424 Dr. T. Muir on an overlooked Discoverer 



which is the identity published by Sylvester in the Philoso- 

 phical Magazine for August 1851 (ser. 4, vol. ii. pp. 142-145). 

 Jn the statement of it there given one of the products is on 

 the one side, and the rest on the other, some affected by the 

 sign + and some by the sign — . A modification of this, 

 which is simpler in that all the signs are positive, is given in 

 my < Theory of Determinants,' p. 124, viz.: — "The product of 

 two determinants of the same order is equal to the sum of like 

 products obtained by interchanging q chosen columns of the one 

 determinant with every set of q columns of the other in succes- 

 sion: the interchange of q columns with q columns being effected 

 by interchanging the first column of the one set with the first 

 column of the other, the second of the one with the second of the 

 other, and so cm." 



12. The only mathematician whom Schweins mentions as 

 having preceded him in handling such identities as are given 

 in this chapter is Desnanot, to whom he attributes the very 

 simple instances 



2± 



a 1 



a 2 





h 1 



\h 



h 



h 



h 





oc 2 « 3 ! 



1 «i 



*2 



"3 



*i 



2± 



h 





b, 



63 



K 





"1 



«2 





«i 



«2 



*3 







= 0, 



0. 



It deserves, however, to be noted that B^zout was much 

 more worthy of being mentioned. We find on pp. 185, 

 186, 187 of the Tlie'orie gendrale des Equations Algebriques, 

 published in 1779, the identities 



1 «cA 1 1 c o^i I — I s c i 1 1 Mi I + 1 Vi 1 1 «cA |=o, 



I « A^ 1 1 d e! I — I aoMs 1 1 c o e i I + I «cAd 2 1 1 Vi 1 — I b Q c x d 2 1 1 «o^i I = 0, 



«o V2 1 1 ^1/2 1 — I aoM2 1 1 o e r f 2 1 + 1 «cAe?2 1 j V1/2 1 — I VA I \a e x f 2 | = 



the second of which is the same as the second of Desnanot's 

 as given above, and the first and third are cases of Sylvester's 

 Theorem. Besides it must be noted that these were viewed 

 by their author as merely simple instances of an unlimited 

 series of identities. His words are (p. 187, § 223): — 



u En voila assez pour faire connoitre la route qu'on doit 

 tenir, pour trouver ces sortes de theoremes. On voit qui! y a 

 une infinite' oVautres combinaisons a faire, et qui donneront 

 chacune de nouvelles fonctions, qui seront zero par elles-memes : 

 inais cela est facile a trouver actuellenient.^ 



There is thus no small share of credit to be assigned 

 to Bezout. It is of no consequence as an objection to say 

 that the identities are not given by Bezout in determinant 



