68 



of the angles made by OQ with the semiaxes. Hence OQ coincides 

 with the perpendicular to the tangent plane at R. 



4. It" a perpendicular at O to the plane POQ meet the surface of 

 the ellipsoid ahc in q, then will OQ and Oq be the semiaxes of the 

 section QOq made by a plane passing through them. 



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

 cular 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 hne OQN is per- 

 pendicular to the tangent plane at R. 



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

 of the ellipsoid abc, let OT and OV be taken equal to OQ and Oq the 

 semiaxes of the section, and imagine the double surface which is 

 the locus of all the points T and V ; then if OS be perpendicular to 

 the plane wjiich touches this surface in T, and OP to that which 

 touches the elhpsoid 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 per- 

 pendicular 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^ 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 semi- 

 axes (that to which OT, is equal) indefinitely near to OQ, and there- 



