Page:Scientific Memoirs, Vol. 2 (1841).djvu/424

 412 and then according to the law just ascertained Draw the line $$F M$$ through $$F$$ parallel to $$A D$$, regard this line as the axis of the abscissæ, and erect at any given points $$X$$, $$X'$$, $$X$$ the ordinates $$XY$$, $$X'Y'$$, $$XY''$$, we obtain their respective values, thus:

In the first place we have, since $$A B = F F''$$ whence follows: or if we substitute for $$G F'$$ its value $$\frac$$If now $$x$$ represent a line such that then Secondly, since $$B C$$ and $$F' X'$$ are equal to the lines drawn through $$I$$ and $$Y'$$ to $$G H$$ parallel to $$A D$$whence or, since $$F' H = G H - G F'$$ If now for $$I H'$$ and $$G F'$$ we substitute their values $$\frac$$ and $$\frac$$, we obtain and if by $$x'$$ we represent a line such that