^80 Account ^’Huygens’ Theory of Bmhle R fraction. 
in which section is the ray RC. Let the semielliptical plane 
passing through Q q and CM be inclined to the first at 6“ 4(y, 
and, therefore, the refracted ray Cl is in that plane. Then, if 
I, a point in the bottom of the crystal is seen by the rays ICR, 
1 cr refracted equally at C, c, and, therefore, equidistant from 
D, and reaching the eye at R, r, the point I must appear raised 
to S, the intersection of R, C, r c, and in the line DP perpendi- 
cular to Q q. If IP is then drawn perpendicular to DP, SP 
will be the apparent elevation of I above the bottom of the 
crystal. 
Let there be described upon Q g a semicircle cutting CR in 
B ; let BV be drawn perpendicular to Q and let the propor- 
tion of the refraction for this section be that of N to CQ. Then 
N : CQ = VC : CD. But 
VC : CD:=VB : DS. Here 
N : CQ VB : DS. 
Let ML be drawn perpendicular upon CL, and because the 
eyes at R and r are supposed to be about a foot from the cry- 
stal, and the angle RS r very small, we may consider VB m CQ 
and DP=i:CL, whence N : CQ =: CQ : DS. But N= 156962, 
CM being lOOQOO and CQrz:105082, consequently, DSrr 70288, 
But CL = 99324::= Cos MCL, or 6° 40', CM being radius. 
Hence, DP : DS = 99324 : 70283, which gives the elevatiou 
of the point I by refraction. 
Let us now consider what takes place in the other section 
LF, Fig. 1. Let GM^, Fig. 3. be the semiellipse already 
considered in Vol. iii. p. 150. par, 2. and 3., and which ds 
formed by a section of a spheroidal wave, having C for its cen- 
tre. Let the point I be seen as before by the eyes at R and r, 
and CR and c r being equally inclined to the surface of the 
crystal. Drawing Cl parallel to CM, the refraction of a per- 
pendicular ray, DC and D c will be equal, (See Vol. iii. p. 150. 
par, 3.) The point I will obviously appear at S in the 
line DP, and if IP is drawn perpendicular to DP, PS will 
be the elevation of the point I. Let there be described 
upon a semicircle cutting CR in B, let BV be drawn per- 
pendicular to G g.) and let the ratio of refraction in this section 
be that of N to GC. Since Cl is the refraction of BC and 
J)I parallel to CM, we have, by Vol. iii. p. 150. par, L 
