36 Mr. A. Cayley on the Carves situate on 



thing, = Xp : p,p : vX : fiv. The four quantities (X, p,, v, p) are 

 for symmetry of notation used as coordinates ; but it is to be 

 throughout borne in mind that the absolute magnitudes of X, 

 and p., and of v and p are essentially indeterminate; it is only 

 the ratios X : p, and v : p that we are concerned with. 

 An equation of the form 



that is, an equation homogeneous of the degree p as regards 

 (X, p), and homogeneous of the degree q as regards (v, p), repre- 

 sents a curve on the quadric surface; and this curve is of the 

 order p + q. In fact, combining with the equation of the curve 

 the equation of an arbitrary plane 



this equation, expressed in terms of the coordinates (X, p., v, p), 



AXp + 'BpLp + CvX + Dpv = 0; 



or, as it is more conveniently written, 



(C,-DJ\,^)(v,p)=0; 

 |A,.B| 



and if from this and the equation of the curve we eliminate X : p. 

 or v : p, say the second of these quantities, we obtain 



(*J\, p,y(—AX-BpL, C\ + Dp,)i = 0, 



which is of the order p + q in (X, p.) ; and X : p, being known, v : p 

 is linearly determined. There are thus p + q systems of values 

 of the coordinates, or the plane meets the curve in p + q points ; 

 that is, the curve is of the order p + q. 



A linear equation A A. -j- B/i = gives a generating line, say of 

 the first kind, of the quadric surface, and a linear equation 

 O + Dp = gives a generating line of the second kind. And 

 by combining the one or the other of these equations with the 

 equation of the curve, it is at once seen that the curve meets 

 each generating line of the first kind in q points, and each gene- 

 rating line of the second kind in p points. 



Consider the curves of the order n : the different solutions of 

 the equation p + q = n give different species of curves. But the 

 solution (n, 0) gives only a system of n generating lines of the 

 first kind, and the solution (0, n) gives only a system of gene- 

 rating lines of the second kind. And in general the solutions 

 (p, q) and (q, p) give species of curves which are related, the 

 one of them to the generating lines of the first and second 

 kinds, in the same way as the other of them to the generating 



