220 Investigation of Determinants. (April, 
either of which involves the other, and each of which is the 
product of +2 constituents involving all the suffixes without 
repetition. Hence the number of such elements is the same 
as the number of ways in which a product of +2 con- 
stituents can be formed involving all the n suffixes without 
repetition. 
Let S, be the number of produé¢ts containing suffixes 
without repetition. ; 
Now the number of constituents containing any particular 
suffix p is clearly (7—1), for these constituents are of type 
A»sy, Where y has every value from 1 to , excepting p. Also 
for every such constituent as as, there are (7—2) suffixes 
a: a n 
remaining to form the remaining (4-1) constituents 
required to form the complete product of +2 constituents. 
Further, these (7—2) suffixes can be arranged into the 
required product of (7-1) constituents of the requisite 
type in S,-2 ways (by preceding notation). 
Hence 8S, =(”—1). Sy_2. 
Similarly S,-2=(”—3).S,-,, and so on. 
Hence S, = (n—1).S,-2= ("—1).(u—3)Sn-4= «se 
= (n—1).(n—3).(u—5) . . » 7.5-3.92 + - | (26) 
= ("=—1).(7—3).(u—5) . « 27.543.L. nee 
since obviously when n=2, there is only one pair of con- 
jugate constituents (viz., a,2,42;), so that S,=1 
_S _ n(n—1)(n—2)(n—3)(n—4) WH nice Mon ee 
x "y nm.  (n—2) (m—4). 5 = sie 3 6.4.2 
n : : - 
=——. , where » is an even integer. > + (27). 
n\n 
22.)5 
It is obvious that, if 2 be an odd number, the proposed 
product of n+2 pairs of conjugate constituents could not be 
formed, z.e. that 10 elements exist of type proposed. 
-, S, =0, when # is an odd integer ss’. Sy ee (28). 
A few values of S, may be recorded for reference, and note 
that S, is always an odd integer (when » is even), being 
itself the product of odd integers, Eq. (26). 
So=1, S2=1, S,=3, Ss=15, Sg=105, Sr= 945 - « » (20) 
VIII. Number of Elements in a Symmetric Determinant. 
Modify the preceding notation in capital letters for ordinary 
determinants by using the corresponding small letters with 
like meanings for symmetric determinants. 
