of the Binomial Theorem » 
309 
x+Fx x-\~e y.x-\-d =pqrts = any other arrangement which 
can take place in the quantities. 
4. It is evident that each of the quantities a, b, c, &c. will 
he found the same number of times in the compound product 
arising from x-\-a x x-\-b xifcx x-\-d x x-\-e, &c. For 
this product is equal to pqrst —pqrs x.x-\-e =pqrt x x -d = 
pqst x x-\-c=zprst x x-f ~b=qrst xx-J-tf, by substituting for the 
compound quantities, x-\-a, x-\-b, Sec. their equals p, q, See. 
Wherefore, in the compound product, each of the quantities 
a, b, c, Sec. will be found multiplied into the products of all the 
others. 
5. These things being premised, we may poceed to the 
multiplication of the compound quantities x-f -a, x-\-b, x-\-c, 
Sec. into one another ; and in order to be as clear as possible 
in what follows, let us consider the sum of the quantities, a , 
b, c, Sec. or the sum of any number of them multiplied into 
one another, as coefficients to the several powers of x, which 
arise in the multiplication. By considering products which 
contain the same number of the quantities a , b, c, Sec. as ho- 
mologous, the multiplication will appear as follows, and 
equations of various dimensions will arise, according to the 
powers of x. 
