THE INDEX SYMBOL IN THE CALCULUS OF OPEEATIONS. 
39 
have only to write ^ for ^ in one of the factors of each term ; the same factor being 
changed throughout. 
uill give 
the y factors being detected at pleasure, provided that they are p in number, and the 
same factors throughout. 
Postscript. 
Eeceived March 3, 1860. 
In continuing my researches on the extended form of the Index Symbol, I have esta- 
blished a variety of formulae relative to the case of many variables analogous to those in 
the case of two. As the expressions occurring in the investigations are frequently of 
great length, 1 propose here to give only the principal results, without entering into the 
details of proof. 
Let /a, • • f*© any permutation of the series I, 2, . , ; and let it be symbolically repre- 
sented thus : 
Then any permutation performed on P,- may be similarly represented , or 
more simply, y^^,y 22 , . . Then^Y,, J^ 2 , . . will represent a new permutation of the original 
series I, 2, ; and the notation above adopted may be extended thus: 
or, dropping the subject of operation. 
G 2 
and so on generally. 
