Prof. Cay ley on Quartic Curves* 105 



For vapour of ether, 



p =(1-210775 + '00564785 tf. 

 For vapour of sulphuret of carbon, 



p = (1-141374+ -005245221 t) 6 . 

 For vapour of hydrochloric ether, 



p= (1-441730 + -005681901 /) 6 . 

 For vapour of oil of turpentine, 



p = (-4734178 + -00396796 1) 6 . 



Fresh researches would probably lead to small modifications 

 of the values of the constants, but I do not think that large 

 changes would be found necessary. 



XVI. On Quartic Curves. By A. Cayley, F.R.S* 



THE expression ' an oval ' is used, in regard to the plane, to 

 denote a closed curve without nodes or cusps ; and, in 

 regard to the sphere, it is assumed moreover that the oval is a 

 curve which is not its own opposite, and does not meet the 

 opposite curvef — that is, that the oval is one of a pair of non- 

 intersecting twin ovals. I say that every spherical curve of the 

 fourth order (or spherical quartic) without nodes or cusps may 

 be considered as composed of an oval or ovals lying wholly in 

 one hemisphere (that is ; not cutting or touching the bounding 

 circle of the hemisphere), and of the opposite oval or ovals lying 

 wholly in the opposite hemisphere ; or, disregarding the opposite 

 curves, that it consists of an oval or ovals lying wholly in one he- 

 misphere. And this being so, the quartic cone having its vertex 

 at the centre of the sphere is met by a plane parallel to that of 

 the bounding circle in a plane quartic curve consisting of an oval 

 or ovals; and thence every plane quartic is either a finite curve 

 consisting of an oval or ovals, or else the projection of such a 

 curve. 



Considering first the case of the plane, aline in general meets 

 the oval in an even number of points (the number may of course 

 be =0); hence as the point of contact of a tangent reckons 

 for two points, the tangent at any point of the oval again intersects 

 the oval in an even number of points (this number may of course 

 be =0). The number of points of intersection by the tangent 

 (the point of contact being always excluded) is either evenly 



* Communicated by the Author. 



f The notions of opposite curves &c. are fully developed in the excellent 

 Memoir of Mobius, " Ueber die Grundformen der Linien der dritter Ord- 

 nung," Abh. der K. Sachs. Ges. zu Leipzig, vol. i. (1852), to which I have 

 elsewhere frequently referred. 



