1890 - 91 .] Dr T. Muir on some unproved Theorems. 
81 
The condition for the existence of the converse theorem is, evi- 
dently, 
l^iAVsl = 0 • 
It is most important to notice that there is no reason for restrict- 
ing the multiplier in the preceding theorem to the form 
\ ^2 \s 
Mi M 2 Ms 
V 1 v 2 *3 
The method of proof which we have used shows that the multiplier 
might be any determinant whatever of the 4th order. This puts us 
in the position of being able to combine Cayley’s two analogous 
theorems into one, as follows : — If an array consisting of r rows and 
n columns (r<n) be such that all the determinants of the r th order 
formed from it vanish , then the multiplication of the array row-wise 
by a determinant of the n th order or column-wise by a determinant 
of the r th order produces a similar array each of whose determinants 
of the r th order will also vanish. 
Or, if 
0, 
then also 
and 
11 
12 
13 • • 
■ • In 
21 
22 
23 • • 
• '2n 
rl 
r2 
r3 • 
• • rn 
11 
12 
13 • • 
• In 
21 
22 
23 • 
■ • 2 n 
rl 
r2 
r3 • • 
• rn 
11 
12 
13 • • 
• • 1 n 
21 
22 
23 • ■ 
• • 2 n 
rl 
r2 
r3 • 
• • rn 
Kl«22 * ’ ' w J = 0 , 
Kl W 22 * * ‘ W rr\ “ 0 . 
VOL. XVIII. 
1 / 4/91 
