a Surface of the Second Order. 37 



lines of the second and first kinds; and they may be considered 

 as correlative members of the same species. The number of di- 

 stinct species is thus — — or x-, according as n is odd or even ; 



for n = 3 we have the single species (2, 1) or (1, 2) • for ?i = 4, 

 the two species (1, 3) or (3, 1), and (2, 2) j for n=5, the two 

 species (4, I) or (1, 4), and (3, 2) or (2, 3) ; and so on. Thus 

 for n=3, the species (2, 1) is represented by an equation of the 

 form 



(a, b } cj\ )f iyv+(a' } b', c'JX, fi)*p = 0, 



which belongs to a cubic curve in space. To show a posteriori 

 that this is so, I observe that the equation expressed in terms of 

 the original coordinates (x, y, z, w) is 



x(a, b, c$x, yy + z(a' } b', c'^jr, ?/) 2 =0, 



which by means of the equation xw—yz — of the quadric sur- 

 face is reduced to 



(a, b, c£x, yf + a'xz -f 2b'yz + c'yw = ; 



and this is the equation of a quadric surface intersecting the 

 quadric surface xw—yz=0 in the line x=0, y = 0; and there- 

 fore also intersecting it in a cubic curve. 



For n = 4, I take first the species (2, 2) which is represented 

 by an equation of the form 



(a, b, cj\, //)V + 2(V, b' c'J\, fi)*vp+ (a", b", c"J\, fi)Y = } 



which in fact belongs to a quartic curve, the intersection of two 

 quadric surfaces. For, reverting to the original coordinates, the 

 equation becomes 



(a, b, cjx, yfx* + 2(a', b', <fjx, y) 2 ^+(a", b", c"Jx } y)V = 0, 



which by means of the equation xw — yz=0 of the quadric sur- 

 face is at once reduced to 



(a, b, cjx, yf + 2a' xz + ^b'yz + 2c'yw + afz* + 2b" zw + c"iv 2 = 0, 



which is the equation of a quadric surface intersecting the given 

 quadric surface xw — yz=0 in the curve in question. 



Consider next the species (3, 1) represented by an equation 

 of the form 



(a, b, c, dj\, fijv+(a', V, c', d'JX, /.cfp = 0, 



which is the other species of quartic curve situate on only a 

 single quadric surface. Reverting to the original coordinates, ths 



