325 



an umbilic ; their difiFerence, tj - 0, has the direction of a cyclic 

 normal ; another umbilicar vector being in the direction of 

 the sum of their reciprocals, tjI + ^S and another cyclic 

 normal in the direction of the difference of those reciprocals, 

 rj-i - 6'^. The lengths of the semiaxes of the ellipsoid are ex- 

 pressed as follows : 



a = Tr,+ T0; 6=T(„-0); c=Trj~Td. (2) 



" The focal ellipse is given by the system of the two equa- 

 tions 



S.pVn^S.pUO; (3) 



and 



TV.p\Jrj = 2SV{r,e); (4) 



where TV.pUrj may be changed to TV .pVd; and which 

 represent respectively a plane, and a cylinder of revolution. 

 Finally, I shall just add what seems to me remarkable, — 

 though I have met with several similar results in my unpub- 

 lished researches, — that the focal hyperbola is adequately re- 

 presented by the single equation following : 



V.rjp. V.p0=(V.„0)V' (5) 



In the same note to the Secretary, it was requested by 

 Sir William R. Hamilton that the Academy might be informed 

 of a theorem respecting the inscription of certain gauche poly- 

 gons, in surfaces of the second degree, which he had lately 

 communicated to the Council. This theorem was obtained 

 by the method of quaternions, and included, as a particular 

 case, the following: — " If the first, second, third, and fourth 

 sides of a gauche nonagon, inscribed in a surface of the second 

 order, be respectively parallel to the fifth, sixth, seventh, and 

 eighth sides of that nonagon, and also to the first, second, 

 third, and fourth sides of a gauche quadrilateral, inscribed in 

 the same surface ; then the plane containing the first, fifth, and 

 ninth corners of the nonagon will be parallel to the plane 



