132 
MAJOR P. A. MACMAHON OR A CERTAIN CLASS OF 
corresponding to the positive and negative signs respectively, and further taking th 
signs all positive, let y^y have the value 
1 / Cxy, 
and taking all the signs negative, let the value be 
1 ! Cyx‘ 
We have the solutions 
(tis) yin y^i) — { r ’ 3 
^13 Ol ^34 
(yi3> yin y24) — i ^ ^ n > 
^31 ^41 ^42 
and we may write either 
(“l33 “l4’ “ 24 ) — 
^12^33 ^12^23^34 ^23^34 
U3 
"14 
24 
or 
(“l3’ “l4> “ 24 ) — 
_ I ^12'^23 ^12‘’‘33‘^34 ^23^34 
^31 % ^42 
The undetermined quantities are thus reduced to the three 
“123 “23’ “34- 
Writing for brevity, 
I 
and also 
l“23’ “23“34’ “l2“22 
;“34) — (As’ 
/^24’ 
x+l - 
= A, x+1, 
1’ ^2’ 
X3, is 
either 
Pi 
fOis 
f-Oi* 
h3 
<a 4 
1 
/dl 2 
P-2 
Asfe 
A4 
— ?24 
^24 
^3 
As 
1 
Ps 
ArS'sr 
O 4 
^ 04 . 
1 
P^ 
A4 
^24 
A4 
or the same matrix with the substitution of Cy.^ for 
The redundant form is 
(1 - .qXi) (1 - s„X„) (1 - S 3 X 3 ) (1 - 5 ^XJ 
of a triply infinite character and of two forms. 
