216 Investigation of Determinants. (April, 
* (E*) = [En] +”Cx. [En-1] +C2. [En—2] +. $C, [En-2] + 
Pe og ape Ie bore mal + per Pa 
um=2) + PByeg Pe +S [En-r) + 
am er ea = Lees a” he's (13). 
Next, changing every term [E,] into its symbolic equiva- 
lent [E}?, see Eq. (5), [E]”-” is seen to be a factor in every 
term of series (13), which may, therefore, be symbolically 
expressed— 
=[E,] +--[E,-1+7— 
Rae = (aes . { [E] ep , (14). 
= (Eel . Eni (See Eq. 6). 
Hence also, by changing m into (1—m)— 
PB TD Ul Oe 0 Ga 
= [Bml. Ey (See grb): 
Again changing [E] into its equivalent (E—1), by Eq. (10), 
and interpreting the result by Eq. (8). 
[Ee] Lee Dee, 
=E,— 2 By S B, a. . +(—1). ap En-rt 
: ee cae n—m= ° (16). 
Formule (13) and (16) furnish the means of calculating 
[E;'] or (E,”] in terms of [E] or E respectively. The 
former is preferable if m <2, and the latter if m > +2. 
The following particular instances of these expansions 
may be recorded for reference :— 
[Ee] — [En] cir [En- 1} 3 ? 
[EA] =E,—En-,: 
[E”,] = [E,) Te [En-:] is [En,-al 3 
Boa = En — 2E na i En-2 
[E,,] = (E,] +3 [En-1] +3 [En-2] + [En-s]; 
a =En—3Es-1+3En-2- | 3] 
The following relations between successive values of [E, ] 
and also [E’’"”] might be obtained by induction from the 
above equations, viz., that— 
[E] = (EY) + (ee 
(18) 
(17). 
ag = (Ea ni £: [Br"4 
n n—-t 
J 
