]90 



There is no line in space through which pass three such planes: 

 3ti=x — 3tiy + z — ti^ = 



3t,=x — Bt.y + z — t.,3 = ti, to, ts, all different, 



3t3=x — Stsv + z — ta' =0 

 ff)r the determinant of the coefficients of x, y, z, is equal to 

 9 (ti — to) (ta— ts) (ts — t,) ^ 



and therefore the three planes are linearly independent. 



§2. 



Giv^en a real* plane 

 E: ax + by + cz + d = a, b, c, not all zero 



and the cubic 

 K: X = t y = t= z = t' 



The equations of the planes which osculate K at any two distinct [)oints 

 Pi ft,), Po (to), ti ± to, determine a line 



L: X = s/3 + u y = p/3 + su z = Spuf 



where s = ti + t2, p = ti to, and u is a parameter. 



That L lie on E, it is necessary and sufficient that 

 (1) a s + b p + 3 d = 



b s + 3cp + a = 



Write the matrix of the coefficients of cciuatioiis (1) 



.M: 



and set 



A = 



b 3d 

 3c a 



§3. 



Suppose A ± 0. Efiuations (1) have the uiii(|ne solution: 

 s = C A p = li .V 



whence ti and to are the roots of the ([uadratic eciuation 

 (2) A t^ — C t— B = 



Therefore if two distinct planes osculat(! K and intersect on E, (in case 

 .\ ± 0), it is necessary that 4 A B + C- > 0, and that the parameter values 

 of their points of osculation be the roots of the (juadratic- (2). 



This condition is also sufficient, for if A =t 0, and if 4 A B + C- X) ecpia- 

 tion (2) determines two real numbers ti and {■•, and if we set s =ti + t;, 

 p = ti t;, these numbers satifsy eciuations (1;, and tlic line x = s/3 + u, 



*Thi.s problem is treat Jt! thro'Jghoul .t* a problem in (ieimo'.ry, not one in AI.rcl)ri. 

 tThes? equations hold even if one of ti, u, is zero. 



