462 MR. A. E. FOESYTH ON INVAEIANTS, COVAEIANTS, AND QUOTIENT- 

 and the one derivative is a factor of the other ; hi fact, we have the relation 



On account of this property the function [s, z\ 2 is called by SYLVESTER a reciprocant. 



In the case of derivatives associated with equations of order higher than the second, 

 the primitive of the differential equation, which is obtained by equating the derivative 

 to zero, is not symmetrical in regard to the dependent and independent variables ; 

 they may not therefore be interchanged, and hence these derivatives are not recipro- 

 cants of any of the known types. It is elsewhere * shown that the connexion between 

 the two classes of functions is constituted by the property that the quotient-derivatives 

 are combinations of homographic reciprocants, such combinations being, however, 

 illegitimate for the preservation of reciprocal invariance. 



Transformation of the Derivatives. 



112. By means, however, of the primitives of the derivative equations, relations 

 are easily obtained which suggest some of the transformations of the derivatives. 

 For, taking the most general change possible, viz., of both the dependent and the 

 independent variables, suppose (i) that $ and z are connected by the equivalent 

 relations (46) and (46'), (ii) that cr and s are connected by the equivalent relations 



[<r, a], = 

 and 



_ C +cy + ... + c_is--i 



= -' 



and (iii) that z and x are connected by the equivalent relations 



[z,xl=o 

 and 



Then the algebraical relation between cr and x is 



- 



= H + K l z + .".r 



where 



/3 -l = (m-l)(n 



and the differential relation is consequently 



[cr, X\ = 0. 



*" Homographic Invariants and Quotient-Derivatives." 'Messenger of Mathematics,' vol. 17,1888, 

 pp. 154-192. 



