282 Jccouni f/‘Huygens' Theory ofBoMe Refraction. 
centre of tlie spheroid HME, in which the light is propagated, 
and of which this ellipse is the section. Let it be required to 
find the refraction of the incident ray RC. 
Through RC draw a plane perpendicular to the plane of the 
ellipse HDE, cutting it in BCK. In this plane draw CO per- 
pendicular to CR, and draw OK so as to be perpendicular to 
OC, and equal to N, which is the space described by light in 
air, while it is propagated in the crystal by the spheroid HDEM, 
then draw KT perpendicular to BCK, and in the plane of the 
ellipse HDE, and conceive a plane drawn through KI, and 
touching the spheroid HME in I, the straight line Cl will be 
the required refraction of RC, as may be easily shewn from the 
demonstration in p. 277, 278, 
In order to determine the point of contact I, let there be 
drawn HE parallel to KT, and touching the ellipse HDE in 
any point H. Having drawn CH meeting KT in T, let there 
be conceived to pass through CH and through CM (the re- 
fraction of the perpendicular, ray,) a plane which forms in the 
spheroid the elliptical section HME. Then it is certain by 
the Lemma demonstrated below *, that the plane passing 
^ Lemma. — Jf a spheroid is touched hy a right line^ and also hy tvso or more 
planes parallel to this Zz'ne, though not to one another^ all the points of contact^ both 
of the line, and of the planes will be in the semiellipse formed hy a plane passing 
through the centre of the spheroid. 
Let the spheroid LED, Fig. 7., be touched by the line BM at the point B, and 
also by planes parallel to this line at the points O and A, we must demonstrate 
that the points B, O, and A. are in the same ellipse, formed in the spheroid by a 
})ianc passing through its centre. 
Through BM and the points O, A, draw planes parallel to one another, and 
which, in cutting the spheroid form the similar and similarly situated ellipses 
LBD, POP, QAQ Tound their centres K, N, R, in the same diameter of the 
spheroid, w^hich will also be the diameter of the ellipse made by the section of a 
plane passing through the centre of the spheroid, and cutting the planes of the 
three above mentioned ellipses at right angles, all wdiich is manifest from 
Prop. XV. of Archimedes’s book on Conoids and Spheroids. Moreover, the two 
last planes drawn through O, A will, in cutting the planes which touch the sphe- 
roid in these points form straight lines OH, AS, which will be parallel to BM, 
and all the three, BM, OH, AS, will touch the ellipses LBD, POP, QAQ in the 
points B, O, A, since they are in the planes of the ellipses, and at the same time 
in the planes wdiich touch the spheroid. If, how’ever, from B, O, A w'e draw BK, 
ON, AR through the centres of the same ellipses, and through these centres, LD, 
PP, QQ parallel to BM, OH, AS, these diameters will be conjugate to BK, ON, 
AR, and because the three ellipses are similar and similarly situated, and have 
