192 



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

 E itself; the other two determine a pair of oscultating planes: 

 3 t2=x — 3 toy + z — t2' = 

 3 t3= X — 3 ts y + z — t^3 = 

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

 would the three osciiltaing planes be linearly depenflent. Therefore, these two 

 planes cut out from E a pair of lines intersecting in P, through each of which 

 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. 



Eqviation (3j 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 t lirough P. 



(3j 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 

 the.se ecjuations follow x = x, y = x'-, z = x^; therefore P is on K and is therefore 

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

 only onef 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: 



// (ill the (lelerminants of ^[ rnriiiih, E itself osculates K. Thromjh every 

 poinl of E at which D>0 there intvj be drawn 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 he drawn a unique line on E through 

 which pass two osculating planes] through every other point of E (including the 

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

 planes. 



§6 



Examples: 



1. E: 3x + 3y — 2z — 5 = 



M: :|3 3 —15 1 



i , A = — 27, B = 54, C =—81. 



'1-3 —6 3 1 



(2) f- — 3 t + 2 = ti = 1, ti = 2, s = 3, p = 2. 



L: X = 1 + u, y = 2/3 + 3 u, z = 6 u. 



Through L pass the two osculating planes: 

 3 X — 3 y + z — 1 = 0, 12 X — 6 V + z — 8 = 0. 



*That is, two equal linear factors distinol from the third hnoar factor. 

 fThat is, three equal linear factors. 



