( J8I ) 



i'+ij) 



— h'=zO.') (C) 



The equation of an osculating plane of C"' is 



^3 _ 3 (.^. J^p)t^ j^Sy t — zzzz 0. 



The equations of a tangent to C^ are 



f - 2 {.V -i-p)t^ y = 0, (./■ + p) e — 2yt-\-z = 0, 

 or 



y = 2 (.. -^p)t~ t\ z = 3 (.. + /)) ^^ - 2i\ 

 Substitution those values of y and z in the equation of S, we find 

 an ecjuation of order n in ,v 



= a^ -f- ^j ,1' -|- ^2 ,r^ -j- <'/a .1'^ -|- etc. 

 where 



a, = 3 pP + 4 ^>2>' f — 2 f — 4: hpf -\- etc., 

 rtj =r 4 hpt + 3 ^^ — 2 hf + 8 hpe — 46^-^ + etc., . ( D) 

 a^z=a-\-4:ht-{-A. hf + etc. 

 The discriminant of this equation is of the form 



As (7„ and a^ contain respectively f and t as a factor, whilst (p 

 and \]i are in general not divisible by t, the discriminant is divisible 

 by f or the discriminant has two roots ^ == 0. As to every root 

 of the discriminant (except the particular n{}i — J)-fold ones) corre- 

 sponds a common tangent of C^ and ^S', the A-axis counts here 

 for two coinnio]! tangents of C^ and S, or the tw^o consecutive 

 tangents of C^ lying iii tJie common tangential plane of D and ;b 

 both touch also ^S'. 



The discriminant is a determinant, which gives when developed 

 according to the elements of the first two columns 



|2 ««„ a.^ — (»— 1) r/^^j ip^ 4- if a;' (f, + mi^ a, <f, + a,^ tp^ . {E) 



§ 6. If the A'-axis does not coincide with one of the intlexional (or 

 principal) tangents of >S' in the origi]i P, then the F-axis can be taken 

 so that h=iO; to this end we have but to take for )'^-axis the diameter 

 of tlie indicatrix conjugate to the A'-axis. The expressions for the 

 coordinates of a point on C^ will not change if we now also 

 take for plane .v = the plane determined by the new I^-axis, and 

 0]ie of the two tangents of C^ meeting the F-axis outside P, and for 

 plane at intinity the osculating plane of C" in the point where C' 

 touches the new plane .v = Ü, whilst for plane y == is taken the 



1) Salmon, Three Dim. § 204. 



2) Salmon, Modern Higlier Algehra. § 111. 



