1871.] Dr. J. Casey on Cyclides and Sphero-Quartics. 



495 



from specimens of portions of the skull in the British Museum, and from a 

 cast and photographs of the entire cranium in the Australian Museum at 

 Sydney, New South Wales. The descriptions of the mandible, and of the 

 dentition in both upper and lower jaws, are taken from actual specimens in 

 the British Museum, in the Museum of Natural History at Worcester, 

 and in the Museum at Adelaide, S. Australia, all of which have been con- 

 fided to the author for this purpose. The results of comparisons of these 

 fossils of Nototherium with the answerable parts in Diprotodon, Meter opus, 

 Phascolarctos, and Phascolomys are detailed. 



Characters of three species,. Nototherium Mitchelli, N. inerme, and N. 

 Victoria, are defined chiefly from modifications of the mandible and man- 

 dibular molars. A table of the localities where fossils of Nototherium 

 have been found, with the dates of discovery and names of the finders 

 or donors, is appended. The paper is illustrated by subjects for nine quarto 

 Plates. 



II. "On Cyclides and Sphero-Quartics." By John Casey, LL.D., 

 M.R.I.A. Communicated by Prof. Cayley, F.R.S. Received 

 May 11, 1871. 



(Abstract.) 



The curves and surfaces considered in this paper are, I believe, some of 

 the most fertile in properties in the whole range of geometry. For the pur- 

 pose of giving a full and comprehensive discussion, I have divided the paper 

 into several chapters. The following is an outline of the method of inves- 

 tigation pursued, together with a statement of some of the results arrived 

 at. 



If we take the most general equation of the second degree in (a, ft, y, £), 

 where these variables denote spheres instead of planes, 



(abcdlmnpq r^a, ft, y, £,) 2 =0, 

 we get the most general form in which the equation of a quartic cyclide 

 can be written. Setting out with this equation, I have proved that a 

 quartic cyclide is the envelope of a variable sphere, w T hose centre moves on a 

 given quadric, and which cuts orthogonally the Jacobian of the spheres of 

 reference (a, ft, y, 3). 



The Jacobian of (a, ft, y, c) can be written in a form identical with that 

 of the imaginary circle at infinity in the system of quadriplanar coordi- 

 nates. The square of the Jacobian can be expressed by an equation of the 

 second degree in a, ft, y, S. This equation assumes a very simple form when 

 a, ft, y, c! are mutually orthogonal. By means of it I have shown that 

 every quartic cyclide can be written in the canonical form, 



aa, 2 + bft 2 + cy 2 + dh 2 + ee 2 m 0, 

 where a, ft, y, c, e are five spheres mutually orthogonal. These are spheres 

 of inversion of the cyclide, and by incorporating constants their equations 

 are connected by an identical relation, « 2 +/3 2 + y 2 +3 2 + e 2 ==0. 



