556 



surface ; for otherwise the polar lines deduced will still be 

 real. It is necessary also, for the imaginariness of the two 

 lines deduced, that the given diameter ab should be itself a 

 real dianoeter, or in other words that it should actually inter- 

 sect the hyperboloid. But even when the given chord cd 

 and the given diameter ab are thus both real, and when the 

 surface is a single sheeted hyperboloid, it does notfollow that 

 the two chords of solution may not be real lines. We shall 

 only have failed to prove their reality by the expressions re- 

 cently referred to. We must resume, for this case, the reason- 

 ings of (12), or some others equivalent to them ; and we find, 

 as in that section of this Abstract, for the imaginarinessof the 

 two sought polar lines, the condition that one of the two ex- 

 tremities of the given and real chord cd shall fall within, and 

 that the other extremity of that chord shall fall without the 

 interval between the two real and parallel tangent planes to 

 the single sheeted hyperboloid, which are drawn at the extre- 

 mities of the real diameter ab. Sir W. R. Hamilton confesses 

 that the case where all these particular conditions are com- 

 bined, so as to render imaginary the two polar lines of solu- 

 tion, had not occurred to him when he made to the Royal 

 Irish Academy his communication of June, 1849. 



15. It seems to him worth while to notice here that instead 

 of the foregoing ifietric processes for finding (when they exist) 

 the two lines of solution of the problem, the ioWow'xng graphic 

 process of construction of those lines may always, at least in 

 theory, be substituted, although in practice it will sometimes 

 require modification for imaginaries. In the diametral plane 

 ABC, draw a chord kd'l, which shall be bisected at the known 

 point d' by the given diameter ab ; and join ck, cl. These 

 joining lines will cut that diameter in the two sought points 

 e', f'; which being in this manner found, the two sought 

 lines of solution ee', ff', are constructed without any diffi- 

 culty. For the sphere, the ellipsoid, and the hyperboloid of 

 two sheets, although not always for the single sheeted hyper- 



