191 



y = p/3 + su, z = 3pu, lies on E and i.s tlie intcrscftion of two planes which 

 o-ifulate K. Moreover, since under these conditions, ti, to, s, p, are uniquely 

 determined, there is no other line on E through which pass two planes which 

 osculate K. 



If 4 A B + C- <0, eciuation (2) has one real root, or no real root, and 

 there exists on E no line through which can be drawn two planes which 

 osculate K. 



§4. 

 Suppose A = but B and C are not both zero. Ecpiations (1) have no 

 solution. There is no line on E through which pass two osculating planes. 

 The results of §3 and §4 may be combined into a theorem: 



// not (ill the dclerminanta of M vanish, there is exactly one line on E, or 

 no line on E, through which pass two osculating planes, according as the 

 equation At- — C t — B = O has or has not two real roots. 



§5. 



Suppose all the determinants of AI vanish. Under these conditions E 

 itself osculates K; for, in order that the equation a t + b t- + c t'' + d = 

 have three equal linear factors, it is necessary and sufficient that A = B = C = 0. 



The plane z = osculates K at the origin. If E osculates K, the point of 

 osculation is (-a/b, aVb^ -aVb^*) if b ± 0, but (0, 0, 0) if b = 0. 



The number of osculating planes which can be drawn to K from a point 

 P (x, y, z) is equal to the number of real roots of the equation in t 

 (3) t^ — 3xt" + 3yt — z = 



Write down the matrix* 

 1 — 



M' 



and set 



y 



B' 



A' = 



— X y 



then the discriminant of (3) is 



12 A' 



C = 



D 



B'l 

 I B' 2C'| 



The points of the plane E may be classified as follows : 

 (1) Suppose at a point P of E, D>0. 



'Reduced from 



— 6 X 3 V 



6 y — 3 z 



