30 
Transactions of the Boyal Society of South Africa. 
bility. On examination, however, it is found that the elements of the last 
frame-line, namely, 
2m -1 2/n-l 
2 m -I 
1(^1 — a^y (a^ — ac^y^ cC' 
(ao - a^- 
-1 » -2 , • . . , -2m - 1 
may be all changed into 1 if we simultaneously annex 
aYa^...a%m-\ 
to the Pfaffian as a factor ; for example, when m = 2 we have 
1 
On the removal therefore of the common factor a^ao...a2w-i we obtain 
I (ai - ^2)2-1 {a^ - a3)2-i . . . (a^ _ a2,«-i)2'«-i 1 
(^2 — Ouz)-'' 
. (a. - a.2,«-i)''^'"-^ 1 
1 
0 1 
= (_ l)iO«+i)0.+2).(2m-l)i(2m-l)o...(2^»-l),,_i. | 
(3) It should be noted in passing that there is a theorem in determinants 
corresponding to the lemma used in the foregoing, namely, when m is 2, 
(^3 + 60H«3 + ^2)H«3 + ^3)'^3 
^1 ^2 ^3 
the determinant on the left being equal to 
(ao + ?.l)H«2 + i^2)H^2+y 1 
{a,^-\)\a, + h,y{a,+ h,yi 
1 11. 
3 
Sal 
3ai 
1 
1 
3 
3ao 
1 
1 to 
hi ht 
3 
3(x^ 
3ci.^ 
1 
• 
1 6, 
>>; K 
1 
1 . 
and the determinant on the right equal to 
a'^^ 3a^ 3a^ 1 
3 
\ 
a!^ 3a" 30,3 1 
1 61 
bl bl 
1 60 
1 63 
K % 
. 1 
