553 



parallel tangent planes, the proportion above written will 

 assign a negative ratio for the squares of the segments of cd ; 

 the points of section e and f, and the two polar chords of so- 

 lution, become therefore, in this case, themselves imaginary ; 

 and of course, by still stronger reason, the four solutions of 

 the problem become then imaginary likewise. If cd be real, 

 but AB imaginary, the proportion in (11) conducts to two real 

 points of section, and consequently to two real chords, which 

 may, however, correspond, as above, either to four real or to 

 four imaginary solutions of the problem. And, finally, it will 

 be found that the same conclusion holds good also in the re- 

 maining case, namely, when the chord cd becomes imaginary, 

 whether the diameter ab be real or not ; that is, when the two 

 fixed polars do not meet, in any real points, the single-sheeted 

 hyperboloid. 



13. Although the case last mentioned may still be treated 

 by a modification of the proportion assigned in (10), whicli 

 was deduced from considerations relative to the sphere, yet in 

 order to put the subject in a clearer (or at least in another) 

 point of view, we may now resume the problem for the ellip- 

 soid as follows, without making any use of the spherical de- 

 formation. It was required to find two lines, reciprocally 

 polar to each other, and ordinately crossing a given diameter 

 AB of the ellipsoid, which should also cut a given chord cd of 

 the same surface, internally in some point e, and externally 

 in some other point f. Bisect cd in g, and conceive ef to be 

 bisected in h ; and besides the four old ordinates to the dia- 

 meter AB, namely cc', dd', ee', and ff', let there be now sup- 

 posed to be drawn, as two new ordinates to the same diameter, 

 the lines gg' and hh'. Then g' will bisect c'd', and h' will 

 bisect e'f'; while the centre o of the ellipsoid will still bisect 

 AB. And because the points e' and f' are harmonic conju- 

 gates, not only with respect to the points a and b, but also 

 with respect to the points c' and d', we shall have the follow- 

 ing equalities : 



2 s 2 



