114 
MAJOR P. A. MACMAHON OR A CERTAIR CLASS OF 
in precisely two ways as an irrational function of its co-axial minors, whilst no deter¬ 
minant of uneven order is so expressible at all. 
Of order superior to 3, it is not jjossible to assume arbitrary values for the deter¬ 
minant itself and all of its co-axial minors. In fact of order n the values assumed 
must satisfy 
2 " — n — 2 
conditions, but, these conditions being satisfied, the determinant can be constructed 
so as to involve n — 1 undetermined quantities. 
§ 1 - 
Aet. 1, In a Memoir on “ The Theory of the Composition of Numbers,” recently 
communicated to the Royal Society (as above-mentioned), there occurred certain 
generating- functions which admitted important transformations to redundant forms. 
I proceed to the general theory of these transformations, and subsequently discuss 
the algebraical and arithmetical consequences. The main theorem is, in reality, a 
theorem in determinants, of considerable interest, as will appear. 
Art. 2. Consider the algebraic fraction 
_ 1 _ 
(1 - s^X^) (1 - S3X2) ... (1 - .s„Xd ’ 
wherein X^, Xo, . . . X,; are linear functions, of n quantities x^, x.^, . . . a;„, as given by 
the matricular relation 
(Xi, Xj,. . . X4 = (rq. a,, . . . a„) x.^, . . . x„). 
61, />3, . 
• '^71 
El,?;., . 
■ n ,, 
I assume the quantities involved to have such values that the fraction is capable of 
expansion in ascending powers, and products of x^, x.^, ■ ■ . Xn by a convergent series. 
Art. 3. A certain portion of this expansion is a function of SoXq, . . . s,iX„, and 
of the coefficients of the linear functions X^, X 3 , . . . X„ only. One object of this 
investigation is the isolation of this portion of the expansion which, for some purposes, 
in the Theory of Numbers is the only portion of importance.*' 
* It will occur to matliematiciaus, who arc familiar with the Theory of invariants, that generating 
functions not unfrequently present themselves in a redundant form. In pai-ticular, it is frequently 
necessary to isolate that portion of a generating function which includes the whole of the'positive terms 
of the expansion, the negative terms, though admitting of interpretation, being of little moment. 
