66 PROCEEDINGS OF THE AMERICAN ACADEMY. 



the point corresponding to A (enveloped by a) is 



5 = X (0 A) F - A. 

 The distance from A to B is 



—^ (pA \X(,I>A)F-A\) ^ 



Thus A B = A a. This shows that ^ a is the distance, measured 

 along A F, from A to the point of intersection of a with A F. 



16. Distance from point to line. We define the distance from 

 a point A to a line r as such a function of their coordinates that one 

 of the quantities being fixed and the distance held constant, the other 

 satisfies a linear relation and such that this distance is invariant 

 under the transformations leaving distance between two points un- 

 changed. Such a function is 



Ar - — - .... (18) 



{4>A){p r) 



If r joins two points, B, C this can be written 



(.4 BC-<l>p) {A <j>) (BCp) + iBct>) {CAp)-{- {C <i>) {A B p) 



Ar 



{A 0) (B C p) {A <t>) {p B C) 



Dividing numerator and denominator by {A 0) {B </>) (C 0), this 

 becomes 



— B~C+C~A + AB , ^. 



Ar = = . . . (19) 



BC 



This expression shows that A r h invariant under the distance 

 transformations. Conversely of 



J7= AW 



there is a transformation changing .1 r into A' r' . For let B, C 

 be two points of r. Take on / two points B', C such that 



AB = A' B' 

 AC = aTC', 



then 



BC = B' C, 



and a transformation of the kind desired can be obtained. 



