Page:Scientific Memoirs, Vol. 2 (1841).djvu/425

$BC:F'X'=\lambda ':x'$,

then

$-X'Y'={\frac {A}{L}}(\lambda +x')-a$.

Thirdly, since $CD=KK'$ and $F''X''$ is equal to the part of $KK'$ which extends from $K$ to the line $X''Y''$, we have

$CD:LK'=F''X'':X''Y''-KF''$,

whence

$X''Y''={\frac {LK'\cdot F''X''}{CD}}+KF''$,

or, since $KF''=KI+IH'-F'H$ and $F'H=GH-GF'$,

$X''Y''={\frac {LK'\cdot F''X''}{CD}}+IH'+GF'-(a+a')$.

If now for $LK'$, $IH'$, $GF'$ we substitute their values

${\frac {A\cdot \lambda ''}{L}},\quad {\frac {A\cdot \lambda '}{L}},\quad {\frac {A\cdot \lambda }{L}},\quad$we obtain

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

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

$CD:F''X''=\lambda '':x''$

we have

$X''Y''={\frac {A}{L}}(\lambda +\lambda '+x'')-(a+a')$.

These values of the ordinates, belonging to the three distinct parts of the circuit and different in form from each other, may be reduced as follows to a common expression. For if $F$ is taken as the origin of the abscissæ, $FX$ will be the abscissa corresponding to the ordinate $XY$ which belongs to the homogeneous part $AB$ of the ring, and $x$ will represent the length corresponding to this abscissa in the reduced proportion of $AB:\lambda$. In like manner $FX'$ is the abscissa corresponding to the ordinate $X'Y'$ which is composed of the parts $FF'$ and $F'X'$ belonging to the homogeneous portions of the ring, and $\lambda$, $x'$ are the lengths reduced in the proportions of $AB:\lambda$ and $BC:\lambda '$ corresponding to these parts. Lastly $FX''$ is the abscissa corresponding to the ordinate $X''Y''$, which is composed of the parts $FF'$, $F'F''$, $F'X''$ belonging to the homogeneous portions of the ring, and $\lambda$, $\lambda '$, $x''$ are the lengths reduced in the proportions of $AB:\lambda$, $BC:\lambda '$, $CD:\lambda ''$. If in consequence of this consideration we call the values $x$, $\lambda +x'$, $\lambda +\lambda '+x''$