232 
Transactions of the jRoyal Society of South Africa. 
(7) If five letters are to be used, only one of them, / say, must occur 
twice ; its companions must be two, d and e say, of the remaining five ; and 
the third row must have two, h and c say, of the remaining three. The 
factorisation of such an auxiliarv is 
1 e,fe 1 
1-^4/6 1 
i ^4/5 1 
1 dj, 1 
1 ^4/0 1 
1 ^^4/5 1 
= 1 ^465/6 i 
1 ^4^5/5 1 
1 ^5^6 1 
1 ^4^6 1 
1 ^4^5 1 
from which we obtain 
1 ^1^2^./^^ 1 
1 ^1^2^'.'}^^5 1 
1 '^l^-2^:^^6 1 
1 (''lh<^-A^4 1 
1 «1^2C3% 1 
1 ^1^2^3^6 1 
= I ^1^2^;^ 1 
• 1 aid^e.^ 
1 ^1^2^3/4 1 
1 ^^1^^268/5 1 
i (^i^z^sfc. 1 
the full number of the type being 
Ce,i X C5.0 X C3,2, i-e. 180. 
(8) When all the letters are used the type of auxiliary equality is 
1 eJe 1 
1 64/6 1 
1 ejr, 1 
1 ^5^6 1 
1 1 
1 i 
1 «5^6 1 
1 1 
1 ^^'4^5 1 
whence we have 
1 a-jD.yp^d^ 1 
! Hh^^-A 1 
1 a^^cd^ 1 
1 Hhejr, 1 
1 ^^1^2^3/6 1 
i ^1^2^3/4 1 
1 ^A^?if^ 1 
1 ^1^^063/6 1 
1 ^4^5/6 1 
'h^Je 1 1 
1 ^h^G 1 1 
(^AA 1 1 ' 
_ 1 1 «'1^2^3 
1 ^1^2^3 1 1 
1 ! ^h^2f\ 
1 ^1^2/^ 1 1 
'6' 
the resolvability in this case being only partial, 
is readily seen to be 15. 
The number of instances 
(9) Summarising the results of this census of the resolvability of 455 
3-line minors of the 4th compound of 
aJ)oC.^ 
0, 
a-fi.jC.^ 
ajhc^c^ 
20 of the type 
60 „ 
180 „ 
180 „ 
15 ,, ,, 
' I ci^As I 
we have- 
As, 
1 ^'1^2^3 1 
1 ^1^2^3 1 
1 ^1^2/3 1 
1 ^1^2/^ 1 
In regard to the second of the six groups it must be carefully noted that 
the 0 (zero) given as the result is not O'Ae? the truth being that the process 
of elimination by instalments here fails, and of course properly so, because 
the 0 arises from the fact of it being possible to choose from our set of 
15 equations in x, y, z three equations that are not mutually independent 
— a possibility that in our original statement in § 2 was only hinted at. 
Further, it is worth noting that these determinants of zero value bear an 
external mark that fully warns us against accepting them as multiples of 
