222 Investigation of Determinants. (April, 
there is a pair of conjugate products of (n—2r) other con- 
stituents, viz. {ajeMgn ni,» ++ + Amp} and {aym, «+ » Gin. Une. Aep.} y 
so that the type of the sum of a pair of such conjugate 
elements is { (@9.aqp).(@ys Asy ) + « (AyzAzy.)} X [ {Ape Agn.Ani + - Amst 
+ {Ajn- + Gin Ang Meg |), which pair reduces in the symmetric 
determinant [¢] to a single element of type— 
Z{Apg.Ayse sree Ay: |. (Ajg Agh.Ani.» + +++ Ams.) 
since by definition ay, =a.). “The number of elements of this 
type in the symmetric determinant [3] is therefore one-half 
that in the ordinary determinant [A], 7.c.is }{ [E,}{—Sn }, 
Adding the number of both classes together, there results— 
[Ey ] =S, +2 { [E,]—S, } = { [Bal +S, } o. ota o ee a ae (30). 
substituting for [E,] and S, from formule (21), (27), (28), t 
(én) =z [En] = e. an when 1 is odd 
(31). 
Js 8 
2 \3 
os ee ere : 
Pikes (LA et Ses Cae 5 gf when n is even 
Also, since [E,] is known to be an even integer when is odd, 
and an odd integer when % is even (see Problem VI.), and 
since also S,, is an odd integer when » is even, it is easily 
seen from (31) that [e,] is always an integer (as of course it 
should be). 
A relation between three successive values of [e,] may be 
deduced from that between three successive values of [E,], 
see Eq. (22)— 
Thus [E,] =(#—1).{ [E,—-1] + [E,-2] } 
sro 2? [en] — 3. (n—1).{2 [én—x] + 
+2 [én—2] —S,-:—Sn-2} ’ by Eq. (30). 
And when is odd— 
|n—1 
,—0; cee = 
u—I, 
2 
—— Sy-2=0, by (27), (28). 
And when 1 is even S,, =(n—1). S,—2) Sn-1=0, by (26), (28). 
+ [én] = (—1).{ [¢n—1] + [€n—2] } = -Sn—s 
when# is odd ./) "7, “21 3. saa 2). 
[én] = (n—1).{ [en—s] + [én—2] } G 
when 1” is even 
Thus the value of [e,] may be calculated for successive . 
