218 Investigation of Determinants. (April, 
VI. Application to Ordinary Determinants. 
Substituting Es = | 4, (Eq. 2) into Eq. (11), there results— 
J = [#-{1-7e+75—Te+ -- wah 
(—a)t* heh 
be Mage ae [u—1 + iz (21) 
A relation between three successive values of [E,] may 
be thus found by Eq. (21)— 
[toa ae [En—al = [m1 (Qc x + |n-2 n—2. x 2 
=\"—I. ae +{@—-n-41}. ln — noes 
*(1—1).{ [En—s] + [En—2l } =(—2)*"*-@—-)+ is = 
=|n. (= ne (- 1)" +S%pie 2 : 
| ~—1 IE 
= |." (=1)* 
| 
= [E,],.. by Eq: (20)... teem 
This relation (22) between three successive values of [E,] 
may also be thus obtained directly :— 
Since the leading constituents of [A] are all zero (by 
definition)— 
[A] =3. Bay cal Pad Glee a . (23). 
in which equation y,z take all positive integral values (except 
d{A]. 
equal values) fromito”. Now ata) is easily seen to be a 
‘yz 
determinant of (n—1) rows containing (”—2) of the leading 
(evanescent) constituents of the original determinant [A], 
and no two of these in the same row or column. ‘These 
(n—2) constituents may by Bees of order of rows or 
columns of the determinant ie be all brought into its 
j2 
leading diagonal without altering its mwmerical value (the 
only alteration being of sign). It follows that the number ~ 
A 
of elements in os a4 is the same as in a determinant of (7—1) 
rows with only one finite sa constituent, which number 
(by notation of Problem IV.) is [E’,-:]. 
