25 
1916-17.] Operators applied to Solution of Equations. 
a coefficient before there is one in the numerator, so that no elements 
become infinite. 
( d ) The above mode of representation soon ceasing to be practicable in 
the case of the quartic, quintic, etc., we adopt another notation. 
For the first solution of the cubic given under paragraph (a) of this 
section, calling D^ 1 . D^ 1 , . D c 2 0 V and D^ 1 . D E 2 . D c | 0 2 , we have 
x — (1 + 4- etc. )(1 + $2 4~ Oc )" + etc.) — — , 
where the operational symbols (1 + 0 1 + 0 1 2 + etc.)(l + d 2 -fd 2 2 + etc.) are 
evidently commutative. Similarly for the other two solutions. 
(e) As the application of this process to the quartic, etc., is strictly similar, 
the only new feature being an additional operator or operators, we shall 
content ourselves by merely indicating these solutions. We shall also 
develop one of the solutions of the quartic to illustrate what is meant by 
saying that the solution of such an equation is a triply infinite expression. 
In the quartic 
Ax 4 + lb* 3 + Cx 2 + Ea- + F = 0, 
we have first 
_F_ C/F V B/F Y _ A/E\ 4 
E E E E E EVE/ EVE/ EVE/’ 
so that the operators are 
D^ 1 . Ey 1 . D e 2 f Db 1 • D E 2 . D E 3 , and Dj 1 . D E 3 . D E 4 , or 0 lt 0 2 , and 0 3 . 
Therefore we have 
x — {\ 4 - 4- etc.)(l + $2 4" 4- etc.)(l 4" 0 3 4- 4- etc.) of — — • 
E 
This solution, corresponding to the first form for the cubic, we may dis- 
tinguish as the direct solution, in contradistinction to the others, which 
are more or less of an inverse form. This direct solution, we shall see, 
always gives the least root. 
For the second solution we write 
E B „ F 1 A o E B/E\ 2 , F C A/EY 3 
c cf c x * c x c cvc) ‘ C x E cvc)’ 
and reach the operators DjY . D E X . D c 2 , D E X . D<Y . D E , and D A : . D E 2 . D c 3 , two 
quadratic ones and one cubic one. The form ^ appears in the evaluation 
F C 
of the term ^ X =. There are two different denominators for the elements, 
E E 
so that, as the solution develops, zero elements will appear as in the third 
form of the cubic given above. 
