( 180 ) 



Aa -{- B C y — B 



II ^=. or X = • — 



^ G A 



must now pass into x = O for / = O, so that C and B must contain, 

 after the change of variables, t as a factor, A not being divisible 

 bj- t. As the projection on the plane ,t = of the tangent lying in 

 this plane can be an}^ arbitrary line and as C vanishes for t = 0, 

 D and E must also vanish for / = 0. In the equation • (j5') the coef- 

 ficient of A"' will be divisible by / and the coefficient of .v^ divisible 

 by t"-'. 



According to Salmon ^) the discriminant of equation (/>') will 

 be divisible by /"("— 0. F'or each one of the 7\ particular tangents 

 of C\ which are at right angles with the A'-axis n{n — 1) roots of 

 the discriminant of equation [B) become equal. That discriminant 

 possessing 2i\ n [n — 1) roots, there are left i\ n {n — 1) roots, to each of 

 which corresponds an equation {B), possessing two equal roots. So 

 there are '}\ n {n — 1) tangents of 6\ which also touch S. As S 

 possesses no multiple curves the class m is 7i {n — 1). The number of 

 connnon tangents of 6'i and S is thus as before mentioned 



r^ rn. 



§ 5. So far we have supposed that C^ occupies no particular 

 position with respect to >S'. For particular positions of C\ two or 

 more of the common tangents of (\ and S cau become consecuti\'e 

 tangeiits of (\. l^et t be a tangent of Ci touching >S' in /-*, and let a 

 tangent of (\ consecutive to t be also a tangent of >S. The developable 

 />! formed by the tangents of C■^ and the surface >S' will touch 

 in P. We shall now investigate when the contact is ordinary and 

 when stationary. 



For simplification I assume for C\ the twisted cubic C^ 



The equation of the developable D^ or jy is now : 



,^ _ 6 (,r 4- />) y z + 4 f + 4 (.. + pr z - 3 U + i>y f = 0, 

 or 



= e--f-^+^te {A) 



4: p 



If we choose for point P where the surface S touches D* the 

 origin of the coordinates the equation of S is 



0=z -]- a.v' + 2 /i ,77/ + hy^ + etc (B) 



The surfaces jy and S have stationai-y contact in the origin when 



1) Modern Higher Algebra, § 111, note. 



