554 



h'f'^ = h'e'^ = h'a . h'b = h'c' . h'd', 



= h'o^ - GA^ = h'g'2 - g'c'^. 



Hence, 



oh'- - g'h'2 = OA^ - c'g'-, 



that is, 



, OA- + Og''^ - CG- Oa' + OC'.OD' 



OH = ^ -, = -, r- • 



2oG OC + OD 



Now each of these two last expressions for oh' remains 

 real, and assigns a real and determinate position for the point 

 h', even when the points c', d', or the points a, b, or when 

 both these pairs of points at once become imaginary; for the 

 points o and G'are still in all cases real, and so are the squares 

 of oa and c'g', the rectangle under oc' and od', and the sum 

 oc'+od'. Thus h' can always be found, as a real point, and 

 hence we have a real value for the square of hV, or h'f', which 

 will enable us to assign the points e' and f' themselves, or else 

 to pronounce that they are imaginary. 



14. We see at the same time, from the values h'o^ - oa^ 

 and h'g'2- c'g'- above assigned for h'e'- or h'f''^, that these two 

 sought points e' and f' must both be real, unless the two fixed 

 points A and c' are themselves both real, since o, g', h', are, all 

 three, real points. But for the ellipsoid, and for the double 

 sheeted hyperboloid, we can in general oblige the points c, d, 

 and their projections c', d', to become imaginary, by selecting 

 that one of the two fixed polars which does not actually meet 

 the surface ; for these two sorts of surfaces, the two polar chords 

 of solution of the problem of inscription of a gauche polygon 

 with an even number of sides passing through the same num- 

 ber of given points, are therefore found anew to be two real 

 lines, although only one of them will actually intersect the 

 surface, and only two of the four polygons will (as before) be 

 real. And even for the single sheeted hyperboloid, in order 

 to render the two chords of solution imaginary/ lines, it is ne- 

 cessary that the two given polars should actually meet the 



