4 The Double Refraction of Light in a Crystallized 



with the semiaxes. Hence OQ coincides with the perpendicular 

 to the tangent plane at R. 



4. If a perpendicular at to the plane POQ meet the sur- 

 face of the ellipsoid abc in q, then will OQ and Oq be the semi- 

 axes of the section QOq made by a plane passing through 

 them. 



For the tangent plane at Q and the plane QOq are perpen- 

 dicular to POQ, and therefore the intersection of the two former, 

 which is a tangent to the ellipse QOq at Q, is perpendicular to 

 OQ ; whence OQ is one semiaxis, and Oq the other. 



If the perpendicular Oq meet the other ellipsoid in r, then 

 OR and Or will be the semiaxes of the section ROr made by a 

 plane passing through them; for (by lem. 3), the straight line 

 OQN is perpendicular to the tangent plane at R. 



5. In a straight line, at right angles to any diametral section 

 QOq of the ellipsoid abc, let 07* and OF be taken equal to OQ 

 and Oq, the semiaxes of the section, and imagine the double sur- 

 face which is the locus of all the points T and V ; then if OS 

 be perpendicular to the plane which touches this surface in T, 

 and OP to that which touches the ellipsoid in Q, the lines OP 

 and OS will be equal and perpendicular to each other, and the 

 four, OP, OQ, OS, OT, will lie in the same plane, which will be 

 at right angles to Oq. 



By the preceding lemma it is evident that Oq is perpendicular 

 to the plane POQ ; and since OT is perpen- 

 dicular to the plane QOq, it follows that OP, 

 OQ, OT, are in the same plane at right 

 angles to Oq. In the surface which is the 

 locus of T, and in the plane TOq, let a point 

 T t be taken indefinitely near to T ; then the 

 plane of the section at right angles to OT, 

 will pass through OQ, and will have one of 

 its semiaxes (that to which OT t is equal) 

 indefinitely near to OQ, and therefore differ- 

 ing from OQ by an indefinitely small quan- 

 tity of the second order, adopting, for brevity, the language of 



