of the Binomial Theorem . 313 
does not enter has the same members as the coefficients of 
the equation immediately before the last, by the 7th article ; 
and when the members of the first part are multiplied by the 
last used quantity, the product becomes the second part of 
the whole coefficient above mentioned. 
Thus the first part of the cubic equation, by the 7th article 
is, x -abx, and as these coefficients are the same as 
the coefficients in the quadratic equation, being multiplied by 
c , and arranged according to the 8th article, we have the co» 
-j -abc. 
efficients of the second part of the cubic, viz. c-f-ac 
4 -be 
Hence it is evident, that there are as many members in any 
coefficient, which have the last used quantity in them, as 
there are members in the coefficient preceding, which have 
not the same quantity. Thus in the 3d term, in the equation 
of four dimensions, there are three members of the whole 
coefficient of x % which have d in them, viz. ad , bd, cd, and 
there are three members of the whole coefficient of x 3 in the 
second term, which have not d in them, viz. a, b, c. In the 
fourth term of the same equation, there are three members of 
the whole coefficient of x , which have d in them, viz. abd, 
acd , bed, and there are three members of the whole coefficient 
of x * in the third term which have not d in them, viz. ab, ac, 
be. Now 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. 
10. From the last article the number of members in the 
several coefficients of any equation may be determined. For 
S s 2 
