144 Mr Swa.u' 8 formulce for constructing 



is to investigate formulae for the construction of these 

 prisms. 



The totally reflecting 

 mirror is represented 

 in fig 1, where D B E 

 is a semicircle, divid- 

 ed into an oc/o? number 

 of equal arcs, a B, 

 B 6, are portions of 

 parabolas having the 

 common focus F, and 

 the axis of each coin- 

 ciding with the line 

 DE;whileI,K,L,M, 

 are figures equal and 

 similar to A B C If, 

 h_ now, the figure D K B 

 revolve round the axis 

 B F, it will generate a solid of revolution, and this, when 

 constructed of glass, forms the totally reflecting mirror. 



It is evident that the portion of the light reflected from the 

 surface generated by the revolution of the semicircle DAE, 

 since it is everywhere incident at an angle of 90°, will be 

 returned to the point F ; and the rest of the light entering 

 the glass at an angle of 90° will suff^er no deviation by re- 

 fraction. Let F a represent one of the rays. Then, since 

 A B D is a parabola whose focus is F, and whose axis is D E, 

 the ray F a, will be reflected in the direction a 6, parallel 

 to D E. Again, because B C is a parabola whose focus is 

 F, and whose axis is D E, the ray a 6, incident upon it in a 

 direction parallel to its axis, will be reflected to the focus, 

 which it will reach ; for it is incident at an angle of 90° 

 upon the surface of the glass, and therefore, will sufi'er no 

 deviation by refraction. 



As the figures I, K, L, are all identical with ABC, their 

 action will be precisely the same ; and if the angle of inci- 

 dence of the rays upon the parabolic surface of the prism 

 A B C, be everywhere sufiiciently great to ensure total re- 

 flexion, all the light will be returned to the point F, except 

 what is unavoidably lost by absorption, or irregularly scat- 

 tered at the surfaces. 



