upon the surface CK.  Make Co perpendicular to rc, and across the angle KCO adjust Ok, equal to N and perpendicular to Co; then draw KI, which touches the Ellipse GSP, and from the point of contact I join ic, which will be the required refraction of the ray rc.  The demonstration of this is, it will be seen, entirely similar to that of which we made use in explaining ordinary refraction.  For the refraction of the ray rc is nothing else than the progression of the portion C of the wave Co, continued in the crystal.  Now the portions H of this wave, during the time that O came to K, will have arrived at the surface CK along the straight lines H_x_, and will moreover have produced in the crystal around the centres x some hemi-spheroidal partial waves similar to the hemi-spheroidal GSP_g_, and similarly disposed, and of which the major and minor diameters will bear the same proportions to the lines xv (the continuations of the lines H_x_ up to KB parallel to Co) that the diameters of the spheroid GSP_g_ bear to the line CB, or N. And it is quite easy to see that the common tangent of all these spheroids, which are here represented by Ellipses, will be the straight line ik, which consequently will be the propagation of the wave Co; and the point I will be that of the point C, conformably with that which has been demonstrated in ordinary refraction.

Now as to finding the point of contact I, it is known that one must find CD a third proportional to the lines CK, CG, and draw di parallel to cm, previously determined, which is the conjugate diameter to CG; for then, by drawing KI it touches the Ellipse at I.

29.  Now as we have found CI the refraction of the ray rc, similarly one will find C_i_ the refraction of the ray rC, which comes from the opposite side, by making C_o_ perpendicular to rC and following out the rest of the construction as before.  Whence one sees that if the ray rC is inclined equally with rc, the line C_d_ will necessarily be equal to CD, because C_k_ is equal to CK, and C_g_ to CG.  And in consequence I_i_ will be cut at E into equal parts by the line cm, to which di and di are parallel.  And because cm is the conjugate diameter to CG, it follows that iI will be parallel to gG.  Therefore if one prolongs the refracted rays CI, C_i_, until they meet the tangent ML at T and t, the distances MT, M_t_, will also be equal.  And so, by our hypothesis, we explain perfectly the phenomenon mentioned above; to wit, that when there are two rays equally inclined, but coming from opposite sides, as here the rays RC, rc, their refractions diverge equally from the line followed by the refraction of the ray perpendicular to the surface, by considering these divergences in the direction parallel to the surface of the crystal.