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II. On an Extended Form of the Index Symbol in the Calculus of Operations. 
By William Spottiswoode, M.A., F.M.S. 
Eeceived November 24, — Bead December 22, 1859. 
In the following paper I propose to resume more in detail the subject of a communica- 
tion made to the Cambridge and Dublin Mathematical Journal (vol. viii. p. 25). The 
present investigations are restricted to the case of two variables. 
Let 
and let 
§ 1. The Theory of\J and Vi- 
d d 
d , d ^ 
y dx^^ dy — ^ C'. 
d\ d' „ 
^ dx^ydy—^^ 
d\ d’ „ 
where the accent indicates that in the combinations of S, Sj the differentiations are to 
affect only the subject of operation, and not x or y, so far as they appear explicitly in 
the values of 3, Sj. So that 3^ 3 3i, 3? are to be understood as follows: — 
d'^ d^ d'^ 
and so on for higher powers. 
It will then be found that 
3^=V^-V =(V-1)V, 
33,=VVi-V,=(V-l)Vi, 
E?=V?-V =V?-V. 
MDCCCLX. D 
