Page:Scientific Memoirs, Vol. 3 (1843).djvu/685

the product of two binomials ${\displaystyle \scriptstyle {(a+bx)~(m+nx)}}$, the result will be represented by ${\displaystyle \scriptstyle {am+(an+bm)x+bnx^{2}}}$, in which expression we must first calculate ${\displaystyle \scriptstyle {am}}$, ${\displaystyle \scriptstyle {an}}$, ${\displaystyle \scriptstyle {bm}}$, ${\displaystyle \scriptstyle {bn}}$; then take the sum of ${\displaystyle \scriptstyle {am+bm}}$; and lastly, respectively distribute the coefficients thus obtained, amongst the powers of the variable. In order to reproduce these operations by means of a machine, the latter must therefore possess two distinct sets of powers: first, that of executing numerical calculations; secondly, that of rightly distributing the values so obtained.