190 



'llioro is no line in space through which pass throe such |)l;mes: 



3tr'x — 3t,y + z — t,^ = 



3trx — 3t.,y + z — t-/ = t,, t,>, t,, all difTerent. 



3t:rx — 3t3y + z — ts' = 

 for the (ictcrniiiiant of the coefficients of x, }', z, is equal to 



9 (t, — t,) (t, — ta) (t3 — t,) =t 

 and therefore the three planes are linearly independent. 



§2. 



Given a real* plane 

 E: ax + by + cz + <1 = a, b, c, not all zero 



and the cubic 

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



The ecjuations of the planes which osculate K at any two distinct points 

 Pi (ti), p.. (ts), ti ± t., determine aline 



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



where s = ti + t2, p = ti t-.., 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 equations (1) 



a 3d I 



C = 



b 3d I 

 3c a 



b a| 

 §3. 

 Suppose A ± 0. Expiations (1) have the unitpie solution: 

 s = C/A I) = — B/A 



whence ti and to are the roots of the cpiadratic etiuation 

 (2) A t^ — C t — B = 



Therefore if two distinct planes osculate 1\ and intersect on E, (in case 

 A ± 0), it is necessary that 4 A V> + C- > (I. .iiidtliat the paramet<>r values 

 of their points of osculation be tlie roots of the (lu.iilrat ic I'Ji. 



This condition is also sulliciciil , for if .\ =±= 0, and if 1 A H + C"-' >() ecpia- 

 tif)n (2) determines two real numlxMs ti and t j. and if we set s =ti + tj, 

 p = ti tj, these numl)ers salifsy e(|Uations (1). and t lie line \ = s 3 + u, 



•This prjblein is troat ■<! iliidjuhoiii ;i-< :i piDhlcin in ( Ir uni: r.\-, iml inu' in Al','!)ri. 

 tTlies ■ ('(jualions hold (•\cn if mu' of li, Ij, is zi"ro. 



