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

and

$\lambda +\lambda '+\lambda ''=L$

then according to the law just ascertained

$GF'$	is	a fourth proportional to $L$, $A$ and $\lambda$
$IH'$		a fourth proportional to $L$, $A$ and $\lambda '$
$LK'$		a fourth proportional to $L$, $A$ and $\lambda ''$.

Draw the line $FM$ through $F$ parallel to $AD$, regard this line as the axis of the abscissæ, and erect at any given points $X$, $X'$, $X''$ the ordinates $XY$, $X'Y'$, $X''Y''$, we obtain their respective values, thus:

In the first place we have, since $AB=FF''$

$AB:GF'=FX:XY,$

whence follows:

$XY={\frac {FX\cdot GF'}{AB}}$,

or if we substitute for $GF'$ its value ${\frac {A\cdot \lambda }{L}}$

$XY={\frac {A}{L}}\centerdot {\frac {FX\cdot \lambda }{AB}}$.

If now $x$ represent a line such that

$AB:FX=\lambda :x$,

then

$XY={\frac {A}{L}}\cdot x$.

Secondly, since $BC$ and $F'X'$ are equal to the lines drawn through $I$ and $Y'$ to $GH$ parallel to $AD$

$BC:IH'=F'X':F'H-X'Y'$,

whence

$-X'Y'={\frac {IH'\cdot F'X'}{BC}}-F'H$

or, since $F'H=GH-GF'$

$-X'Y'={\frac {IH\cdot F'X'}{BC}}+GF'-a$.

If now for $IH'$ and $GF'$ we substitute their values ${\frac {A\cdot \lambda '}{L}}$ and ${\frac {A\cdot \lambda }{L}}$, we obtain

$-X'Y'={\frac {A}{L}}\left(\lambda +{\frac {F'X'\cdot \lambda }{BC}}\right)-a$;

and if by $x'$ we represent a line such that