ny THEORY OF COLLINEATIONS. 
The cross-ratios along the sides of the invariant triangle 
AA’A” are as follows: k is the cross-ratio along AA’, k’ is 
that along AA”; let that along A’A’ be k,, then k’=kk, 
(Art. dei): These three k’s each approach the limit 7; the 
limit of = is t, as above. 
To find the value of eee , we proceed as follows: re- 
place k’ by kk,, and find the values of k and k,. Let g be 
the line joining P and P,, a pair of corresponding points, and 
let g cut the sides of the invariant triangle in L, M, N, respec- 
tively, Fig. 20. Projecting the cross-ratio k from A” on g 
LP MP, 
LP MP* 
 —MP NP _ LP+ML | MP+NM 
7a MPN WLP Lome (MPAONMe 
=) eae 
we have k= In like manner we have 
(ML.MP:+LP.NM+ML.NM) —k (ML.MP+NM.LPi+NM. ML) 
LP:\.MP+ML.MP+LP:.NM+ML.NM 
ML.MPi+NM.MP-k(ML.MP+NM. MP) 
ale aly MP\. NP ; 
When A, A’ and A” are consecutive points on s, ML and NM 
are infinitesimals of the same order as AA’ and may be 
. k—2k+1 
made equal to each other. The ae becomes 
eds? 
k2—-2k+1 , k (MP+MP:\ NM _ 1-k 
eds? as \eenea) FE ds 
MP+MP,\ NM 
MP. cl ds * 
= — +ht where —/h is the 
é k/-k ; 
The expression ——— easily be- 
limit of > (; oa 
t 
comes _. 
. . 1 9 
Since c is the curvature of s at A, | =a, the radius of cur- 
vature of s at A, —— ean ag ore the direction cosines of the 
tangent to s at A, oa may be replaced by « and £, respec- 
d?A 
, d2B foe ye 
tively ; also a?+(6’=1. a, and a, are the direction 
cosines of the normal tos at A, and may be replaced by «’ 
