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II, On the Interchange of the Variables in Certain Linear Differential Operators. 
By E. B. Elliott, M.A., Fellow of Queens College, Oxford. 
Communicated hy Professor Sylvester, F.R.S. 
Received June 5,—Read .June 20, 1889. 
Contents. 
Introduction 
I. Binary Operators. 
Definitions ......... ... 
Symbolical basis of method of transformation. 
Application to cases in which sum of degree m and step n is not less than unity 
The special case of m = 0. Transformation of V. 
Cases ofm + w = 0. 
Cases ofm + M<[0 . . 
II. Ternary Operators. 
Definitions and basis of method... 
Proof and exemplification of formulie of cyclical transformation .... 
Transformation of w^, Aj, . 
Operators free from first derivatives. 
Transformation of Qj, Ag, A_j, Vg. 
Section. 
1 
2 
. 3, 4, 5 
6, 7 
. 8, 9, 10 
11, 12, 13 
. 14, 15 
IG, 17, 18 
19, 20, 21 
22 
23 
24 
1. The operators to be considered, include or involve all those which have presented 
themselves as annihilators and generators in recent theories of functional ditferential 
invariants, reciprocants, cyclicants, &c. The general form of the binary operators, 
operators wliose arguments are the derivatives of one dependent with regard to one 
independent variable, which I propose first to consider, is adopted in accordance with 
that used in two remarkable papers by Major MacMahon,* They are his operators 
in four elements. The analogous ternary operators to which I subsequently devote 
attention, are distinct from his operators of six elements. Their arguments are the 
partial derivatives of one of three variables, supposed connected by a single relation, 
with regard to the two others. 
* “ The Theory of a Multilinear Partial Differential Operator with applications to the Theories of 
Invariants and Reciprocants,” ‘London Math. Soc. Proc.,’ vol. 18, 1887, pp. 61-88. “The Algebra of 
Multilinear Partial Differential Operators,” ‘ London Math. Soc. Proc.,’ vol. 19, pp. 112-128. 
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