54 THOMAS MUIR ON GENERAL THEOREMS ON DETERMINANTS. 
and so on. Hence the sum of the 2 determinants is 
(& ae S - S Ethie wt é,) D (din) 
as was to be shown. 
From this simple theorem there follows at once Mr MAtet’s theorem* 
regarding the multiplication of 
1 1 1 1 
a” B® y" oO” 
a” p” y" 6” 
a? B? y o” 
or F( Ms, 1, p) 
by a+B+y+6, the product in question evidently being 
F(a mM, N, ?) + F(o, m+1,n, p) + F(o, Mm, 2 + 1,7?) + F(o, Mm, N, p + 1) F 
and the theorem is seen to be capable of extension not merely as regards the 
order of the determinant (as Mr Mater indicated), but as regards also the 
degree of the multiplier, which, instead of being 2a, might as easily be 2a". 
* “ Hducational Times” Reprint, vol. xxviii. p. 51. 
