1874.] Investigation of Determinants. 217 
but are more readily obtained in a general manner by 
symbolic work, thus— 
[E, J ot [Ea = [En—m41) “Em1+ [En—m] -Em—1 by Eq. (14). 
= [Exn-m]. En-1 { [E] +1} by Eq. (5). 
=[(F,2nleBm— [E,] bysEq. (no. 7,.and x4). 
San - noes = Bigerl-Epmta— Pen denen DY G5). 
7 [Em-—xz] i j—-m.(E —1) by Eq. Cae 
= [Ey] En-m= (Bey by Eq. (Io, 5» & I5)- 
V. Addendum to Problems (II1.) and (IV.) 
The expressions (11) and (16) for [E,] and [E/~”] in 
terms of E obtained by a symbolic inversion of the formule 
(4) and (13) previously obtained for E, and [E”’] in terms of 
[E,] may also be obtained by algebraic inversion of the 
same formule, but the process is very tedious. They may 
' also be obtained directly from the properties of determinants 
by establishing expressions for [A] and [A”~”] in terms of A 
inverse to the expressions (3) and (12) used in the text. 
The required expressions are easily seen to be— 
dA d2A 
[A] =A- %{49p-aa,,} + 3 {496 Va0- Ga aay | 
ep 
dA 
pda gg Aa,} t 
Pe es eae (= 1)%. ads, 4. = Gan)» (TQ) 
a | =a-3 {andes} +3 ep | 
{4 pp%aqrr- da 
pp 4% aq 
dma 
) 
dayx. ddzg. da33.-- damm | (20). 
ae +(- 1)". 5 { AirF2a%33 ©» Ainme 
Equations (11) and (16) may be derived from Eq. (19) and 
(20) respectively by considerations precisely similar to those 
used in obtaining Eq. (4) and (13) from Eq. (3) and (12) 
respectively in the text. 
The relations established in Problems (III.) and (IV.) 
between E,, [E,], and [E;,] are evidently derived in a 
manner which shows them to be generally true of all determi- 
nants whatever which are related to one another similarly to 
those styled A, [A], [A”], @.¢., differing only in their leading 
constituents (those of A being all finite, those of [A] all zero, 
those. of [A”] being m finite, and the rest zero). 
