1874.] Investigation of Determinants. 219 
Further, the number of terms in the sum in Result (23) 
is clearly the same as the number of constituents of type 
Ayz in [A], t.e. (n?—n), see par. 4, Problem I. 
Hence, since [E,] is (by the notation) the number of 
elements in [A], it follows from (23) that— 
[E,] = . (n? —n). [E’,-1] 
=(n—1).{ [E,-:] + [E,_,] }, by Eq. (17) . . (22). 
Thus, the value of [E,] may be calculated for successive 
values of » by formula (22) from the known values of 
‘{E,], [E.], &c., or may be directly calculated from the 
series (21). 
It will be useful to record a few values of [E,] for 
reference, thus— 
[E,] =0, [E,] = 1p [E,] = Zoe [F,] =9Q, [E,] = 44, 
[E«] =265, [E,] =1854, [Es] =14,833,/- - (24). 
[E,] = 133,496, [Exo] = 1,334,961. 
Corollary. Substituting for [E,] from Eq. (21) into 
Eq. (4) and (2), and separating the symbols of operation 
and quantity— 
[a=En= | *{20+77.20 +7230 +e (oe 
oa eee sa aeeee ed 
from which may be deducted Hees sues 
Me de (OF arash 
[= tna La www 0 (25) 
? 
It is easy to see that all the terms of the series (21) for 
[E,] are even integers except the two last, which are 
(—1)""? (n—1), so that [E,] will be an integer, and odd or 
even according as 7 is even or odd. 
VII. Number of Elements which are Products of n+-2 Pairs of 
Conjugate Constituents in a Determinant whose Leading 
Constituents Vanish. 
This problem is a preliminary to the problem of finding 
the number of elements in a symmetric determinant. 
By ‘conjugate constituents”? are meant pairs of type 
Ba xy « 
The type of element in question is— 
{ (Apq.qp)+ (Ars.Asr) + oe os (Ayz. Ary.) } 
containing »+2 products of pairs of conjugate constituents, 
such as (@yz, az). This element may be separated into two 
conjugate factors, ViZ. (@pq.rs- « + + Gyz)» (Aqp. sre « 2° » Apy) 
