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IV. AN INVESTIGATION OF THE NUMBER 
OF CONSTITUENTS, ELEMENTS, AND MINORS 
OF A DETERMINANT. 
By Captain ALLAN CUNNINGHAM, R.E., 
Honorary Fellow of King’s College, London. 
constituents, elements, and minors, in four classes of 
determinants, viz., in ordinary, symmetric, skew 
symmetric, and skew determinants. In calculations and 
investigations relating to determinants it is often useful to 
know these numbers independently of the actual formation 
of the individual constituents, elements, and minors. The 
investigation of these numbers will be found an interesting 
exercise in the theory of combinations, and leads to some 
remarkable symbolical relations. 
The references are to Salmon’s “‘ Modern Higher Algebra,” 
2nd edition. The type of a constituent of a determinant of 
n rows will be written a?* and the corresponding first minor 
as A,,., (as in Salmon, Art. 2.) All the constituents and all 
the minors of determinants in the following problems are 
supposed finite and unequal, except when expressly stated to 
be otherwise. Without this limitation the investigations to 
be given are not necessarily applicable. 
A ae following paper is an investigation of the number of 
I. Number of Constituents in a Determinant. 
1. The number of constituents in an ordinary determinant 
of n rows isin general (1.e. if the constituents be all unlike) n*. 
2. In a symmetric determinant of m rows, the (1*~—%) con- 
stituents not in the leading diagonal occur in pairs, and are 
equivalent to only 3(”*—n) different constituents, making up 
with the 2 different constituents in the leading diagonal a 
total of 3(n?—n)+n=3n(n+1) different constituents im 
general. 
3. In askew determinant* the » constituents of the leading 
diagonal are in general different, whilst the remaining (n*—n) 
constituents occur in pairs equal i in magnitude, but of opposite 
sign, so that there are im general— 
n? different constituents, but only 
4(n?—n)+n=3n(n+1) constituents of different magnitude. 
* Saumon, Art. 37. 
+ A distin&tive name would be convenient for this variety: the author 
suggests ‘sub-determinant.” 
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