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same sign as $\varphi$, the fringes appear in front of the mirrors; if $i$ has the opposite sign, the fringes appear behind the mirrors.

To find the form of the curves as viewed by the eye at $E$, let $SE=D$; call $T$ the distances between the surfaces at $E'$, the projection of $E$. From $P$ draw $PR$ parallel to $DP'$, and $RS$ at right angles, and let $RS=c$. We have then $t_{0}=T+c\tan \varphi$, whence, substituting for $c$ its value $D\tan i$,

$\Delta =2{\frac {T+(D+P)\tan \varphi \tan i}{\sqrt {1+\tan ^{2}i+\tan ^{2}\theta }}}$

(5)

If on a plane perpendicular to $EE'$ at distance $D$ from $E$, we call $x$ distances parallel to $P'C$, and $y$ distances parallel to $P'D$, reckoned from the projection of $E$ on this plane, then, putting $\varphi =K$ and $D+P=S$, we have for the equation to the curves, as they would appear on this surface to an eye at $E$,

$\Delta =2{\frac {DT+SKx}{\sqrt {D^{2}+x^{2}+y^{2}}}}$

or

$\Delta ^{2}y^{2}=\left(4S^{2}K^{2}-\Delta ^{2}\right)x^{2}+8TSKDx+\left(4T^{2}-\Delta ^{2}\right)D^{2}$

(6)

If, numerically,

$\Delta <2SK$	the curve is a hyperbola,
$\Delta =2SK$	the curve is a parabola,
$\Delta >2SK$	the curve is an ellipse,
$K=0$	the curve is a circle,
$\Delta =0$	the curve is a straight line,

All the deductions from equations (4) and (6) have been approximately verified by experiment.

It is to be observed that in the most important case, and that most likely to occur in practice, namely, in the case of the central fringe in white light, we have $\Delta =0$, and therefore also $t_{0}=0$; and in this case the central fringe is a straight line formed on the surface of the mirrors. Practically, however, it is impossible to obtain a perfectly straight line, for the surface of the mirrors is never perfect.

It is to be noticed that the central fringe is black, for one of the pencils has experienced an external, the other an internal reflection from the surface $b$, fig. 1. This will not however be true unless the plate $g$ (which is employed to compensate the effect of the plate $b$) is of exactly the same thickness as $b$, and placed parallel with $b$. When these conditions are not fulfilled, the true result is masked by the effect of "achromatism" investigated by Cornu (Comptes Rendus, vol. xciii, Nov. 21st, 1881). This remark leads naturally to the investigation of the effect of a plate of glass with plane parallel surfaces, interposed in the path of one of the pencils.