It is obvious, that according as A, is positive or negative, the excentric, as compared with the central rays, will be incident unfavourably for total reflection on the first or second surface, and favourably on the second or first. It is evident, also, that as e is to be determined so as to secure the total reflection of all the rays, we must compute it for the point where the incidence is unfavourable. From these considerations it therefore appears that e is to be found from the first formula, or from the second, according as A, is positive or negative. For this purpose, if we substituted, according to circumstances, the value of ', or ', from last article, in the first or second formula, we should more readily find e by trial at once than by deducing and solving the resulting biquadratic equation in sin e or cos e. But it will be still better to ascertain, graphically, by help of Mr Balfour's refraction protractor, such a value of e as will no more than insure total reflection of all the rays. From e, so obtained, we can then compute 91, or 9, and ▲, and ascertain what value of e, along with these quantities, satisfies the appropriate equation of condition. If the two values of e do not agree, the computation must be repeated with a new assumed value until the equation of condition be well enough satisfied for practical purposes; and for this end a very few trials will suffice. As in the case of a monesoptric prism, where, as observed in article 22, the value of the angle i may be within limits arbitrarily varied, provided the other dimensions be made to suit, so here also the performance of the prism does not depend upon a very exact determination of e, but solely upon the accuracy of the subsequent computation.

When the prisms are to be employed for lighthouse purposes, it is necessary to prevent the rays from the flame falling on the sides BD, fig. 10, where they would be lost; and for that end they must be arranged as shown in the figure, where it will be seen that, with the exception of the lowest, the face of each prism, on which light ought not to fall, is masked from the luminous origin by the prism which is placed below and somewhat in front of it.

47. The uppermost diesoptric prism ACBD, represented in figure

10, has been protracted to ths of its dimensions computed by means of the formulæ of articles 45, 46.*

The following are the data and results of the computation :

48. Besides the particular form of diesoptric prism now described, I have, among others, devised one which reflects. at its first surface the incident light parallel to the axis, along with the rays which have been totally reflected at the posterior surfaces of the prism, thus saving the superficially reflected light. Its range of application, is but limited; but it might sometimes be usefully employed. A prism of this kind, protracted to scale from computations made for an index of refraction 16, is represented in figure 11, where Oabu, ODC,C,Dv, are the paths of the extreme totally reflected rays of the incident pencil, while OAw, ODv, are the paths of the portions of the light in these rays which are superficially reflected. The portion BD of the first surface.

It is perhaps superfluous to observe that, compared with other dimensions, the radii of curvature of the reflecting sides of lighthouse prisms are so great, that, in small diagrams constructed accurately to scale, the profiles of the sides would generally appear as straight lines, and are here and elsewhere actually so drawn.

I may also say that, for the sake of brevity, I have omitted, meantime, the formulæ for the co-ordinates to be employed in constructing the prisms represented in figs. 7, 8, and 9.

The indices and μ are for the mean and extreme red rays respectively; the angle e is computed for, but all other dimensions for fx.

of the prism is intended to be masked from the luminous origin by another prism placed below, whose angle is shown at Y in the figure.

49. It has been shown in article 12, that in a prism in which the rays have been only once totally reflected, a pencil of small divergence on emerging will suffer little chromatic dispersion, provided its angles of incidence and emergence are nearly equal. The demonstration might have been extended so as to prove that all prisms in which the rays suffer an odd number of reflections, possess, under the same condition of equality of the angles of incidence. and refraction, and provided the deviations by refraction are both in the same direction, the like property of achromatism. Prisms, on the other hand, with an even number of total reflections do not share this valuable property unless the deviations by refraction be in opposite directions. If the deviations at incidence and emergence be in the same direction, the prisms will not be achromatic, and in that respect will be like Fresnel's lenticular apparatus. As with lenticular apparatus, so with such prisms, if we cannot prevent chromatic dispersion, we must try to keep it as small as possible by avoiding large angles of incidence or emergence. For this reason it is not desirable to extend the application of the prism described in article 45 beyond a

range of angles of deviation from 180° down to about 130°.

Triesoptric Prisms.

50. But if for any purpose we desired to continue a series of prisms with deviations from about 180° to about 130°, so as to include rays having still smaller deviations, it might be done by employing triesoptric prisms, or prisms with three totally reflecting sides. Of these I have devised more than one kind. In the meantime it may be enough to show in a diagram, fig. 12, the general forms and arrangement of a

Fig. 12.

series of four such prisms totally reflecting the intromitted light through angles varying from 110° up to 140°, and, moreover, reflecting superficially the incident rays back through the luminous origin. The central rays are all normally incident and emergent, so that the prisms possess the notable property that their forms are independent of the index of refraction of the glass of which they are made. The refractive power of the glass has no other influence except to determine the critical angle, and thus to limit the