140 Proceedings of Royal Society of Edinburgh. [ sess . 
and 
[OjOgOgOj = [^i] [^ 2 ] [^3] [^4] 
- [O J[ Oj[Oj[Oj - [^1] [^2] [ ^3] [^4] ” [OJMO'JC 
+ [O'Jf'O 2 ] [^3] [^ 4 ] + P i][ ^2] P'3] [ ^4] + [^i][^ 2 ][ Q sit ^4] 
The general theorem is described as giving an expression for 
[OjOg . . . Oj in terms of 
py, [oj. • • • > [o^],, [oj 
[O'l] , [O'J , . . . , p'i-i] 
['OJ, . . . , [' 0 *-i], ['Oj 
['o' 2 ], . . . , ['OV-J; 
that is to say, in terms of all the unaltered O’s, all the curtailed 
O’s except the last, all the beheaded O’s except the first, and all 
the “doubly-apocopated” O’s except the first and the last; and it 
is pointed out that the number of products (or terms) in the ex- 
pansion is 2* _1 “ separable into i alternately positive and negative 
groups containing respectively 1 , (t- 1 ), - \)(i - 2 ) , . . . , 
i — 1 , 1 products.” Further, it is noted that “ in every one of the 
above groups forming a product the accents enter in pairs and 
between contiguous factors, it being a condition that if any O 
have an accent on the right the next O must have one on the left, 
and if it have one on the left the preceding O must have an 
accent on the right, and the number of pairs of accents goes on 
increasing in each group from 0 to i - 1 .” 
In a footnote the case where each O has only one element, and 
where, therefore, each singly-accented O becomes 1 , and each 
doubly-accented O vanishes, is stated to be identical with the 
“rule ” 
<h a 2 * ' 
a 
l 
M 
1 
®e a e+l 
+ 2 
1 
• a x a 2 , 
■ • • a i 
a e a e+i ( t'f a f+\ 
