348 Proceedings of Royal Society of Edinburgh. [sess. 
Here the new column is headed by the triple of the sum of the 
numbers in the previous column ; and the rest of the column is got 
by adding the doubles of the adjoining figures. 
Thus it seems that the comparison of several quantities in con- 
tinued proportion leads us to this general conclusion that, if r 
be the n th root of a number or of a rational fraction, and if the 
powers of 
r n ~ l 4 - r n ~ 2, + . . . . + r 2 + r + 1 
be expanded, the numerical coefficients of the several surds are 
approximately proportional inversely to the values of the surds 
themselves : the approximation being closer as the index of the 
power is augmented. 
