goo Mr. Robertson's Demonstration 
in the XLIId Volume. In effecting it he had recourse to the 
doctrine of combinations of quantities, in that part of his in- 
vestigation which relates to the raising of integral powers ; 
and by extending this to the involution of a multinomial, and 
employing an assumed series, he made out the most general 
case, or that in which the exponent is a fraction. In neither 
of the cases, however, in my opinion, is the law of continua- 
tion proved with sufficient perspicuity. In the XLVIIth Vol. 
of the Transactions there is a paper, not expressly on the bi- 
nomial theorem, by the celebrated Mr. Thomas Simpson, in 
which the case for raising integral powers is demonstrated by 
fluxions.^ 
With respect to the following ^demonstration, I submit it to 
your inspection, with the most perfect confidence in your 
judgment and candour ; and if it appears to you not unworthy 
of the attention of the Royal Society, by presenting it to that 
learned body you will add to the favours which you have al- 
ready conferred upon me. 
I am, &c. 
A. ROBERTSON. 
1. The product arising from the multiplication of any num- 
ber of quantities * into one another, continues the same in 
value, in every variation which may be made in the arrange- 
ment of the quantities which compose it. Thus/> x q x r x s= 
pqrs — spqr—psqr=pqsr— any other arrangement of the 
same quantities. 
* When I speak of the multiplication of quantities into one another, I mean the 
multiplication of the numbers into one another which measure those Quantities. 
