36 Geometrical Propositions applied 



arriving at the first surface, will each be divided, by a new re- 

 flection, into two rays parallel to T, OT' ; and so on, for any 

 number of reflections. Any of the rays emerging at the lirst 

 surface, after internal reflections, is parallel to the ray Os pro- 

 duced by ordinary reflection at the point of incidence ; and any 

 ray emerging at the second surface is parallel to the incident 

 ray S'OS. 



35. This construction may be changed into another that will 

 be found more convenient both in theory and practice. 



Through 8 draw Sit perpendicular to 01, and meeting OG, 

 OH, produced, in the points P, M. Then as the angles at G 

 and R are right angles, the points J, R, G, P, are in the circum- 

 ference of a circle, and therefore OP x OG = 01 x OR = OS 2 = k* ; 

 and similarly, M x OH = k*. If then we take for the fixed 

 origin, or pole, and k* for the constant rectangle (Theorem I.), 

 and describe the surface which is reciprocal to the wave surface, 

 it is evident that the points P and M will be points of the sur- 

 face so described, and that OT, OT', will coincide in direction 

 with perpendiculars let fall from on planes touching the sur- 

 face at P and M, and will be inversely proportional to these 

 perpendiculars. It follows in the very same manner, that if per- 

 pendiculars Og, Oh, let fall from on the tangent planes at t, t', 

 be produced to meet SR in the points p, m, these points will also 

 be on the surface reciprocal to the wave surface. 



In the present case, it is manifest that this reciprocal surface 

 lies wholly without the sphere 08. 



36. The surface reciprocal to the wave surface, the pole 

 being at 0, we shall call the surface of refraction. 



It is hardly necessary to observe that the surface of refrac- 

 tion has a centre at the point 0, round which it is symmetrical ; 

 that it is a sphere in a singly refracting medium, a double sur- 

 face in a doubly refracting medium, and a surface of three sheets 

 if we suppose a case of triple refraction. 



37. In the case that we are considering, let the figure 

 (Fig. 15) represent a section made in the double surface of re- 

 fraction and its attendant sphere by the plane of incidence. 



