252 Geometrical Propositions 
by a new reflection, into two rays parallel to O 7,07"; and so on, for any number of 
reflections. Any of the rays emerging at the first surface after internal reflections, is 
parallel to the ray Os produced 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 con- 
venient both in theory and practice. 
Through S draw SR perpendicular to OJ, and meeting O G,OH, produced, in the 
points P,M. Then as the angles at Gand R are right angles, the points J, R, G, P, 
are in the circumference of a circle, and therefore OP x OG@= OI x OR=OS=Kk 
and similarly, OMx OH=k'. If then wetake O 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 O7,O7", will coincide in direction with perpendiculars 
let fall from O on planes touching the surface at P and M, and will be inversely pro- 
portional to these perpendiculars. It follows in the very same manner, that if per- 
pendiculars Og,Qh, let fall from O on the tangent planes at ¢,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 OS. 
86. The surface reciprocal to oy wave surface, the pole bing at os we shall call 
the surface of refraction. {fy Ly f ie ede Kearse het tet / VA 4 
It is hardly necessary to observe that the surface of refraction has a centre at the 
point O, round which it is symmetrical; that it is a sphere in a singly refracting me- 
dium, a double surface 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. 8.) representa section 
made in the double surface of refraction and its attendant sphere by the plane of in- 
cidence. Through the point S, where the incident ray S'O prolonged cuts the cir- 
cular section of the sphere, draw SR perpendicular to the face of the crystal, or to 
FA; and let SR produced cut the circle again in the point s. Then Os is the di- — 
rection of the ray given by ordinary reflection at the first surface of the crystal. 
Produce the right line Ss both ways, to cut the surface of refraction in the points 
P,M, behind the crystal, and in the points p,m, before it; and conceive planes to 
touch the surface of refraction at the points P,M/, p,m. Suppose also that perpendi-— 
culars OP', OM’,Op',Om’, are let fall from O upon these tangent planes, and that 
they intersect the planes in the points P’,J7, p',m’, respectively. ; 
Then from the preceding observations (33, 34, 35), it is manifest that OP’,OM’, 
are the directions of the rays into which S’O is divided by refraction ; that each of 
these refracted rays, on arriving at the second surface of the crystal, is divided by in- — 
; 
