-1874.] Investigation of Determinants. 215 
B,.E,=((E] +1).((E] +1)" 
=((B) +1) 
= Epig eae wh Ce ics We Sela ee sin ee (7). 
that is to say, the suffixes of E also follow the index law, 
so that the notation may be further modified with the 
interpretation— 
(12) sed De gar ae oie ean ac Cor er (8). 
Hence, from Eq. (6) and (8)— 
GE)t= = (EP +1) sates ee (g). 
and this Equation being general for all (positive integral) 
values of n— 
Bs hea, and je) = Bw <a. 3 (2O)s 
These are symbolic relations between {E] and E of re- 
markable simplicity, and lead to an explicit formula for 
calculating [E,]. For— 
LE,,] == [E] e—(K—1)". 
n(n— | 
= E,—*.E,-1+ =e a Bye oe +(- reo eer E,y-+t+ 
7 
—— e 
(eat 
——. 
4 (—1)t-2, 9). B+ (—1)"- 12. E,+(—1)" Ct). 
IV. Number of Elements in a Determinant out of whose n 
Leading Constituents only m are Finite. 
Generalise the notation of Problem III. as follows :— 
mera), [A’], [A], .-» = fA”) represent the values of 
A (an ordinary determinant with finite constituents), when 
out of its leading constituents (of type @,,) there are re- 
Spectively none, one, two, &c.,... . m fintie, and the 
remainder zero. 
Memes), (,), [E's],> . . [E"] represent the corres- 
ponding values of E,: by this notation [E”] =E,. 
Then, by the same reasoning as in Problem III., and 
noting (in addition) that all terms involving the continued 
product of more than m leading constituents vanish 
necessarily in the case of the determinant [A”], it follows 
that— 
[A”] = [A] +35 an-Laa,, |} +3 1p i eosge all, cat 
dma ) 
$Y (i223 le oe dnm)-| dagedice don ae a= ||, S (12). 
