208 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
From these cases of compound determinants of the second class, the 
author passes to those of the 'i th class, but contents himself with stating 
only two results. The first is that the complete determinant of this class, 
denoted as above by 
where 
(n-l)(n-2) ... (n — i + 1) 
1 -2 . 
- 1 
result which agrees with that 
obtained on putting n = 0 in Sylvester’s general theorem of March 1851; 
and the other is that any first minor of A *, say the minor 
1/2, 
2 2 .. 
• 2i ) 
A 
3 2 . . . 3A 
.... (Vl /*2 * • ■ 
■ Ml 
w 
2 2 . . 
2 ./ 
• -i 
A 
3, • • • 3/ 
V 1 P 2 • * ■ 
• IV 1 
f 1, 1 2 ... M fi 2 ... rcV- 1 
K 1 2 ... 1 J (12... n\ . 
He appends, however, the words “and so on,” and tells us that “other 
formulae may be written as required.” 
Two theorems of Sylvester’s are next given, the one being the general 
theorem just alluded to, and the other that contained in the paper of 
16th December 1852. In the case of the former he varies the notation, and 
probably by reason of the above-mentioned serious misprint of an m for an 
r in the original he misses Sylvester’s meaning, and makes an incorrect 
statement. In the case of the other no risk of this kind is incurred, because 
he takes the unusual course of reproducing Sylvester’s words letter for 
letter to the extent of almost two pages (pp. 361-363). The original two 
pages, however, being, as we have seen, not without evidence of Sylvester’s 
carelessness, this course also was unsafe. 
Lastly, he shows how the multiplication-theorem may be deduced from 
Sylvester’s first theorem of March 1851 regarding compound determinants, 
by taking as an example of the latter the identity 
1 . a 
1 . b 
1 a 
. 1 V 
a /3 
a /3 
1 . a 
1 . b 
1 a 
. 1 V 
a /3’ 
a' fi . 
1 . a b 
.lab' 
a P . . 
a ft . 
