Page:Optics.djvu/137

 glass, $CD$ of the mirror (sections, of course, as usual), $QR$, $QR′$ are two rays of an oblique pencil, of which the former is incident on the glass at $R$, refracted to $S$, reflected to $T$, and refracted towards $V$; the other in like manner. Now let $QR$, $TV$ meet in $s$: the refractions at $R$ and $T$ being equal, the effect on the whole is the same as if a reflexion took place at a mirror parallel to $CD$, placed at $s$; in like manner $QR′$, is as it were, reflected at $s′$. So far all is regular enough, but are $s$, $s′$ equidistant from the mirror $CD$? because if not, the case is not the simple one of a reflexion at $s$, $s′$, instead of one at $S$, $S′$. Now strictly speaking $Ss$, $S′s′$ are not equal, for it will be seen by referring to p. 47, that there is an aberration in these refractions, of the amount of $m^{2}−1⁄2n·∆·tanθ^{2}$, $∆$ being the thickness of the glass, and $m$ the ratio of refraction out of glass into air, which being less than unity, the aberration is negative, and as it increases with $θ$, the angle of incidence, $s′$ must be nearer the surface than $s$, and the rays will not diverge from $q′$, as if reflected at a plane mirror at $s$, but will form a caustic to which $q′T$, $q″T′$ will be tangents. Hence, the image of any object situated obliquely, with respect to the mirror and the eye, will be more or less confused, besides being situated in a different place from that which it would occupy, in the case of a plain metallic speculum without glass.

There is another irregularity attending these looking-glasses, which is easily perceived, when the incidence of the rays is oblique; it is an additional reflexion at the first surface of the glass. Indeed, when the obliquity is great, there may be observed a series of reflexions, at the two surfaces of the glass, each of the alternate ones attended with an emergence of part of the rays, as shown in Fig. 155, where $QR$ represents a pencil of rays, partly reflected, and partly refracted at $R$; the refracted rays are reflected at $S$, and in part emerge at $T$, in part are reflected at $T$, and at $V$, and meeting the upper surface again at $X$ are divided, some escaping in the direction $Xx$, some suffering another reflexion, and so on.

When the image of a candle is seen in a common looking-glass, the following phænomena may be observed, as the obliquity of the incidence is increased: at first one image is seen, then two, then several in a row, apparently decreasing in magnitude and brightness