36 Proceedings of the Royal Society of Edinburgh. [Sess. 
The equation x 4 — 12x4-7 = 0 has three roots belonging to a; 3 = 12. 
have 
(12 7 m 
x== \T~i-xf’ 
and then 
__ ( 
~ r 
1 
3 
We 
The operator is therefore D ( _{, . D x 3 . D 12 V 
This gives 
X 5 = 
/12\i,M- 7 ). 1 
VI/ 3 1 L2 
I 4 (_7)2 u is 5 ( - 7) 3 
3*3* ' 1 24 + 3 ’ 3 ' 3 ’ 3! 
I I 1 16 13 10 7 ( - 7) 5 1* 
' 12'# + 3 ’ 3 ‘ T ’ T ’ 3 ' 5! 12V 9 
1? 
12V 
etc. (8) 
These operators no longer obey the Index Law (as did the operators 
with which we began), and each individual term must be derived directly 
from the operand. We bring the indices of 12 and 1 to the required 
value, and then the multiplying factor may be written down. The operator 
for (8) is D ( _\j . Dj 3 . D 3 , so that the integration with respect to 1 is a real 
integration. 
The last coefficient (dropping the constant -J is thus 
19 , ~ 
- ■+■ o 
19 ±f 
~T + 6 
1 
etc., 
and the practical rule is that the infinite factor to the right here, from 
H 9+b ). 
and after the term is to be evaluated as + 1. (It may in reality 
5 
be + 1 or — 1 as we please to include in it an odd or even number of 
terms.)* Thus the above multiplier is eventually +etc., being 
Turning now to (7) we find the operator here to be . Db .D a * , 
so that the operation carried out with respect to E (as in every case when 
— < 1) is really a differentiation, or, as it is perhaps better to drop this 
word, it is an operation reducing E’s index, and thus of essentially the 
same character as that carried out with respect to A. The elements in 
the numerator and denominator must therefore be represented as either 
* But see later. 
