[ *58 ] 



LXIX. On Qtiatcniions; or on a New System of Imaginarics 

 in Algebra. Dtj Professor Sir William Rowan Hamilton, 

 LL.D., F.P.il.I.A., F.R.A.S., Corresponding Member of the 

 Institute of France, and of other Scientific Societies in British 

 and Foreign Countries, Andrews' Professor of Astronomy in 

 the University of Dublin, and. Royal Astronomer of Ireland. 

 [Continued from vol. xxix. p. 328.] 



29. TF we denote by a and /S two constant vectors, and by 

 Jl p a variable vector, all drawn from one common origin ; 



if also we denote by n and v two variable scalars, depending 



on the Ibregoing vectors «, /3, f by the relations 



t;2=_4(V./3p)2=-(/3^-p/3)2;/ • • • • U-J 



we may then represent the central surfaces of the second de- 

 gree by equations of great simplicity, as follows: — 



An ellipsoid, with three unequal axes, may be represented 

 by the equation 



?<^ + w2=i (2.) 



One of its circumscribing cylinders of revolution has for equa- 

 tion 



^2=1; (3.) 



the plane of the ellipse of contact is represented by 



u=0; (4.) 



and the system of the two tangent planes of the ellipsoid, 

 parallel to the plane of this ellipse, by 



ifi=.l (5.) 



A hyperboloid of one sheet, touching the same cylinder in 

 the same sheet, is denoted by the equation 



n^-v'^=-\', (6.) 



its asymptotic cone by 



M2_y2^0; (7.) 



and a hyperboloid of two sheets, with the same asymptotic 

 cone (7.)» and with the two tangent planes (5.)> is represented 

 by this other equation, 



,,2_l;2=l (8.) 



By changing p to p— y» where y is a third arbitrary but con- 

 stant vector, we introduce an arbitrary origin of vectors, or an 

 arbitrary position of the centre of the surface, as referred to 

 such an origin. And the general problem of determining that 

 individual surface of the second degree (supposed to have a 

 centre, until the calculation shall show in any particular ques- 

 tion that it has none), which shall pass through ni7ie given 



