/E 
c cvc 
A/E 
= + ^ = 
CVC 
The conversion of — g into — and is the work of the operators 
Dg 1 . De 1 . D c 2 and D^ 1 . D^ 2 . D c 3 respectively. By the aid of these operators 
we develop x as follows : — 
_ E 
C 
x = 
_B/E \ 2 A/E \ 3 
cvc/ + c\c7 
c 
E\5 
a 
B\ 3 /E \ 4 
C 7 VC 
B\ 2 /A\/E \ 5 
-5 ^ m +21 ^ “ ^ -28 
C 7 VC/VC/ 
BVA YYSV+12 
C7VC7 VC 
AWE 
C 7 VC 
-14 
B\ 4 /E \ 5 
~) +84 W ~ 
C 7 VC7 
B\ 3 /A\/E \ 6 
C 7 VC7VC 
B\ 2 /A\ 2 /E \ 7 
C 7 VC7 VC 
etc., etc. 
-iso ; S , +165 
B\/AWE 8 
C7VC7 VC 
A\ 4 /E \ 9 
55 C7 \C 
To get the third horizontal line here, pass the first operator over all 
the elements in the previous line. Then pass the second operator over 
