50 KANSAS UNIVERSITY QUARTERLV. 



AC'+AB'— 2AC.AB cos<^ 



I sin^^ AC ' cos ■-<^ — 2 AC . AB cost^+AB" 



sin2^~"" AC2 sin3<^ 



^ AB— AC cos<i 



... cot 6^ -^^. 



AC sin<^ 



In like manner we find, 



AB— AC, cosd) 



cot e, 



AC^ sin<^ 



AB AB 



.-. a = COt6'. COt^=-— ;; : — COt^ . ^^. --rC0t<i = 



^ ACi sin<^ ^ AC sin<^ ^ ^ 



AB/ I I \ AB , , sin</) 



-r — I — ^ ■— =; 1= -." -a . or a ===—-— a. 



sinpVACj AC/ sin^ AB 



s a I 



multiplier being 



(9) 



Hence a is a constant multiple of a; thus a =pa, the constant 

 sin^ 

 AB" 



10. Relations Unaltered by the Transformation T'. The 



cross-ratio of the pencil A(BCPP,J is k and it can l)e shown 



that k is also equal to the cross-ratio of the four perpendiculars 



from P and P, on the invariant lines AB and Al. 



Let the feet of the perpendiculars from P and Pj on AB be p and 



p,, and the feet of the perpendiculars on Al from P and P, be 1 and 



, sinPAB sinP.AB Pb P.b, 

 1,. ilien k== , -_^- ^ • ^— ts aTT-^' ?=rr : vs , • 1'^ another 



^ smPAC sinP,AC PI P,l, 



P 1 PI . 



form this becomes ^-^-^ ,= k -r- This means that the cross-ratio 

 P, b, Pb 



of these four perpendiculars is constant for all pairs of correspond- 

 ing points in the transformation T'. 



There is another very useful relation among certain perpendicu- 

 lars which can be shown to be constant for all pairs of correspond- 

 ing points. Let Bl' be a line through B parallel to Al. Let the 

 feet of the perpendiculars from P and P, in Bl' be h and h,. We 



P h Ph 

 proceed to show that -,\ ^ - v^rr^^'^^j where m is constant for all 

 Pjb, Pb 



pairs of corresponding points in the transformation T'. Denote 



the angles PBl' and P^Bl' by y and y, respectively. Then 



P,h, siny, , Ph siny ^^ P.h Ph siny siny, 



— ^ — Lr=: ^^" and :^-=-^^ Hence ', — - =-- -^ — l ~^—- 



P^bi sm^i Pb sm^ P^b, Pb sm^ sm^, 



Since y=i8o"--((94-(^) and yj =i8o» — (6*1 +<^), we have, 



