1874.] Investigation of Determinants. 223 
values of ” by formula (32), from the known values of [e;], 
fe], &c., or may be directly calculated from formule (31). 
\ 
It will be useful to record a few values of [e,] for reference— 
[20] =I, [e,] =o[e,] =1, [e,] =, [e,] =6, [es] =22, 
[és] =140, [e,] =927, [es] = 7469, [e,] =66,748,+..- - (33). 
[ero] = 667,953 
IX. Numbers of Elements in a Symmetric Determinant. 
This may easily be calculated from the known number 
[én] of elements in the symmetric determinant [6], whose 
leading constituents are zero, by help of the relation between 
eand [e] (see Eq. 6). 
Thus é, =(1+ [e] )” 
_— id 
=1+ = [e] + enor eee Eel 
+ = [e.-1] + [én] +--+ (34)- 
It will be useful to record a few values of e, for reference, 
thus— 
=I, =I, &,=2, 6,=5, e,=17, €;= 73, &=398, ) 
¢, = 2636, €g= 20,542, €s= 182,750, €,,-=1,819,148 J 
~ » (35). 
X. Number of Elements in a Symmetrical Determinant, out of 
whose Leading Constituents only m are Fimte. 
This may be easily calculated, either directly from the 
relations (13) and (16), or for successive values of ~ from 
the relations (18) which have established for all deter- 
minants. Thus changing E to e— 
ie fe, | Fd 
m 
ag | 7. | m—r* -[én-r] +. +7. [En — m+xl aa [en pleas (36). 
m( m— 
— (J2)5 hm 
=€y SN ee eee spe atee+ [7 Tr [mar -€n-rt - 
m 
+(-1)"- hs » Cn- m+r+(—I)" Cn—m 2(37)- 
(P) = 61 + ME), and (5) = fe) — IT. . G8). 
On account of the great use of symmetric determinants 
in modern geometry, it will be useful to record the values 
of [e"] in a few cases. Thus, observing that fe. | = (eo); 
and that [e” ] = [e ], (by notation) — 
