ON A SIMPLE PLAN OP EASING RAILWAY CURVES. 93 



Let the tangent offset be represented by t = UP. 



t x (2 R - t) = (*)* (Euclid, Book iii., Prop. 35) 

 When t is very small, t x (2 P - r) == t x 2 P 



o -®" 



6 \2/ -ffi 



(I) 



6 \2/ ~ P 

 3 



• 6 5 



<f)' 



ButZ= 4 



6 P 

 4 

 . • . Y = s t where t is small. 

 o 



Proposition : — To find the position of the circle. 



f = 3mx 2 



^| = 3 m X 2 . (at P) 



But m = 



6 RX 



dy _ ZJ& 3X _ X 



'"' cfe 6-B-T 6 P ~ 2P 



PL _^dy PL ^dy PL X^ 



LG dx° r R ' ~dx° r R ' 2R 



To show that JOT = J TP = J F = 7* 



Let a? x and 2/1 be co-ordinates for points in circle for origin ; 

 and let MN =h 



(* - f ) 



But y x == h + 



2R 

 Y = h + %f£ but 3TI 6 iSJF 



