126 



Raihvay Curves. 



Fig. 0. 



P0S=a-S 



ADO = FOS = "" 5 



2 2 



•ADB = «-ADO = a Jl? 



ABp = MPB == /3+^ 



2 2 



ABD = /3-ABP=^l! 



— 2 



The angle BAD = supt. of a _lA 



Fig. P. 

 ABP = ^_~L! = ABD + B 



— 2 



ODA= a + i80°- a = ADPi + n 



180°-a-S 



ADB = 



• . • remaining [_, BAD = 180°- 



180°-(a + B) 



2 



or, the angle BAD = 90° - '1±£ 



2 



The same in effect as Fig. M, if 

 the same |_ be taken as a, for 

 90° + « + /3 = 180 o_(180-a) + y3 



In like manner it might be proved that the same values respec- 

 tively attach to the [_ BAD, at whatever other point A', two arcs, 

 tangt. at B and D respectively, may osculate. If therefore a segment 

 of a circle be described through the points BAD, that is, if there be 

 drawn a segment cai)able of containing an angle of the proved value 

 of BAD, its arc will be the locus of osculation. 



Q. E. D. 



Cor. — From these two things, viz., the [_ of segment and the 

 length DB of its chd., the position of the tangents of the arc of con- 

 tact and its radius are determined. Thus — 



For setting out this arc on the ground. 



BD is the chd., and the position of the tangents of the arc of con- 

 tact is at once determined, for they intersect the line DB at angle 

 (T DB or T V BD) = BAD,— (the latter being the [_ in the alterut. 

 Segmt.) 



* Or it might have been proved by means of Fig. M, that BA"T) 

 has the same value as BAD — put x', y' for the contiguous angles, in 

 Fig. M, of the isosceles triangles whose bases are the chords BA'". 

 A'"D (as x and y are in Fig. P), and let h' in Fig. M serve as 

 auxiliary to the proof. 



Then,/ = ( 180 °-") + * 



y 



and BA'"D is their sum : 



,,_(180°— B) + p 



360° ■ 



a+ (3 , n a — B 

 L = suppt. 01 Z 



