PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 5g9 



process of examining the results of each particular form, and retaining only that 

 one which gives no absurdity as its result. 



We are aware that the whole substance of an Academic memoir is usually, 

 as far as is known, origirial, either in its results or in its mode of arriving at them. 

 We trust, however, that we shaU offer a sufficient reason for deviating from the 

 usual custom to a slight extent, when we state that the analysis which we treat 

 of is hardly known by name even in this country ; and even where it is known, 

 from M. Greatheed's paper, its value is far from appreciated. The latter cir- 

 cumstance is undoubtedly owing to its want of generality ; for we confess that, 

 before we were so fortunate as to discover a general form for all the differential 

 coefficients, we subscribed to Mr Peacock's views of the memou- in question, and 

 agreed with him in treating it as almost or altogether erroneous. 



It is right at the same time, that we should state our conviction that M. 

 LiouviLLE himself clearly saw, in his second memoir, the unity of his calculus ; 

 for although he appears to rest his conclusions, even in his very last memoir, on 

 the complementary function, yet, from one phrase in the 15th volume of the 

 Poly technique Journal, we are induced to infer that he considers the fundamental 

 form to be not merely generally true as a definition, but also as a means of opera- 

 tion. We had not seen either this memoir, or those in Crelle's Journal, until 

 long after our own views had been settled. It is strange, however, that M. 

 LiouviLLE does not appear to be able to a^^ply his ideas in the establishment of 

 the theorems requisite for the foundation of an analytical calculus. 



We trust that no apology will be requisite for introducing into the present 

 memoir a considerable portion of analysis, derived from M. Liouville's memoir, 

 as we fear there is no other way of making the subject interesting, if indeed we 

 can otherwise make it intelligible. We trust, too, that the nature of our addi- 

 tions to the theory wiU plead our apology, and we hope that the importance of 

 our generalization will suffice to make some amends for the introduction of bor- 

 rowed materials. Not to spend more time, then, on apology, we proceed to the 

 definitions and first principles of the science, leaving it to the reader to discover, 

 in many instances, what is original and what not. 



Section I. — Fundamental Principles. 



1. There are two ways of arriving at the fundamental formula : we shall ex- 

 hibit them both ; the latter is, however, merely the converse process to that con- 

 tained in the former. 



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