Anglin — On sotne Theorems in Determinants. 



661 



Professor Malet, P.R.S., who speaks of the general result {m) in 

 the main proposition as a very pretty one, has suggested the following 

 direct proof oi it, and has kindly given me permission to state it here : — 



Taking the m letters a, h, c, . . . I, the determinant 



a; 



1 i«-i 



a, 

 1, 



Im-l 

 lm-2 



1 



= (^hc . . . I) = J), say, 



where (aic . . . I) stands for the product of the differ ences^in pairs of 

 the m letters a, h, c, . . . I. 

 Now let 



{a-h){a- c) ... {a -I) 

 X = 



, P = 



{b-a){h-c) ... {b-iy 



1 



{l-a){l-h)... {l-ky 



so that 



• . A. = — ; then 



according as ^ is < = or > m - 1 , where h„ is the sum, includiiig powers, 

 of the homogeneous products of n dimensions of the letters. 

 Now let 



then if we call 



aa'^-\ 



/5r-S 



. . . A^'"-i 





aar-^, 



fih^\ 



...XL 



m-2 





' 



I 





; 



1 



aa, 

 a, 





.. XI 

 .. X 







or, 



b% .. 



. ^" 







«", 



¥, .. 



. l^ 







«', 



¥, .. 



. l^ 











- 



= V, 





a", 



b% .. 



. I' 







1, 



1, . . 



. 1 







R.I,A. PROC. , SEK. IT., VOT,, IV. — PflENCF. 



3e; 



