in the Theory of Determinants. 419 



changing the superfixes when finding the development, we 

 may do the opposite with the same effect : and theorems III. 

 to VI. hold when " suffix " is put for " superfix." 



VIII. (Development of a determinant in terms of binary 

 products of a row and column.) 



This is given incidentally, but is quite generally stated and 

 proved. 



IX. (Development of a determinant in terms of products 

 of complementary minors.) 



This also is enunciated and proved in all its generality. 



X. If two suffixes or superfixes be identical, the determi- 

 nant vanishes. 



XI. If the first element of a row of a determinant be mul- 

 tiplied by the cofactor of the first element of another row, the 

 second element of the former row be multiplied by the cofactor 

 of the second element of the latter, and so on, the sum of the 

 products is equal to zero. 



Of these theorems only two, those here marked IX. and X., 

 are claimed by Schweins as his own. This of course they 

 really are not ; X. having been enunciated by Vandermonde, 

 and IX. being due in some considerable part to Vandermonde 

 and Laplace. To Schweins, however, belongs distinctly the 

 credit of the formal and general enunciation of the latter 

 theorem, and a systematic proof of it. 



But although the individual results of the chapter were not 

 new when published, the chapter viewed as a whole (that is 

 to say, as an orderly arranged and rigidly demonstrated body 

 of truth) was undoubtedly "a new thing." 



Sect. I. Chap. 2. 



5. The title of this chapter is not sufficiently definite ; it 

 should be Transformation of a series whose terms are products 

 of pairs of determinants into another similar series. 



The first identity which is given showing such a transfor- 

 mation is in modern notation, 



I ajbtcz&i 1 1 e 5 f 6 g-; \ — | a^c^ \ \ d 5 f 6 g 1 1 



-f | aj> 2 c 3 ft \\d 5 e 6 g 7 \ — | «iVa^ || ^5^/7 1 



= I a x b 2 c z J I d 4 e 5 f 6 g 7 | — [ aAc 4 1 1 d i e 5 f 6 g 1 \ 



+ ! ajw* || d 2 ej 6 g 1 \ — | a 2 b^ \\ d&f^ \. 



This Schweins establishes by expanding the first factor of each 



product on the left-hand side in terms of the elements of its 



ast row and their complementary minors, and then, by means 



of the same theorem, combining in new sets of four the 



