550 



because when we return from the sphere to the ellipsoid, or 

 other surface of the second order, the condition of bisection of 

 the given dihedral angle cabd is no longer fulfilled by the 

 sought plane abe, a slight generalization of the foregoing 

 process becomes necessary, and can easily be accomplished as 

 follows. 



10. Conceive, as before, that on the diameter ab the or- 

 dinate ee' is let fall from the internal point of section e, and 

 likewise the ordinates cc' and db' from c and d ; and draw also, 

 parallel to that diameter, the right lines cc", dd", ee", from 

 the same three points c, d, e, so as to terminate on the dia- 

 metral plane through o which is conjugate to the same dia- 

 meter ; in such a manner that oc", od ", oe" shall be parallel 

 and equal to the ordinates cc, d'd, ee ; and that the segments 

 ce, ED of the chord CD shall be proportional to the segments 

 c"e", e"d" of the base c'd " of the triangle c"od", which is 

 situated in the diametral plane, and has the centre o for its 

 vertex. For the case of the sphere, the vertical angle c'od" of 

 this triangle is, by (9), bisected by the line oe"; wherefore 

 the sides oc", od", or their equals, the ordinates c'c, d'd, are, 

 in this case, proportional to the segments c'e", e "d" of the 

 base, or to the segments ce, ed of the chord : while the 

 squares of the ordinates are, for the same case of the sphere, 

 equal to the rectangles ac'b, ad'b, under the segments of the 

 diameter ab. Hence, ^r the sphere, the squares of the seg- 

 ments of the given chord are proportional to the rectangles 

 under the segments of the given diameter, these latter seg- 

 ments being found by letting fall ordinates from the ends of 

 the chord ; or, in symbols, we have the proportion, 



CF^ : df^ :: ce^ : ed^ :: ac'b : ad'b. 



But, by the general principles of geometrical deformation, the 

 property, thus stated, cannot be peculiar to the sphere. It 

 must extend, without any further modification, to the ellipsoid; 

 and it gives at once, for that surface, the two points of har- 



