of the Binomial Theorem. 311 
the second term of any equation is made up of members, each 
of which contains only one of the quantities a, b, c, &c. and 
the whole coefficient of the second term is the sum of all 
these members, or the sum of all the quantities a , b, c, &c. 
used in the multiplication by which the equation, under consi- 
deration, was produced. Thus in the equation of four dimen- 
sions, the whole coefficient of the second term is a+b+c+d, 
and a, b, c, d, were used in the multiplication in obtaining the 
equation. The coefficient of the third term, of any equation, 
is made up of members, each of which contains two of the 
quantities a , b, c, &c. used in the multiplication in obtaining 
the equation. Thus in the equation of four dimensions, the 
whole coefficient of the third term is ab + ac -j- be -f- ad -f- bd 
-J- cd. And indeed, not only from inspection, but also from 
considering the manner in which the equations are generated, 
it is evident that each member of any coefficient has as many 
of the quantities a , b, c, &c. in it, as there are terms in the 
equation preceding the term to which the coefficient belongs. 
Thus each member of the coefficient in the second term of 
any equation is one quantity only, and only one term precedes 
the second term. Each member of the coefficient in the third 
term, of any equation, consists of two quantities, and two 
terms precede the third, &c. 
7. When any equation is multiplied in order to produce the 
equation next above it, it is evident that the multiplication by 
x produces a part in the equation to be obtained, which has 
the same coefficients as the equation multiplied. Thus, mul- 
tiplying the equation of three dimensions by x we obtain that 
part of the equation of four dimensions which has the same 
mdcccvl S s 
