194 



through which pass osculating planes 



3 X — 3 y + z = 1, 48 x — 12 y + z = 64 



c) Consider the point ( 0. 0, 1) on E: 



II o; 



M': ! A' = 0. B' =— 1. C' = 0, D = — 3. Xolinc. 



;0 —1 



In case E is an osculating plane different from z = 0, and P is on E,t = — a/b 

 is a root of equation (3), which can consequently be depressed to the ciuadratic 



(4) (bt)2— (a + 3bx) bt + a ( a + 3 bx) + 3 b= y = 



and the number of osculating planes through P which are distinct from E is 

 ecjual to the nimiber of real roots of this equation which are different from — a/b. 



d) Consider again the point P ( — 2. 1, 10) on 3x — 3y + z = 1 

 (4) t2 + 7t + 10 = t, = — 2 to = —5 



both different from 1: therefore two lines as before under a), 

 c) Consider the point P ( — 1, 0, 4) 



(4) t= + 4 t 4- 4 = t = — 2 

 one root flifferciit from 1; therefore one line 



L: x = — 1 — u, y = 11, z = 4 + G u 



through which pass osculating planes 



3x — 3y + z — 1 =0, and 12x + Gy + z + N = 



§■• 

 The case where E is an osculating plane may also be treated geometrical- 

 ly by making use of certain considerations given in a later chapter of Eisenhart's 

 book. The ecjuation of the envelope F, of the osculating planes to K is ob- 

 tained by equating to zero the discriminant D of eciuation (3): 



(5) 3x-y- -I- Gxyz — 4x'z — z- — 4y' = 



Since x = t, y = f-, z= t', satisfies (5) for all values of t, K itself lies on F; 

 in fact, K is the edge of regression of F. 



A given osculating plane E not only touches F, but in general cuts out from 

 F a plane curve H. which passes through the point where E osculates K. Every 

 osculating plane different from E, cuts E in a line tangent to H; conversely 

 through every straight line on E tangent to H passes an osculating plane w^hich 

 is distinct from E.* The curve H divides E into two or more regions throughout 

 each of which D is always positive or always negative and therefore serves to 

 classify the points of E into those through which can be drawn two lines, or one 

 line, or no line respectively, which is the intersection of two osculating planes, 



'Unleis perchance this line is a part of H, as is the case with the x-axis on the plane z = 0. 



