i3._A GENERAL THEOREM GIVING EXPRESSIONS FOR 

 CERTAIN POWERS OF A DETERMINANT. 



By Thomas Mum. LL.D. 



1. In the "Philosophical Magazine" for October, 1902, use 

 was made of an unproved theorem giving the («+i)th power of 

 any determinant of the nth order in the form of a determinant of 

 the order hii{ii + i), any reference to the proof being purposely 

 delayed because of uncertainty as to the history of the theorem. 

 The object of the present note is to remove this uncertainty, and at 

 the same time to merge the theorem in a much more general one 

 the demonstration of which is given. 



2. It would appear that the first noted case of the theorem, 

 viz., that in which /z=3, is due to A. Brill, who in his paper " Ueber 

 diejenigen Curven eines Biischels, welche eine gegebene Curve 

 zweipunktig beriihren," {Math. Annalen, III., pp. 459-468, year 

 1870-1) made use of it for the purpose of effecting a transformation 

 upon a certain covariant intimately bound up with the geometrical 

 subject he was dealing with. Denoting the determinant 



rti6i ^262 i^zbz ^70634-^362 ^361 -1-^7163 67162 + ^7961 

 by Ag, and multiplying row-wise by 



2I62C3I . . . I61C2I I63C1I 



2I63C1I . I61C2I . I62C3I 



2\biC2\ J63C1I 1626-3! 



2ai|ai62C3| 2a2\aib2Cz\ 2«3|ai62f3| 



C\ i ^162^3 1 C2 1 a-JjiCz I C3 1 ^162^3 1 



rtl6i (7262 ^7363 6i|rti62C3| 62|rti62C3! 63|rti62r3| 



