Mr Stevunson's Totally Reflecting Mirror. 146 



As it is impossible in practice to grind and polish parabolic 

 surfaces in glass, it becomes necessary to make the arcs A B, 

 B C, portions of circles instead of parabolas ; and this may be 

 done either by finding a circle having a common tangent with 

 the parabola at the points A and B, or by finding the oscu- 

 lating circle to the parabola at a given point in the arc A B. 

 In order, then, to construct the apparatus, it becomes neces- 

 sary to find the radius and centre of curvature of one of the 

 arcs A B, vv^hich may be taken to represent all the others ; 

 and to avoid the loss of any of the light incident on the sur- 

 face A B, it is necessary to find the greatest arc of the circle 

 DAE, which a single zone may subtend, consistently with 

 the total reflexion of all the rays. 



I. The Radius and Centre of Curvature. 

 We shall first find the radius and centre of the arc A B, 



fig. 2j on the supposition that 

 the rays incident at its ex- 

 tremities, A, B, are reflected 

 K in the proper direction, which 

 is at right angles to B F ; and 

 next, on the supposition that 

 A B is the osculating circle 

 to the parabola at the point 

 B. By the latter assumption 

 it will be found that the for- 

 mulae become extremely sim- 

 ple , and will probably be suffi- 

 ^ ciently accurate in practice. 

 1. To find the Radius of Curvature on the supposition that the 

 rays incident at the points A and B are accurately reflected. 

 If A B be assumed, so that the rays at A and B will be 

 reflected in a direction perpendicular to B F ; the problem to 

 be solved is to determine a circular arc, so that rays inci- 

 dent upon its extremities in given directions may be reflected 

 in given directions. This problem may easily be extended 

 to the case of a refracting instead of a reflecting surface ; 

 and as it may be useful in constructing other sorts of light- 

 house apparatus, it may be proper to give a general solution. 

 Since the du-ections of the rays at the extremities of the arc 



VOL. LI. NO. CI.--JULY 1851. K 



