322 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Denoting it by (0, 1, 2, ... , n), since it is a function of the ^(^+1) 
quantities, 
(01) (02) . . . (On) 
(12) ... (In) 
(n- 1 ,n), 
he first shows with some prolixity that the cofactor of (01) in it is 
(0 + 1, 2, 3, ... , n), next that the cofactor of (01) (02) . . . (0 i) is 
(0 + 1 + ... +i , i + 1 , i + 2 , . . . , n) , 
and finally that 
(0,1,2,.:.,*) = 2( 01 Xb2, ...,») 
+ 2(°l)(0 2 )( T +2- 3 ,...,») 
+ 
+ Z( 01 )(° 2 )- • • (0*)(i + 2+ . . . +*,*+1, . . . ,n) 
+ 
+ (01(02) . . . (0 *). 
Resuming consideration, but proceeding on a different tack, he arrives at 
Sylvester’s “rule,” namely, that (0, 1, 2, . . . , n) is “gleich der Summe 
aller nicht-cyclischen Producte, die aus je n jener 1) Elemente (i Jc) 
gebildet werden konnen.” Unlike Sylvester, however, he is careful to give 
a justification of it based on four observed facts, namely, (1) that 
(0, 1, 2, . . . , n) is unaltered by interchanging any two of the umbrae ; (2) 
that the coefficient of the term (01)(02) (0^) is 1 ; (3) that none of the 
terms is free of the umbra 0 ; (4) that, as already mentioned, the cofactor of 
(01) is (0 + 1, 2, ... , n)* As the proof, which extends to two pages 
(pp. 119-120), applies only to the case of axisymmetry, it need not be given. 
Lastly, the number of terms in the development of (0, 1, 2, . . . , n) is 
investigated, the result obtained agreeing with Sylvester’s. 
We may note for ourselves in passing that the first three of the basic 
facts of the proof are, like the last, most readily appreciated by observing 
the determinant form, the case where n — 3, namely, 
10 + 12 + 13 -12 -13 
-21 20 + 21 + 23 - 23 
-31 -32 30 + 31 + 32 
being amply sufficient. Thus, increasing any column by all the others, and 
thereafter increasing the corresponding row by all the other rows, we 
* As (01) occurs only in the element ^ - (11), its cofactor is the primary minor obtained 
by deleting the first row and the first column, and this is seen to be (0+1, 2, . . . , n) by 
definition. 
