34 
Proceedings of the Royal Society of Edinburgh. [Sess. 
generally to get the y> th power (or extract the p th root) of any root we have 
simply to raise the operand to the p th power (or take its p th root) and act 
on this with the same operator or operators unchanged. 
Regarding (5) as an attempt on the part of the surviving terminal 
multiplier in (3) to reach its terminal partner which has disappeared, then 
n 
in (6) we see that this quest will be successful when — is an integer, and 
m 
n 
the operator D m coincides with the ordinary symbol of differentiation. If — 
m 
be not an integer in (6), however, and always in (1) where the root has been 
extracted, the terminal partner is never reached, the quest is perpetuated 
to infinity, and determinateness, tangibility, real quantity thus is only to 
be attained by the aid of a similar operator acting inversely. Thus 
n 
unless — be integral, the occurrence of D" 1 necessitates the presence of 
m 
D m , its “ conjugate ” operator, and in this way we avoid all reference 
n 
to fractional differentiation (the occurrence of D m alone), speculation about 
which must be endless till we can control it by concrete results. (With 
n 
this interpretation of D ,n , cf. Gauss II function.) 
To illustrate this Method of Operators further, in possession of which, 
it will be seen, we have left both Lagrange’s Theorem and the Differential 
Calculus behind, suppose we ascertain, by a test to be given presently, that 
the cubic Aaf 5 -f- B,r 2 + Or — E = 0 has three roots belonging to the operand 
E N 
A, 
We write therefore 
3 E B 9 C 
rytO ,yt 
*Ay tAy %Ay • 
A A A ’ 
EA 
or with x — ( J as initial value, 
x 
f E BE* CE^) i 
’ lA _ AS _ "aE/ ‘ 
We now go through the formal process of extracting the root once. This 
oaves 
o 
/E'i IB 
X== W "3A 3 
1 C E-* 
' ' 1 ‘ AT 
The conversion of the operand y j J i nto “ g ^ 
EV 
1 B 
-r and — ^ — 
ICE-* . 
0 . . is a de- 
3 1 A 
terminate problem, with only one solution.* The necessary operators are 
DJlg) . De ( i_1) • D a s and DJIq, . . D a j , and these will give a rigorously 
* Dropping tlie constant of course. 
