of the Binomial Theorem. 
3* 1 
n — qr 
tiplicand be denoted by z J x r , the term of the multiplier 
l ~q'- 
immediately under will be expressed by z ? x ' , according to 
the nature of the two series ; and upon multiplying the first 
term of the multiplicand by this term of the multiplier, the pro- 
duct will be z*x r , which is equal to that term of the 
multiplicand immediately over that in the multiplier, after 1 
is added to the numerator of the exponent of x. And the other 
terms in the multiplicand, successively to the right hand, being 
multiplied by the same term of the multiplier, the terms will 
n-f-i — qr — r n 1 — q r — i r »-|-i — qr — 3*- 
be z? +1 x r , z*+* x 7 , 2?+3.r " 7 ", 
&c. in arithmetical progression, which are equal to those terms 
of the multiplicand immediately over them, after the numera- 
tors of the exponents of x are increased by 1. And from hence 
a general rule is obtained for the arrangement of any hori- 
zontal line of products. For when the first term in the mul- 
tiplicand is multiplied by a term in the multiplier, the product 
is placed immediately under that term of the multiplier ; and 
the products which arise from multiplying the other terms of 
the multiplicand, successively towards the right, by the same 
term of the multiplier, are placed successively towards the 
right of the first mentioned product. 
15. The several products, therefore, arranged under one 
another in a perpendicular line, arise in the following 
manner. The first arises from multiplying the term in the 
multiplicand directly over it into the first term in the multi- 
” + » — 3? 
plier. Thus * x — x n -=-±L z 3 x r is the product of - x 
r zr $r r 
Ss 2 
