18 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
III. — The Application of Operators to the Solution of the Algebraic 
Equation. By James Littlejohn, M.A. Communicated by The 
General Secretary. 
(Complete MS. received May 9, 1916 ; final abridged form, July 31, 1916. 
Read July 3, 1916.) 
§ I. Foundation of the Calculus. Extended Significance 
of the Symbol D _1 . 
The solution of the common quadratic 
ax 2 - bx + c = 0, 
which is usually written 
b — Jb 2 — 4 ac 
2 a 
is, when expanded, 
a 2 c 3 ,.a 3 c 4 
a 4 r 5 
a 5 c G 
c ac 
x=: T) + ~b 3+2 ¥ + ® ¥~ + 1 ^ + etc ' 
This expansion will be found to be the work of the operator D' 1 . Dy 1 . D^ 2 , 
each term being derived from its predecessor by this operator’s agency, 
so that the whole solution may be written 
(1 +0 + 6 , '- + 6 3 etc.) £ where 6 = D” ] . D“* . D,, 2 . 
b 
In this operator the symbols have their usual signification, but with the 
following important exceptions : — 
1 (jC 
In the first place, Dj 1 . is not log a, but is n ; and again, D a of q or 
a u 
any expression not containing a is to be written 
I), t of <x° x q , or - f q, 
a 
and if, as always happens when we have one quantity (such as a) passing 
downwards, we have, sooner or later, another passing upwards, then there 
occurs the indeterminate form jj, whose correct evaluation, we shall find, 
is -1. 
In the solution written above, this form does not yet occur. 
