200 Proceedings of the Royal Society of Edinburgh, 
is equal to the following product : 
where 
and 
— y y 
^m+\ O'm+2 • • • ^ m+n ^1 * ' * ^m+n 
b m +l ^ot+2 • • • ^ m+n ^1 ^2 * * * ^ m+n 5 
l)(m — 2) ... (m - r + 1) 
M ... (r-1) ’ 
[Sess. 
, _ (m - l)(m -2 ) ... (m - r) „ 
** 1^2 ... r • 
In reference to this one must remark at the outset on the inappropriate- 
ness of the notation 
%x%, • • • ,'0m 
for the q th combination of r integers taken from 1, 2, . . < , m. Manifestly 
J ? ^2 ’ 
q o„ 
though equally awkward, would have been less misleading. Indeed, as 
there is one clear misprint in the enunciation, namely, “ less than n ” for 
“ less than m ” ; and as Sylvester is known to have been inaccurate in the 
correction of proofs, we might suspect 6 m to be a misprint for 0 r , were it 
not that 6m occurs four times in the short passage and 6 r not once.* For q 6 p 
it would have been much more convenient to write pq , which would thus 
have stood for “the p th integer of the q th combination”; and Sylvester's 
theorem might then have been written 
*81 ‘ 
. . a rl 
Um+l 
a m+2 
• • • a m+n 
a 12 
a 22 . . 
. Ct r2 
a m+ 2 • • • 
Um+n 
*21 ‘ 
. . b rl 
^m+1 
b m +2 
“m+n 
^12 
b 22 • • 
. b r2 b m _^i 
^m+2 • • 1 
• b m + n 
a m 
a 2U * • 
. O rp 0 > m + 1 ^m+2 • • 
• &m+n 
^2U * ’ 
• b rfJi b m+ 1 b m+ 2 . . 
• b m j ( . n 
= 
a m+l 
a m + 2 
a m+n 
On— 1, r 
&Y ^2 * 
• • ^ m+n 
C m _i, r— 1 
b m +l 
^m+2 
^m+n 
\ 2 • 
• • b m+n 
5 
am being used as before for C m> r . To help towards clearness let us illustrate 
by means of the case where m = 4, n = S, r = 2. We then have jul = 6 ; each 
set of 6’s equal to a binary combination of the first four integers, that is to 
say, 
^ 2 . 8 V*8. •••> %% 
equal to 
12, 13, 14, 23, 24, 34: 
* e m was actually a misprint. Sylvester himself had to draw attention to it a year later 
in the Cambridge and Dub. Math. Journ ., viii. p. 61. 
