1899-1900.] Dr Muir on the Theory of Shew Determinants. 201 
Immediately following this there comes the announcement : — 
“ J’ai trouve recemment une formule analogue pour le dd- 
yeloppement d’un determinant gauche horde, tel que 
a!234 | £1234 - 
Cette formule est : — 
a 1234 | £1234 
a/? 
al 
a2 
a.3 
H 
1/3 
11 
12 
13 
14 
2/3 
21 
22 
23 
24 
3/3 
31 
32 
33 
34 
4(3 
41 
42 
43 
44 
a £ 
■ 11 
22 
•33- 
44 
+ a£ 
• 12 
12 
• 33. 
44 \ 
+ a£ 
• 13 
13 
• 22 • 
44 
+ a £ 
• 14 
14 
• 22* 
33 ( 
+ a£ 
• 23 
23 
• 11 • 
44 
+ a£ 
• 24 
24 
• 11 • 
33 
+ a£ 
• 34 
34 
• 11 • 
22 ' 
+ a£1234 
• 12 
34* 
+ al 
■ (31 
22 
• 33- 
44 \ 
+ a2 
■(32 
11 
• 33 ■ 
441 
+ a3 
■ /33 
11 
• 22 • 
44 r 
+ a4 
■(34 
11 
• 22 • 
33 j 
+ al23 • £123 ■ 44 \ 
+ al24 • £ 1 24 - 33 [ 
+ al34 - £134 * 22 T 
+ a234 • £234 • 11 j. 
Naturally enough it is noted by Cayley that the writing of a = £ = 
5 gives us the less general theorem with which we started ; but he 
does not explain why a third way of arranging the terms of the 
development is adopted. Stranger still, he does not remark on 
the fact that by making 11, 22, 33, 44 all vanish there is obtained 
the identity 
al234 | £1234 = a£1234 • 1234, 
rs— - si', rr= 0 
which is the twin theorem to one given in his previous paper 
A serious misprint in the original is here corrected. 
