42 
MR, E. B. ELLIOTT ON THE INTERCHANGE OF THE 
by the equalities of ratios at the end of the last article. Accordingly 
{0, 1, 0 ; m, n, ri}^ — — 4- 1){1, 0, 0 ; n -{■ I, n, m — l]y 
= (0, 0, 1 ; n + 1, m — 1, n}~. . (85) 
And once more, precisely in the same way, 
a ^ ci^ 
are equivalent operators in x, y, z respectively dependent, so that also 
{0, 0, 1 ; m, n, = {0, 1, 0 ; n -{■ I, n, m — l}y 
= — {ll -f" 1) {1} b ; 11 7Yl — 1, 71 j. 
( 86 ) 
Of these sets of equalities (85) and (86) may in reality be deduced from (84) by 
cyclical interchanges of the variables and alteration of parameters. The independent 
investigation above is justified by the verification it affords. 
The general formulse of transformation, including (84), (85), (86), follow from them 
by (79), and are 
{/r, V, v '; m, n, u'}^ = { — i^(n -j- 1), i'', — -; n -j- 1, n, m — l]y 
= { — V {n' ^ , V n' in — 1, 7 i }-. . (87) 
TYt/ 
Included, it is interesting to notice that we have three distinct classes of self 
reproductive or cyclically persistent operators, of characters corresponding each to one 
o± the cube roots of unity, viz,: 
(— 777, 1 , 1 ; 777, 777 — 1, 777 — 1 }^ ={-- 777, 1, 1 ; 777, 777 — 1, 777 — 1 ]^ 
= { — 777, 1, 1 ; 777, 777 — 1, 777 — 1 }. . . (88) 
f — 777, CU, CD® ; 777, 777 — 1, 777 — 1 ]^ = CD [ — m, CD, CD® ; 777, 777 — 1, 777 — 
= CD® { — 777, CD, CD®; 777, in — 1, 777 — 1 }- . (89) 
I — 777, CD®, CD ; 777, 777 — 1, 777 — 1 = CD® { — 777, CD®, CD ; 777, 777 — 1, 777 — \}y 
= CD { — 777, CD®, CD ; 777, 777 — 1, 777 — l}j . (90) 
20. Some of the simplest, and most important so far as actual experience goes, 
examples of the formulm now proved will be considered in what follows. 
