292 
on schubert’s method for the 
[762 
intersections with F of the line of intersection of their planes, the order of the scroll 
generated by the lines which meet three of the sections is 2n 3 — 3ft 2 ; this scroll meets 
the fourth section in n( 2ft 3 — 3ft 2 ), = 2w 4 — 3n 3 points; or we have this number of lines 
meeting each of the four sections. But among these are included 3n 2 (ft — 1) lines 
which have to be rejected, viz. the sections 1 and 4 meet in n points, each of which is 
the vertex of cones through the sections 1 and 2 respectively; these cones meet in n 
lines, which are to be disregarded, and in n 2 — n other lines, and we have thus n (n 2 — n), 
= n 2 (n — 1) lines; and similarly from the intersections of 2 and 4, and from the inter 
sections of 3 and 4, n 2 (n— 1) and ft 2 (ft — 1) lines, in all 3ft 2 (ft — 1) lines. Hence the 
number of lines meeting the four sections is 
2ft 4 — 3 n 3 — 3 n 3 + 3 n 2 , = 2ft 4 — 6ft 3 + 3ft 2 ; 
taking any one of these for the line of the subject, the remaining points 5, 6, ..., i are 
any i — 4 of the remaining n — 4 intersections, or we have the required formula 
Pip 2 p 3 Pi = ft 2 (2ft 2 — 6ft + 3) (ft — 4)...(ft — i + 1). 
The four numbers p 2 p 2 , p^p^ps, PiP^PsPi, & for any line of the table being now 
known, we can at once calculate the required values e 2 g s , &c., as the case may be ; for 
instance, 
i = 5, e 5 = —10p 2 p 2 = — 10ft 2 (ft — 2) (ft — 3)(ft — 4) 
— \0p l 2 p 2 p s — 10w 2 (ft — 1) (ft — 3) (ft — 4) 
+ ^PiPiP?,Pi + 5ft 2 (2ft 2 — 6ft + 3) (ft — 4) 
+ 106r + 10ft (ft — 1) (ft — 2) (ft — 3) (ft — 4) 
= 5ft (ft — 4) (7ft — 12). 
In fact, throwing out ft (ft —4), the remaining terms give 
— 10 ft 3 + 50ft 2 — 60ft 
— 10?? 3 + 40ft 2 — 30ft 
-1- 10ft 3 — 30ft 2 + 15ft 
+ 10ft 3 - 60ft 2 + 110ft - 60 
35ft-60, =5 (7ft-12). 
And we obtain in like manner the other formulae of the table. 
The remainder of § 33 contains investigations of less systematically connected 
theorems, and I quote the results only. 
25. If on the surface F n there is a curve order r, then of the tangent planes of F n 
along this curve there pass r(n — 1) through an arbitrary point of space; aliter, 
class of torse is =r(n — 1). 
In particular, for curve of 4-pointic contact, r = ft(llft—24), class of torse is 
= ft (ft-1) (lift - 24). 
No. of taDgent planes through line, or class of surface, =n{n — l) 2 .