1 54 Proceedings of Royal Society of Edinburgh . [sess. 
The first matter of interest is the expansion of as a sum of 
products of a K , a K _ 1 , . . . , a n , e.g ., 
N lt8 = 1 4- {a 1 + a 2 + a 2 ) + a x a s . 
This is written in the form 
1 + a Kn + a Kn = ... . 
where, he says, u af n die Sunime alter moglichen (als Producte 
aufgefassten) Comhinationscomplexionen ohne Wiederholung be- 
deutet, welche sich aus a Ki a* +1 , . . . , a n so zu je r Elementen 
bilden lassen, dass nicht zwei neben einander stehende Elemente 
a s , a s+1 dieser Reihe in den einzelnen Producten zugleich vor- 
kommen.” By way of proof it is pointed out (1) that the term 
independent of all the a’s is 
1 1 
0 1 1 
0 1 1 
0 1 
i.e. + 1 ; 
(2) that the cofactor* of ( - a r )( - a s )( — a t ) .... when two of 
the a’s are consecutive is 
1 1 
0 1 1 
0 1 1 
0 1 1 
0 1 1 
0 1 
* To obtain the cofactor of the product of a number of a set of elements in 
a determinant Worpitzky puts a 1 in the determinant in place of each element 
occurring in the said product, 0’s in all the other places of the rows to which 
these elements belong, and 0’s for all the other elements of the set. 
