1874.| Investigation of Determinants. 213 
4. An important variety* of the above classes of determi- 
nants is that in which the constituents of the leading diagonal 
are all zero, in which case the above numbers are to be all 
reduced by 7: thus the number of constituents will be :— 
(x). In an ordinary determinant whose leading con- 
stituents vanish, (”?—7). 
(2). In a symmetric determinant whose leading con- 
stituents vanish, $n(7—1). 
(3). In a skew determinant whose leading constituents 
vanish (this is styled a skew symmetric determi- 
nantt), (”?—m) different constituents, but only 
3n(n—1) of different magnitude. 
II. Number of Elements in an Ordinary Determinant. 
Let E, be the number of elements in an ordinary deter- 
minant of ” rows. 
Then a determinant of 7 rows oe be expressed i general 
as a sum of different terms, viz., A= 3°" (a>,y. As,y), where 
A,,, is the first minor of A STON ae to ay, and is 
therefore itself a determinant of (n—1) rows, and contains 
(by above notation) E,_, different elements! Moreover, 
these E,,_; elements of each term of type (ap,y. Ap) in the 
whole sum, which together make up A, are in general different 
from all the elements in every other such term. 
Hence, E,=1. Ex-:- 
Similarly, E,-;=(#—1). E,-., and so on. 
Hence, E,=2. E,-,="(n—1). Ey,-2= 
| 2 
=—~———" En-+ (x): 
[n—r 
= |. E, 
SST CE BoA pe ete een ey ie eee me) 
Since it is obvious that E, »=1, E,=1. 
III. Number of Elements in an Ordinary Determinant whose 
Leading Constituents Vanish. 
dA , —_ A be Bila MAP AO, ? & 
A, dappy dapp.dagg dapp. dagg. day Cos we 
represent an ordinary determinant, and its successive first, 
second, third, &c., leading minors, the leading constituents 
(which are a type asp) being finite. 
lees F lteg aac! leaeta ae: |e 
Ts da pp da pp. da gq da pp. da gq. da yy C. « 
» See note on preceding page. 
+ SALMON, Art. 40. 
VOL. IV. (N.S.) aE 
