192 



Equation (3) has three real roots, ti, to, t^; one of these ti say, determines 

 E itself; the other two determine a pair of oscultal iii^ planes: 



3 t2= X — 3 t.. y + z — t./ = 



3 t3= X — 3 t, y + z — th = 

 distinct from E and from each other; their intersection does not lie on E, else 

 would the three oscvdtains planes be linearly dependent. Therefore, these two 

 planes cut out from E a pair of lines intersect iiiji in I', throujili each of whiidi 

 passes a pair of osculating planes. E itself and one other. 



(2) Suppose at a point P of E, D = but A', B', C, are not all zero. 

 Equation (3) has only two *roots, both real; one of these determines E 



and the other determines an osculating plane distinct from E which intersects 

 E in a line through P. 



(3) Suppose at a point P of E, A' = B' = C = 0; then is D = 0. 

 There is in fact only one point on E at which A' = B' = C = 0, for from 



these equations follow x = x, y = x-, z = x'; therefore P is on K and is therefore 

 the point of osculation of E and K. Under these conditions equation (3) has 

 only onet root and that determines E. 



(4) Suppose at a point P of E, D < 0. 



Equation (3) has only one real root and that determines E. 



These results may be combined into a theorem: 



// all the deter niinantsi of M vanish, E itself osculates K. Thruuijh every 

 point uf E at trhicli D>0 there niaij he ilrawn a unique pair of lines on E, through 

 each of which pass two osculating planes; through every point of E at which D = 

 {except the point where E osculates K) there may be drawn a unique line on E through 

 which pass two osculating j)l(ines] through every other point of E (inchuling the 

 point of osculation) there exists no line on E through which pass two osculating 

 plancK. 



§6 



Examples: 



1. E: 

 M: 



3 X + 3 y — 2 z — 5 = 



3 3 — 15 1 



, A = — 27, B = 54, C = —81. 

 2, s = 3, p = 2. 



1 3 — () 3 



(2) t- — 3 t + 2 = t, = 1, t 



L: X = 1 + u, y = 2/3 + 3 u, z = u. 



Through L pass the two osculating planes: 

 3 X — 3 y + z — 1 = 0, 12 X — 6 y + z 



8 = 0. 



*That is, tw.> (■<iii:il linear factors distinct from tin- third linear factor. 

 tTliat is, three ei|iial linear factors. 



