162 PROFESSOR CHRYSTAL ON 
proved as follows:—Let A,, A,., &c., the first minors corresponding to 
@, Go, &e., then (see Satmon, “ Higher Algebra,” p. 29) we have 
Aa; Ao — Ay" = (a4, vere an) (Bes eee nn) . 
Hence, if Ay, , %.0. (Gao +» On), Vanish, 
(qq + + + Gan)(Ugg + + + Gin) = Ayo”, 
z.e., the determinant and its second principal minor must have opposite signs ; 
and similarly for the other cases, since all the principal minors are symmetrical 
determinants. 
2. In the next place, we have by the multiplication of matrices, 
D(a, dg... Ay)PE(a, Qe . . « Ar)(@—A;)(t—ay) . . . (4—a,) 
SL 1) | aed Seay dee o ata 
Ade (hy cl vey ah 0 a? re Oe 
BP ae ne ee Onaga 
af as... at | | O atl... agetio} 
(= 1)? Mle Sot sph 21 eS Se asay, 
Sp+1 rh Chi Sp+r 
e Sp42 + 6 + Sptr4) 
L” Spry s+ + Sppor—1 
Here a,...a, are any n quantities real or imaginary; C(a, a... a,) denotes, 
according to SYLVESTER’s notation, the product of the squares of all possible 
differences of a, a....a,; and 2 denotes summation with reference to all 
possible groups 7 at a time of the 7 quantities. 
It is obvious that, if7 >, then S,v)=0; and ifr=n, 
Sn(z)=(—1)" | 1 Lume ak 
Ss Sp+1 Sp+2 «+ + Sptn 
Spt+n—1 Spin Sptn+1 + « « Sp+2n-1 
=(ayay. . . An)?&(ay. . « An)(X—a,)(H@—ay) . . . (W—an) 
= (aay «08 an)’ Kay aes an) (a" + pa"! + pv"? + m Te) +),) 5 
