in the Theory of Determinants. 421 



Then from the gnomon (4, 7, 4) he would pass to the 

 gnomon (p, n, q), n being unequal to p + q, and finding, as 

 before, its two developments in terms of complementary 

 minors with non-zero elements, would at once arrive at 

 Schweins' most general theorem. 



7. The essence of the chapter would thus be seen to be the 

 equating of the tivo possible developments of a quadrate gnomon 

 in terms of products of complementary minors ivith non-zero 

 elements ; and if we should wish to attach to the theorem a 

 depreciatory technical label, we might characterize it as but 

 the statement of the " dimorphic expansibility of a quadrate 

 gnomon." 



The very reverse of depreciation must, however, be our 

 feeling when we recall the date 1825 ; and when we bear in 

 mind the fact that a formal statement of the general identities, 

 independent of the way in which they may be established, has 

 not even yet found its way into our textbooks, such modern 

 mathematicians as Sylvester (1851) having enunciated only 

 special cases, and the masterly memoir of Reiss (1867) having 

 been as nearly as possible altogether neglected. 



8. The following is Schweins' statement of the most general 

 of the theorems : — 



2±|A 1 ,..A^ f B' 1 ...B'J-|BV H ...B' ro | 



I u'i a'(n— q) I a'(n— q+l) . . . a'n, b { . . .b(m — q)\ 



= 2±[A 1 ., A„_ q \ Bx B„|. 



The only points about it requiring explanation are the exact 

 effect to be given to the symbol 2 and the meaning of the 

 dashes affixed to certain of the letters. The two symbols are 

 connected with each other, the dashes not being permanently 

 attached to the letters, but merely put in to assist in explain- 

 ing the duty of the 2. On the left-hand member of the 

 identity the two symbols indicate that the first term is got by 

 dropping the dashes, and that from this first term another term 

 is got if we substitute for B 2 . . B q some other set of q B's chosen 

 from B x . . . B m , and take the remaining B's to form the B's 

 of the second determinant, the two sets of B's being in both 

 cases first arranged in ascending order of their suffixes. On 

 the other side of the identity the use of the symbols is exactly 

 similar, n — q of the n superfixes a ly . . ., a n being taken for the 

 first determinant of any term of the series and the remainder 

 for the second determinant. The number of terms in the 

 series on the one side is evidently m ! fq l(m — q)l, and on the 

 other n ! jq ! (n — q)\ 



