81 



ray exceeds sixty-two degrees this difficulty cannot be obviated ; and 

 it is obvious that in a prism of only thirty degrees such an obliquity 

 of the internal incident ray as sixty-two degrees is impracticable. 



For the foregoing reasons I conclude, then, that the construction 

 as in fig. 5 is the best that can be adopted ; consequently the side of 

 the prism, d c, fig. 7, must be drawn so as to make the angles d w v 

 and c w z equal to one another, and the side, q b, so that the angles 

 q v A and b v w are also equal. 



Having thus determined the inclinations of the various planes, let 

 us now consider their respective dimensions; any two contiguous 

 ones being settled, the others follow as a matter of necessity. There 

 are two conditions to take into consideration, viz., that the prism 

 shall be capable of revolution without throwing the light out of the 

 field of view ; and that as much light (and no more) shall be capable 

 of entering the prism as is capable of emerging from it. Draw 

 d t, fig. 7, equal to d q, at right angles to z A, and let z A bisect d t 

 at k, and y A bisect d q in I. Draw t c parallel to A z ; then let d q 

 be any convenient known size, which we will call s, and d c an 

 unknown size, which we will call x. If c d be taken as the radius, 

 t d, equal to d q, equal to s, is the sine of the angle ted. But ted 

 is the complement of the angle deb (called in fig. 2 the angle c) ; 

 consequently the cosine of the angle c : s : radius x, the side 

 required. 



The focus of the lens required is evidently A I, less the thick- 

 ness of the lens. Join A d; then, because d I equals d k (by 

 construction), and the angles A k d and Aid are right angles, 

 the angles d A k and d A I are also equal ; consequently the angle 



d A I equals an& e a ' Now if d I be taken as the radius, A I will 



2 



i, i-i, i. -~~~j. ~4 a ija7 the angle a ; and as d I equals s 

 be the cotangent ol the angle dAlor s — ' ^ — > 



& 6 2 a 



the radius : — : : cotangent — e an £ e a . f ocus -f. thickness of the 

 3 8 2 



lens. A square diaphragm, whose side is equal to s (or d q), must 



be placed under the prism, so that one of its angles is coincident 



with the point c, fig. 2 ; the opposite angle of the same side will 



then be exactly perpendicularly under the point d, and thus prevent 



the entrance of any rays nearer to b, which would otherwise reach 



the surface, d q, without any reflection. 



TRANS. MIC. SOC. VOL. in. M 



