GRAY, ON A DOUBLY-REFLECTING PRISM. 275 



the length of the path traversed by the pencil in the prism, 

 the shorter we can make the path the less will be the loss of 

 light. The sides of the prism ought, therefore, if the arrange- 

 ment is practicable, to be just sufficient to receive and trans- 

 mit the given pencil, and no more. 



Let RiA, RgB be the extreme rays of the pencil, and its 

 diameter R^R.^^^P- Now, AB is obviously equal to p sec. i,, 

 and, the pencil being the same diameter at emergence as at 

 incidence, BC ought to equal p. In order to this, R B 

 ought to be reflected to C ; in other words, ST, the axis of 

 the pencil, must be parallel to BC, which it will be if the 

 angle AST be equal to the angle B of the prism. 



Now, zAST=zASR+ znST:=90°-i^ + 2i^=90° + i^; 

 and the angle B = 180^— i^ + d. Equating these, 



90°-f-ii = 180-i,-|-f/, 

 we get, ii=45°-f^f/. 



Employing this value of i-^, therefore (which gives 45° for the 

 angle 1.2, and also for the angle C), the prism will be the least 

 possible, and the loss of light by absorption will consequently 

 be a minimum. 



The angles of the prism will now be — 



A= 45°-fif/, 

 B = 135° + i^, 

 C= 45°, 

 D = 135°-c?. 



Sum . 360°. 



From the values of A and B above, and also from that of i^, 

 we learn that 90° is the greater limit of d ; so that we may, 

 by means of such a prism as is here described, obtain any 

 amount of obliquity or deviation short of 90°. The value 

 obtained for i^, the less of the two angles of incidence, namely, 

 45°, is an admissible value, being sufficiently removed from 

 the critical angle in a prism of glass (about 41° 28') to afford 

 total reflexion. 



It would, I believe, be practicable to assign the values of 

 AD, DC, the remaining sides of the prism, in terms of ^ and 

 d ; but the angles, and the two adjacent sides, AB, BC, now 

 known, suffice for its geometrical construction; and it will 

 be, perhaps, quite as easy specially to compute the remaining 

 sides, if they should be required, in any particular case to 

 which the formulae may be applied. 



A prism of the form now under consideration was first used 

 in connexion with the microscope by M. Nachet, for the pur- 



