1889.] Variables in certain Linear Differential Operators. 359 



linear differential operators, whose arguments are the derivatives 

 of one with regard to the others of a set of variables connected by 

 any single relation. By aid of such operators, in their quality of 

 annihilators or generators, the forms of classes of functions of the 

 derivatives having properties of persistence in form after various 

 classes of transformations have been discussed with some completeness, 

 and great light has been thrown on the properties of other functions 

 in connexion with such transformations. Often, however, in cases 

 where the transformations dealt with have not been symmetrical in 

 all the variables, the investigation has presupposed that a certain one, 

 or one of a restricted set, of the variables has been chosen as the 

 dependent variable. A complete theory of the interchange of the 

 variables in the classes of functions has been a desideratum, and 

 towards the attainment of that end a theory of the interchange of 

 the variables in the operators has been a first requirement. Such a 

 theory it is the aim of the present memoir to supply for the cases of 

 two and of three variables. I speak of the operators appertaining to 

 the two classes of cases as binary and ternary operators respectively. 

 For the binary operators dealt with I adopt a general form, which is 

 a slight extension of one introduced in an able investigation of Major 

 MacMahon's, and for the ternary operators one that is closely 

 analogous. 



I. Binary Operators. 



By x and y are denoted two variables connected by any relation. 



By x r and ii r are meant -i ^-^ and ~~ ^ respectively. Let £ and ?? 

 J J r\ dy r r\ dx r r J b ' 



be corresponding finite increments of x and y, so that 



£ = X l r } +X^+Xzr ] Z+ , 



and consequently 



p = ( XlV + X2 f + Xd7] 3 + )m 



= XtV+XHi V m+1 +^ l A ^ +2 + , say. 



In like manner let Y { ™\ Ygi, Y { J$ 2 , be defined. 



Denote the operator 



[ O + V8 )XP j by fa v ; m, n} x , 



the summation being with regard to s, which assumes in turn all 

 integral values not less than the least of m and —n-\-l. Fractional 

 values of m and n are not admissible, but their integral values may 



