The Resolvahility of the Minors of a Compound Determinant. 231 
In the case of three letters the determinant is simply an adjugate. We 
have, for example, 
1 65/0 1 
1 64/6 1 
1 ^4/0 1 
3 
1 ^hfo 1 
1 dj, 1 
- 1 clj, 1 
1 1 
1 d^e^ i 
1 ^^4^5 1 
whence by taking complementaries we deduce the desired factorisation- 
a-fi.yC.^d^ 1 1 
aJ).,Godr-, 1 1 
'2^ A i 
a^h.c.^e^ 1 1 
a^hoc-^e^ 1 i 
ail 
^ is written for 
1 ^A^A^Jc> 
The number of instances of this type of resolvability is manifestly 
Co,3, ie. 20. 
In 
(6) In the case of four letters it is necessary to make a subdivision, 
the first place we have to consider the type 
I 65/6 I I 64/6 I I ^4/5 I 
! (hfe I I 'hfe I I (hA I 
I ^rife I I ^jt/o I I ^4/5 I 
in which the four letters are so distributed that a selected one of the four 
appears in every row. Here the determinant vanishes, and the Law of 
Complementaries consequently gives us 
a^b.^Cod^^ I I a-Jy^c.^d^ \ \ ob^^^c.^dQ 
the number of instances of the type being 
Cg^ i X 4, i. e. 60. 
In the next place we have to consider auxiliaries such as 
0, 
1 \e:J'cA 
1 ^4/6 1 
^4/5 1 j 
i %/6 1 
1 ^4/6 1 
1 ej\ 1 
! dj, 1 
i dj, 1 
1 ^^4/5 1 ! 
1 i 
1 i 
1 C4/5 1 
1 1 ^^5^0 ! 
1 ^A^C. 1 
1 ^4^5 1 1 . 
1 ^^5^6 1 
1 f^4^'6 1 
1 ^4^5 1 ' 
in which each of two letters, e and / say, occurs twice, and the other tv/o, 
c and d say, each once. The factorisation for both of these is the same — a 
fact worthy of separate note— namely, 
^^4^5/^ 
^4^5/6 I 5 
and consequently we have 
I a^:,c.^d^ I I aih.yC.dr^ \ \ a]h:,c./\. 
= ...= I aih.Co\-\aihd.^ 
the number of instances of the type being 
C,;,o X C4., X 2, i.e. 180. 
