TBAJVSACTIONS OF SECTION A. 559 



MONDAY, SEPTEMBER 14. 



Department op Mathematics. 



The following Papers were read : — 



1, On the Differential Invariants of Surfaces and of Space. 

 By Professor A. R. Forsyth, F.B.S. 



2. On Spherical Curves, By Harold Hilton, M.A. 



If the stereographic projection of a curve on a sphere from any point is a 

 rational algebraic curve, so is its projection from any other point. One projection 

 is derived from another by an inversion followed by a reflexion in a straight line. 

 In general the projection of such a spherical curve is a plane curve intersecting 

 the line at infinity only in a multiple point at each circule. If the plane curve is 

 of the n-ih. degree, the spherical curve is said to be of the n-t\\ degree. If it has 

 5 nodes and k cusps, touches r great circles in two distinct points, and has t great 



circles of curvature, th&a m= -^v? -2b-ZK, n= m{m - l)-2r-3t, i. = ^n{n - 2) 



- 68 - 8k, K = 3??i (m - 2) - 6r - St. The deficiency is Mi-2f-b- k. The foci of 



a spherical curve (i.e. the intersections of generating lines which touch the curve) 

 are (pn-n)- in number, m-n being real, and project into the foci of the pro- 

 jection of the curve. Every focus of a spherical curve is a focus of its evolute. 



Very many properties of spherical curves may be deduced from known 

 properties of plane curves, and vice versa ; the simplest cases are the circle and 

 the spherical quartic with zero deficiency. For instance :— ' If three small circles 

 are drawn passing through the cusp of a spherical quartic and touching^ the curve, a 

 circle can be drawn through the focus and their other three points of intersection.' 



Some striking theorems can be proved for the real intersections of a real cone 

 and a sphere. If the vertex of the cone lies outside the sphere, we can by two 

 projections reduce the curve to the intersection of a sphere and a cylinder; if the 

 vertex lies inside the cone, we can reduce the curve to the intersection of a sphere 

 and a cone whose vertex is at the centre of the sphere. In either case we can 

 deduce properties of a spherical curve of the 2/)-th degree which lies on a cone by 

 means of known properties of plane curves of the^-th degree. If the vertex of 

 the cone is at the centre of the sphere, properties of the curve may be derived 

 from the fact that the equation of the cone may be put into the shape 



= ai ua . . . Qp + fli(.v- +y" + 3') /3i ^2 . . . i3p -a + «3 (x'+y"' + z^fji 72 • • . r;>-i + • ■ -, 



where the a's are constants and the as, ffa, y& are real linear functions of .r, y, s. 

 The foci of the intersection of the sphere and the reciprocal cone are the 2p poles 

 of the great circles in which the planes ai = a, = ... = a,, = intersect the sphere. 



Projecting on to the plane we have properties of a curve which is its own 

 inverse" with respect to a circle whose centre is real, and whose radius is real or 

 purely imaginary. In particular many interesting properties of bicircular quartics 

 whose four" real foci are concyclic may be obtained. For example : ' If P is any 

 point on a bicircular quartic whose four real foci S, S', H, H' lie on a circle, and 

 are such that the lines S S', H H' meet inside the circle, the circles S P S'. H P H' 

 make equal angles with the tangent at P.' 



To many of the properties of spherical curves correspond properties of curves on 

 a conicoid. To obtain these we project stereographically from the sphere on to a 

 plane, project orthogonally on to another plane, and then project stereographically 

 on to a suitable conicoid. For example : ' If two curves of the fourth degree on an 

 ellipsoid both touch the generating lines through four given real coplanar points, 

 the tangents at a point of intersection of the curves are parallel to conjugate 

 diameters of the indicatrix at that point.' 



