Dr T. Muir on the Theory of Determinants. 
521 
so geht diese in folgende Gleicliung liber 
/n k Kn+l 
a P( a ; a , a) x 
n+l \n h > h' 
C n ni n\ k / n k\ 1 /n 1 
+ •< a P( a ; s , « j + . . . + « P( a ; s , a j + . . . + a V(a ; s , a 
(. 71+1 \7i h liJ ?i+l \n h iJ ?i+l \n h 
( ” / 
/ 77 77 + 1 77\ 
k 
^ 77 77+1 k\ 
1 7 
7" 77 77 + 1 Iv 
^ «+l 
a ; a , a) . 
. . + a Vi 
a ; a , a) + . . 
, . + aV( 
a; a , « ) 
1 X 
1 n+1 ' 
^77 h h/ 
77 + 1 ' 
\ 77 h h/ 
77+1 
'' n h hJ 
i 
+1 
X 
folglicli 
- a P( a ; s , a ) 
71+1 \n h h/ 
1 / n 71+1 1 \ 
- a F[a ; a , a) 
71+1 \i li h* 
2 , 
/n 
77 
7' 77 
77\ 
/ 
7 77 
k 
k\ 
a P( 
a 
; ^ 7 
a)-. 
a Vt 
a 
; s 7 
a) 
I + 
. P 
a ; 
a , 
, a) 
77 + 1 
^7? 
h 
iJ 
77+1 
'' 77 
h 
iJ 
77 + 1 ' 
h 
hJ 
^ / 
7 77 
77 + 1 
77 j 
''77 
77+1 
ii\ 
77+1 , 
''77 
k 
k\ 
a P( 
a 
i « 7 
a)-. 
. . - «P 
a 
; a , 
a 
) + 
a P( 
a ; 
a 
,a) 
77 + 1 ' 
h 
iJ 
77+1 ' 
'' 77 
k 
ih 
77 + 1 
77 
h 
iJ 
die first theorem here made use of and formulated, viz., 
( n 77+1 77 + 1 l'\ /?7 k\ / 77 77+1 77 + 1 
a \ s- a X ^ a] = T[a ] s , a) - Via ] a x , a] (xlvii.) 
77 h h iJ \ 77 /7 iD \ 77 ll 1l ) 
is the now familiar rule for the partition of a determinant with a 
row or column of binomial elements into two determinants, or for 
the addition of two determinants which are identical except in one 
row or one column. The second theorem, viz., 
( 77 77+1 77 + 1 lC\ 77+1 / 77 77 + 1 k\ 
a ] a 5 a ) = X via ; a , a) (xLViii.) 
n h hJ '^77 h h ' 
is the now equally familiar theorem regarding the multiplication of 
a determinant by means of the multiplication of all the elements 
of a row or column. That these two very elementary theorems 
should not have been noted until the time of Scherk is rather 
remarkable. 
The consideration of the constitution of 
Ic 
a ; a , a) 
77+1 h ll/ 
is next entered upon, with the object of showing that the terms are 
exactly the terms of the denominator 
