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VIII. On a General Method in Analysis. By George Boole, Esq. Communicated 
by S. Hunter Christie, Esq., Sec. R.S. fyc. 
Received January 12th, — Read January 18th. 
Much attention has of late been paid to a method in analysis known as the 
calculus of operations, or as the method of the separation of symbols. Mr. Gregory, 
in his Examples of the Differential and Integral Calculus, and in various papers 
published in the Cambridge Mathematical Journal, vols. i. and ii ., has both clearly 
stated the principles on which the method is founded, and shown its utility by many 
ingenious and valuable applications. The names of M. Servois (Annales des Mathe- 
matiques, vol. v. p. 93), Mr. R. Murphy (Philosophical Transactions for 1837), 
Professor De Morgan, &c., should also be noticed in connection with the history of 
this branch of analysis. As I shall assume for granted the principles of the method, 
and shall have occasion to refer to various theorems established by their aid, it may 
be proper to make some general remarks on the subject by way of introduction. 
Mr. Gregory lays down the fundamental principle of the method in these words : 
“ There are a number of theorems in ordinary algebra, which, though apparently 
proved to be true only for symbols representing numbers, admit of a much more ex- 
tended application. Such theorems depend only on the laws of combination to 
which the symbols are subject, and are therefore true for all symbols, whatever their 
nature may be, which are subject to the same laws of combination.” The laws of 
combination which have hitherto been recognised are the following, tt and § being 
symbols of operation, u and v subjects. 
1. The commutative law, whose expression is 
•jr^u—g’xu. 
2. The distributive law, 
t(u- f-v) —nu-^KV. 
3. The index law, 
K™K n U = K rn + n U. 
Perhaps it might be worth while to consider whether the third law does not rather 
express a necessity of notation, arising from the use of general indices, than any pro- 
perty of the symbol rr. 
The above laws are obviously satisfied when t and § are symbols of quantity. 
They are also satisfied when t and § represent such symbols as A, &c., in combi- 
nation with each other, or with constant quantities. Thus, 
2 g 2 
