552 



jugate point f; and through these two points of section of cd 

 we can still draw two real right lines, which shall still ordi- 

 nately cross the real direction of ab, and shall still be two re- 

 ciprocal polars, satisfying all the transformed conditions of the 

 question, and coinciding still with two chords of real and 

 imaginary solution. For the double-sheeted hyperholoid, there- 

 fore, as well as for the ellipsoid, the problem of inscribing a 

 gauche chiliagon, or other even-sided polygon, whose sides 

 shall pass successively, and in order, through the same given 

 number of points, is solved by a system of two polar chords, 

 which we have assigned geometrical processes to determine; 

 and the solutions are still, in general, ^wr in number; two of 

 them being still real, and two imaginary. 



12. If the given surface be a hyperholoid of one sheet, then 

 not only may the diameter ab be real or imaginary, but also 

 the chord CD may or may not cease to be real ; for the two 

 fixed polars will now either both meet the surface, or else both 



Jail to meet it in any two real points. When ab and cd are 

 both real, the proportion in (10), being put under the form 



CF^ : DF^ : : ce^ : ed^ : : oa^ - oc'^ : oa^ - od'^, 



shews that the point of section e and its conjugate f will be 

 real, if the points c' and d' fall both on the diameter ab itself, 

 or both on that diameter prolonged ; that is, if the extremities 

 c and D lie both within or both without the interval between 

 the two parallel tangent planes to the surface which are drawn 

 at the points a and b : under these conditions therefore there 

 •will still be two real right lines, which may still be called the 

 two chords of solution ; but because these lines will still be 

 two reciprocal polars, they will now (like the two fixed polars 

 above mentioned) either both meet the hyperholoid, or else 

 both/ail to meet it ; and consequently there will now be either 

 Jour real, or else Jour imaginary solutions. If ab and cd be 

 still both real, but if the chord cd have one extremity within 

 and the other extremity without the interval between the two 



