VARIABLES IV CERTAIN LINEAR DIFFERENTIAL OPERATORS. 
21 
of 
and of 
CIq, rto, C<'3, . . . 
0, .X’j, a? 3 , x^, . . . in the one case, 
0. yn Ui’ • in the other. 
MacMahon himself generally takes them as meaning 
Vi, ^3. yn yo • • • • > 
a fact which must not be forgotten in connecting his results with those to be here 
obtained. 
The essential difference between the cases of m + <{: 1 and m n <l should be 
noticed at the outset. Tn the former case, the complete set of coefficients appears 
in the operator (p,, v; m, In the latter, one or more of those coefficients 
(a number of them equal to the excess of — n 1 over m) is wanting at the 
beginning. 
3. The aim in view is to express any MacMahon operator {p, v, m, in x 
dependent as an operator or sum of operators of like form {p', v ; m, ')i'}y in y 
dependent. We need the linear expressions in djdy^, dldy. 2 , djdy^, . . . which, when 
operating on any function of y^, y^, 3 / 3 , .. . are equivalent to djdx^, djdx^, d/dx^, . . . 
operating on the equal function of Xy, x.^, x<^, . . . The expressions in question I have 
obtained in the second"'' of a series of paper’s on Cyclicants, &c. The best form for 
present purposes is hardly there given to the conclusions. It will therefore result in 
a gain of clearness and no loss of brevity if in the present article the proof is given 
rather than the result quoted. The same remark will apply to Article 17 below. 
We may look upon Xy, x.y, x.^, ... as a number of independent quantities, upon 
3/1, 3 / 2 , ys? ■ • • as determinate functions of these quantities, and upon ^ and y as two 
other quantities connected with one another and with Xy, x^, ajg, . . . by ( 1 ) or its 
equivalent ( 2 ). 
Give Xr alone of all tlie quantites Xy, x^, x^, . . . an infinitesinnal variation. Keep 
7] constant. In virtue of ( 2 ) or its equivalent ( 1 ) ^ will vary in consequence of the 
variation of a;,.. Also, as some or all of yy, y^, 3 / 3 , .. . are functions of x,-, some or all 
of those quantities will vary. Thus, from ( 2 ) we shall obtain 
and from (1) 
0 = {yi + 23/2^ + 33/3^3 q_ 
= rj’’ hXr, 
dy^, >3 , dy? 
dXr dXr 
Sap = 0 . 
* “ On the Linear Partial Differential Equations satisfied by Pare Ternary Reciprocants,” ‘ Loudon 
Math. Soc. Proc.,’ vol. 18, 1887, pp. 142-164. 
