150 
MAJOR P. A. MACMAHON OR A CERT AIR CLASS OF 
the third row by divide the third column by ; multiply the fourth row bv a.-,„ 
divide the fourth column by ; the determinant is then | \ ^ 
Similarly it is sliowu that the equation is satisfied by the system 
yxy — C,yx ^ — byxjUy^x- 
The equations, involving determinants of higher order, can similarly be shown to be 
. / 
satisfied by both systems of values, and since the j quadratic equations have each 
but two roots, it follows at once that 
are the only systems. 
-xy 
-1 
■yx 
-1 
§ 7. Arithmetical Interpretations resumed. 
Art. 55. The arithmetical interpretations drawn from the theory have been so far 
of two kinds. In the examples taken from the “ Memoir on the Compositions of 
Numbers” we had a redundant form of generating function aud an exact or condensed 
form; the redundant form and the exact form could be differently interpreted, and 
this led to an arithmetical correspondence which was duly noted in the memom 
quoted. The interpretations, subsequently considered in this paper, were single, and 
there was no arithmetical correspondence; the condensed forms did not admit of easy 
and useful interpretations, but only the redundant forms. The redundant forms were 
not considered in the most general form which, as we have seen, involves n — I 
undetermined quantities, but each of these quantities was given a special numerical 
value ; this process led to simple and useful arithmetical results but it will be obvious 
that the possibility of interpretation does not stop here. 
Art. 56. In proceeding from the condensed form to the redundant form we met 
Avith n — 1 undetermined quantities 
ai2, a^S’ • • • 
As before remarked, we may, if we please, put these quantities equal to certain 
functions of the quantities 
/y» /yt 
tA-Oj • . • 
We are not at liberty to choose any functions. The functions must satisfy certain 
conditions, otherwise the coefficient of 
r/* ^2 /y» 
in the particular redundant product Avill not remain unchanged. 
I propose to examine this question. 
