194 



tlii-oufili wliicli pass osculat iiifi planes 



3 X — 3 y 4- z = 1 , 48 X — 12 y + z = 04 



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



1 o; 



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



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 consecjuently be depressed to the quadratic 



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



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

 equal to the number of real roots of this ecjuation which are different from — a b. 



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



(4) t^ + 7 t + 10 = ti = — 2 t2 = — o 



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



e) Consider the point P (— 1, 0, 4) 



(4) t^ + 4 t + 4 = t = — 2 

 one root different from 1; thei-efore one line 



L: X = — 1 — u, y = u, z = 4 + 6 u 



through which pass osculating planes 



3x — 3y + z — 1 = 0, and 12x + Oy + z + 8 = 



§7. 

 Tiie case where E is an osculating plane nuiy also be treated geometrical- 

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

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

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



(5) 3x-y- + 6xyz — 4x^z — z- — 4y' = 



Since X = t, y = t'-, 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 fiom 

 F a plane curve H, which passes through the point where ]•; 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 which 

 is distinct from E.* The cinve 11 divides 1'] into two or more regions throughout 

 each of which D is always |)ositi\i' or always iicgati\(' and therefore serves to 

 classify t he points of I'] into t hose t h rough w liiili laii be drawn two lines, or one 

 line, or no line respect ixcly, wliirh is the intei'sect ion of two osculating planers, 



•Unle<s pcrchanco this line is ii purt of 11, as is tlio case willi the x axis on the plane z = 0. 



