1874.] Investigation of Determinants. 221 
Thus.A, [A], [A”], E,,[E,],[E”] become— 
6, [8] , (8"], ¢,, [e,], [1 - 
All the relations between E,,[E,],[E),] established in 
Problems (III.) and (IV.) obtain also between ¢,, [e,], [e, J, 
having been established in a general manner, as properties 
common to all determinants. 
The number of elements [e,] in the symmetric deter- 
minant [6], whose leading constituents vanish, will first be 
investigated. 
Number of Elements in a Symmetric Determinant whose Leading 
Constituents are Zero. 
In the ordinary determinant [A] whose leading constituents 
vanish, the elements may be divided into two classes :— 
(z), Of type {(apq.@ 4). (Grsasr,) - +» (@yz4zy)} Consisting 
of the product of +2 pairs of conjugate constituents (such 
aS (dyza:,) only. This number has been investigated in 
Problem (VII.), and denoted by S,.. 
(2). Of type— 
{(Apq.%qp)-(ArsMsy) « « « (yz. Azy) | X {Aye Agn.ani. » + + Amz}, 
consisting of the product of + pairs of conjugate constituents 
(such as ayz. @zy), and (1 — 27) other constituents (containing 
no conjugate pair; that is to say, consisting in part at least 
of the product of constituents containing o conjugate pair. 
The number of this type is clearly {[{E,]—S,},[E,] 
being (by the notation) the whole number of elements in the 
determinant [A]. 
In the corresponding symmetric determinant [6], in which 
(by definition) a,,=a.,, these classes become— 
Bet type: {(ap7.Grs:. «.- + . @y;}*, each element being 
the square of the product of +2 constituents, without 
repetition of any suffix. Also the number of these is clearly 
the same, viz., S, as in the corresponding ordinary deter- 
minant [A]. 
(2). Of type{as.a,s. . . . Ay, | x {Ajg.dghAni. + + + Ang} 
consisting of the product of the square of the product of 7 
constituents, without repetition of any suffix, and the product 
of (n—z2r), other constituents containing no conjugate pair, 
which therefore involves the remaining (n—2r) suffixes in a 
cyclic change, 7.¢., each occurring twice. 
Now in.the ordinary determinant [A], elements of this 
type (2) occur in pairs, 7.e., for every product of a particular 
set of conjugate constituents, as {(a pq Aap) -(ArsQsy,) + (ays, Gxy)}, 
VOL. IV. (N.S.) 2F 
