516 MR EDWARD SANG ON THE THIRD CO-ORDINATE BRANCH 



In the process of differentiation, the variable quantity is regarded as a func- 

 tion or dependent of some primary variable, and the ratio of its change to the 

 change of this primary is sought for ; this ratio being the differential co-efficient, 

 or, as Lagrange calls it, the derived function. Here we have three connected 

 variable quantities, the primary, the function, and the derivative; and thus we 

 have three distinct problems characteristic of the three great branches of the 

 calculus. 



In the first of these branches the leading problem is this, — having given the 

 relation between the primary variable and its function, to discover the derivative. 

 In the second branch, the relation between the primary variable and the derived 

 function being given, the primitive, or, as it is called, the integral, is sought. 

 While the province of the third branch is to discover the primary, when the rela- 

 tion between the primitive and derivative functions is given. This third branch 

 may thus be called the Calculus of Primaries. 



This classification of the different branches of the calculus of variables is com- 

 plicated by the existence of various orders of derivatives. Besides the three 

 variables quantities, viz., the Primary, the Function, and the Derivative, there is 

 the order of derivation to be considered, and, therefore, it may be argued, there 

 must be four complementary problems, the fourth problem being that in which 

 the order of derivation is the qucesitum, the Primary, the Function, and the 

 Derivative, being the data. The mutual relations of these problems have been 

 obscured partly by circumstances incidental to the subject, and partly by the 

 peculiarities of the notation employed. The notation used by Newton, and 

 revived in a slightly modified form by Lagrange, is essentially defective ; while 

 that contrived by Leibnitz, and now commonly in use, is cumbrously redundant. 

 In order to denote the derivative of a function, Lagrange places an accent over 

 it ; to indicate the second derived function he places two accents, and so on. The 

 radical defect of this notation is apparent, when we consider that a variable 

 quantity U may be regarded as a function of one or of another primary, and that 

 the symbol 'U contains no indication of the primary ; thus ^U is simply the fourth 

 derivative of the variable U. On the other hand, the notation of Leibnitz is explicit 



on this score, the symbols -3-4- and -pj- indicating the fourth derivatives of the 



variable U, regarded in the one case as a function of x, in the other case as a 

 function of y ; these symbols also paint, as it were, the process of derivation ; 

 they, however, contain a redundancy of parts : the symbol of derivation is twice 

 written, and so also is the index of the order. Now, essentially, there are four 

 indications to be made ; we need the sign of derivation, its order, the function to 

 be operated on, and the primary in regard to which the derivation is to be made. 

 The sign of derivation may be given by a conventional arrangement of the 

 characters, just as in products and powers ; there remain, therefore, three things 



