577 



u = 0, - = X, and — = Y. 



As P, Q, R, are proportional to the cosines of the angles 

 which the normal at the point (x,y) makes with the axes ; it 

 is easy to shew that if we describe a sphere with its centre 

 at the origin and radius = 1, x and Y will be at the same time 

 the rectangular spherical coordinates of that point on the 

 sphere at which the tangent plane is parallel to the plane 

 touching the surface u = o at the point (x,y). Thus, to 

 every point on the latter corresponds a point on the former : 

 and a succession of points, or a line of any kind, on the sur- 

 face u = o is in general represented by a succession of points, 

 or a line upon the auxiliary sphere. 



As plane curves have been classed according to the de- 

 grees of the equations by which they are represented, so 

 curves traced on any given surface may be advantageously 

 distinguished by the degrees of the equations in x and Y 

 which define them. For the properties of a curve, traced 

 on the surface u=o, and characterized by an equation of 

 the «th degree between x and y, are, so far as we regard 

 only the relations of normals or tangent planes along it, 

 identically the same as those of the spherical curve which 

 has the s^me equation. But the analogy between spherical 

 and plane curves of the nth degree has been already estab- 

 lished. Instead, then, of looking upon the shortest ihies on a 

 surface as analogous to the right Hne, Mr. Graves directs 

 his attention to the line defined by the equation 



ax + br + 1 = 0, (A) 



the geometrical character of which is, that the normal at any 

 point on it is always parallel to a fixed plane. Systems of 

 such lines upon any surface possess, in general, those pro- 

 perties of right lines which have been termed projective. 



Thus, for instance : "If four lines of the first degree, di- 

 verging from the same point on a given surface, be cut in 



