[45] The truth of this equation may be easily ascertained as follows (See fig. 27):--Let AP be the mirror, F its principal focus, and PH the line of reflection of the ray FP; then an object at I will be reflected at P to the conjugate focus O, where the screen is supposed to be placed. But by construction, FPI = HPO = POF, and the angle at F being common, the triangle FPI is similar to FPO, and hence FO ∶ PF ∷ PF ∶ FI, and PF² FI = ---; FO and substituting the letters in the text, we get _d_ ∶ _r_ ∷ _r_ ∶ _b_, and _r_² _b_ = ----. _d_ [Illustration: Fig. 27.] [Illustration: Fig. 28.] [Illustration: Fig. 29.] [Illustration: Fig. 30.] [Illustration: Fig. 31.] ~Argand Lamps used in Reflectors.~ The flame generally used in reflectors, is from an Argand fountain-lamp, whose wick is an inch in diameter. Much care is bestowed upon the manufacture of the lamps for the Northern Lighthouses, which sometimes have their burners tipped with silver to prevent wasting by the great heat which is evolved. The burners are also fitted with a sliding apparatus, accurately formed, by which they may be removed from the interior of the mirror at the time of cleaning them, and returned exactly to the same place, and locked by means of a key. This arrangement, which is shewn in figs. 28, 29, and 30, is very important, as it insures the burner always being in the focus, and does not require that the reflector be lifted out of its place every time it is cleaned; so that, when once carefully set and screwed down to the frame, it is never altered. In these figs. _a a a_ represents one of the reflectors, _b_ is the burner, and _c_ a cylindric fountain, which contains 24 ounces of oil. The oil-pipe, the fountain _c_ for supplying oil, and the burner _b_, are connected with the rectangular frame _d_, which is moveable in a vertical direction upon the guide-rods _e_ and _f_, by which it can be let down, so that the burner may be lowered out of the reflector, by simply turning the handle _g_ (as will be more fully understood by examining figs. 28 and 29), which has the effect of forcing a _thread_ (like that of a screw) on the outside of the guide into a groove in the frame, or withdrawing it, and thus allows it to slide down or locks it at pleasure. An aperture of an elliptical form, measuring about two inches by three, is cut in the upper and lower part of the reflector, the lower serving for the free egress and ingress of the burner, and the upper, to which the copper tube _h_ is attached, serving for ventilation; _i_ shews a cross section and a back view of the main bar of the chandelier or frame on which the reflectors are ranged, each being made to rest on knobs of brass, one of which, as seen at _k k_, is soldered to the brass band _l_, that clasps the exterior of the reflector. Fig. 28 is a section of the reflector _a a_, shewing the position of the burner _b_, with