551 



monic section, e and f, of the given chord cd, through which 

 points the two sought chords of real and imaginary solution 

 are to pass; these chords of solution are therefore completely 

 determined, since they are to be also ordinates, as before, to 

 the given diameter ab. The problem of inscription for the 

 ellipsoid is therefore fully resolved ; not only when, as in (6), 

 the number of sides of the polygon is odd, but also in the 

 more difficult case (7), when the number of sides is even. 



11, If the given surface be a hyperboloid of two sheets, 

 one of the two fixed polars will still intersect that surface, and 

 the fixed chord cd may still be considered as real. If the 

 given diameter ab be also real, the proportion in (10) still 

 holds good, without any modi6cation from iraaginaries, and 

 determines still a real point E, with its harmonic conjugate F, 

 through one or other of which two points still passes a chord 

 of real solution, while through the other point of section still 

 is drawn a chord of imaginary solution, reciprocally polar to 

 the former. But if the diameter ab be imaginary, or in other 

 words if it fail to meet the proposed hyberboloid at all, we 

 are then led to consider, instead of it, an ideal diameter a'b', 

 having the same real direction, but terminating, in a well- 

 known way, on a certain supplementary surface; in such a 

 manner that while a and b are now imaginary points, the 

 points a' and b' are real, although not really situated on the 

 given surface ; and that 



OA^ = ob^ = - oa'2 = - ob'2. 



The points c' and d' are still real, and so are the rectangles 

 ac'b and ad'b, although a and b are imaginary ; for we may 

 write, 



ac'b = ots? - oc'2, ad'b = OA^ - od'2, 



and the proportion in (10) becomes now, 



CF^ : DF^ : : ce^ : ed^ : : oc'^ + oa'^ : od'^ + oa'^. 



It gives therefore still a real point of section e, and a jea/ cow- 



2s 



