ELISHA MITCHELL SCIENTIFIC SOCIETY. 109 



DL 



DL. Then LOG = a = LTK and = number of loo 



100 



arc DL 



feet stations in arc DL .'.0° = total angle turned 



100 

 D°j 



in length DL =: a := by ^10); whence, 



200 



arc DL = >^ ^ = >< length of spiral SEL .... (11). 



From this we have, since for flat arcs, LD = LE nearly, 

 LE = y2 SEL, or point E is nearly at middle of spiral. 

 The distance CD = q between SY and the circular curve, 

 may then be regarded as the offset at the middle of the 

 transition curve from circular curve to tangent SY. 



By aid of (9) above we can deduce two more useful 

 formulas: 



20 N 



^ = (12), 



D° 



20 N^ 



N^ = .9 =r (13). 



D° 

 From eq. (3) we have, developing sin (as^) by a well- 

 known formula, 



dx = ds (as- — \ (as^f + -^ (as'^f — , etc. ) ; 

 whence integrating and noting that the constant is zero, 

 since x := when s ^ o^ and placing for brevity a for its 

 equivalent as^ (eq. 2), we have, 



a' a' 



X = sa{% \ , etc.) (14). 



42 1320 



Developing dy = ds cos {as^} we deduce similarly, 



a^ a' a' 



y = ^{^ \ f-, etc. ) (15). 



10 216 9360 



