293 
762] CONTACTS OF A LINE WITH A SURFACE. 
26. e 3 b 3 g = e 3 b 3 2 + e 3 g e = 2n + 3n (n — 2), =n (3n — 4). 
e 3 b 3 g, = ft (3ft — 4), is the order of curve of contact of the 3-pointic (chief) tangents 
which meet a given line. 
Parabolic tangents are coincident chief tangents. 
No. of 4-pointic parabolic tangents = 2n (n — 2) (lift — 24). 
27. Order of parabolic curve = 4n (n — 2). 
Order of regulus formed by parabolic tangents 
= 2 n (n — 2) (3« — 4). 
The parabolic curve and curve of contacts of an e 4 tangent meet in 
4n (ft — 2) (lift — 24) 
points, i.e., they touch in 2ft (ft — 2) (lift — 24) points. 
28. Umbilici. No. is = 2ft (5ft 2 — 14ft+ 11). 
29. No. of points at which the chief tangents being distinct are each of them 
4-pointic, or, what is the same thing, No. of actual double points of 
curve e 4 , 
= 5ft (7ft 2 - 28ft + 30), 
ft = 3, No. is 15 (63 — 84 + 30), = 135, viz. this is the number of points of 
intersection of two of the 27 lines; or, what is the same thing, the number 
of triple tangent planes is = 45. 
30. No. of parabolic tangents which have besides a 2-pointic contact is 
= 2 ft (ft — 2) (ft — 4) (3ft. 2 + 5 ft — 24). 
31. No. of double tangent planes such that line through points of contact is at one 
of these points 3-pointic 
= ft (ft — 2) (ft — 4) (ft 3 + 3ft 2 + 13ft — 48). 
32. No. of points where one chief tangent is 4-pointic, the other 3-pointic and (at 
another point of the surface) 2-pointic is 
= n (ft — 4) (27ft 3 — 13ft 2 — 264ft + 396). 
33. No. of points where chief tangents being distinct are each of them at another point 
of the surface 2-pointic is 
= ft (ft — 4) (4ft 5 — 4ft 4 — 95ft 3 + 99ft 2 + 544ft — 840). 
34. The curve of contacts b 3 of an e 32 tangent has with the parabolic curve 2- 
pointic intersections only, and these are at the points for which the chief 
tangent is (at another point of the surface) 2-pointic. 
35. The curve of contacts b 3 of an e 32 tangent has, with the curve of contacts of 
an e 4 tangent, 2-pointic intersections at the contacts of an e 5 tangent; and 
has also simple intersections with the same curve, 1° at the contacts b A of 
an e 42 tangent, 2° at the points where the chief tangents are e 4 and e 32 .