36 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The particular matter to be investigated is the circumstances under which 
the determinant vanishes. It is closely connected with an important 
question in geometry, which has recently been engaging the attention of 
Professor Hayashi of Sendai. 
(2) Unfortunately, from the examination of this first case very little 
can be learned regarding the higher cases, there being no conditions 
attached to it at all for evanescence. We only need indicate, therefore, 
that multiplication of the determinant by 
1 c cf 
- 1 -a -ad 
1 e eb 
in column-by-column fashion gives a product which vanishes by reason 
of being skew and zero-axial, and that in other less compact ways a 
similar result can be obtained.* 
(3) If -We denote, the 4(4 — 1) variables in the next case by 
a, b, c, d, e,f, g, h, i,j , k, l, 
the twelve rows thence constructed are 
1 a + b + c ab + ac + bc abc 
1 b + c + d be + bd + cd bed 
1 l -j- a -4- b la -f- lb -1- ab lab , 
the arrays are 
j 1 
a -p b + c 
. . abc 
1 
b-\-c + d . . 
, . bed 
1 
e+f+g . 
■■ »f 9 
1 
f+g+h . 
■ ■ fgh 
1 
i +j + k . 
■ ■ \i k 
5 
1 
j + k + 1 
. . jU 
1 
c + d + e . . 
, . ede 
1 
d + e +/ . 
. . def 
1 
g + h + i . . 
. ghi 
1 
h + i +j . , 
. . hij 
1 
Jc + l -{-a 
kla 
5 
1 
l+a+b . , 
. . ■ lab 
and only want of space prevents the immediate visualising of the de- 
terminant, (y) say, to be dealt with. All, however, that remains to be 
mentally pictured is that it has for its (r, s) th element the minor got 
from the r th array by deleting the s th column. 
(4) At the outset it is of the utmost importance to note, if for no other 
* From a purely algebraical point of view the following generalisation is a more 
interesting result : The determinant of the primary minors of the arrays 
II 1 a + b ab | l| 1 b + c | [ | 1 c + d cd [I 
|| 1 d + e de | , !| 1 e+f ef || , | 1 f+g fg || 
is equal to 
(a-g).(b- d)(c - e)(d -f){e - b)(f- c). 
