30 6 Mr. Robertson's Demonstration 
coefficient, which have the last used quantity in them, as there 
are members in the coefficient preceding, which have not the 
same quantity; and as it has been proved that each of the 
quantities a , b , c, &c. enters the same number of times into the 
coefficient of the same term, what has here been proved of the 
last used is applicable to each. 
8. From the last article the number of members in the several 
coefficients of any equation may be determined. For if we 
put s = the number of times each quantity is found in a co- 
efficient, n = the number of quantities a, b, c, &c. and p = 
the number of quantities in each member ; then as a is found 
s times in this coefficient, b is found s times in this coefficient, 
&c. the number of quantities in this coefficient, with their re- 
petitions, will be s x n, and as p expresses the number of 
quantities requisite for each member, the number of members 
in the coefficient will be ~. 
P 
9. Using the same notation, we can, by the last two articles, 
calculate the number of members in the next coefficient. For 
as S y expresses the number of members in the abovementioned 
coefficient, and s the number of times each quantity is found 
in it, ^ — s = the number of times each is not found in it. 
P 
By the 6th article, therefore, a will be found — — s times, b will 
be found — — s times, &c. in the next coefficient, and 4 " — as 
P P 
__ s n —psn __ t ^ e num b er 0 f quantities, with their repetitions, 
in it. But as the number of quantities in each member of a 
coefficient is 1 less than the number in each member of the 
coefficient next following, each member of the coefficient 
