AS COMMONLY INVESTIGATED. 221 



and equalities, and having apparently an obscure notion of that most valuable 

 part of analysis, the theory of correlations, vv^as led to remodel the differential 

 calculus, and to discover R by a detailed investigation of each special case. 

 Such a process led to rules that seemed to multiply infinitely; the unity and 

 power of analysis became frittered away into — the limited propositions — the par- 

 tial clearness — and general obscurity of synthesis ; and yet so great was the in- 

 fluence of this clever writer, that the injury done to science may be traced 

 even in the works of analysts the most profound,* and greatly disfigures the 

 works of many continental writers of less reputation. The extent of this in- 

 jury having been seen by Mr. Peacock, he was led, in opposition, to assertions 

 that err in a contrary direction, laying down as a laM^, of which he regards 

 himself as the discoverer, that not only is every development an equality, but, 

 proceeding farther in this method of regarding the subject, that every proposi- 

 tion of numerical logic, once established as true within limits, is thenceforth to 

 be looked upon as universally true — an assertion which would reduce all ar- 

 rangements to one arrangement, and deny to correlation the power of restrict- 

 ing, or of referring propositions not conveniently grouped together, to distinct 

 classes. 



We trust that, after the preceding remarks, it will be clear that both of these 

 views are erroneous; that developments considered 'per se, as, for example, in 

 the theory of generating functions, involve no idea of equality between the 

 members of the proposition, and are even, in many cases, independent of con- 

 nexion between the terms, as in the proposition 



12 3 X 



P /l;^ P, P, P, P, . . P. . : 



an independence that does not prevent the equality between the numbers of 

 the original development 



P = Pk" + Pk^ + Pk' + . . Pk^ . . -^ R 

 whence the preceding may have been derived, from having its use in tracing 

 the nature of the latter, or regulating its applications. 



The development whereon the dijfferential calculus is founded requires that we 

 should keep these remarks in mind. The great analysts to whom that method 

 of reasoning is due perceived that, treating F (x + h) as a generating function, 



12 X 



F X becomes the first term of a series of functions P, P, P . . . P . . . having 



* The second edition of Poisson's Mechanics may be cited in proof. 

 VII. — 3 F 



