134 Railway Curves. 



VW 2 — 4- h 2 

 . \ VW 2 = d?-2 drn+t* n 2 ; but as r 2 = v 



4 -TO 2 



,_ d 2 — 2 c£ro+« 2 ^ 12 

 4- TO 2 



, » 2 "■« j 2 ?ze£ d 2 - /i 2 



)' 



4 — to 2 J 4 - to 2 4 m 2 



Let p be put for d? — h 2 = (d+2 h) (d — 2 h)a,nd cancel com. den. r. 

 4 — (m 2 + n 2 ) r 2 = — 2 ?icZ r + p 



. • . r 2 = — r + tJ 



if q be put for 4 - (m 2 + n 2 ) q q 



.-. r= Vnci\ 1 + M — n d (the rule required.) 

 For the length and position of the chords we have — 



Sin. 



g = 



PN = 



rm 





= 



m 



cos. 







OP 



2 r sec. 



9 



2 





Cos. 



6 = 



r 



■ _ or 



h 



tan. 







r 







The angles at the bases of the isosceles triangles DPE AOD, are 

 severally the mean between a and 8 + 9 and (3 and 8 + 9. (See Fig. V.) 



(i.e) EDP = a + 8 + 9 and OBA = /3 + 8 + 9 



Chd. AD = 2 7- cos. EDP and AB = 2 r cos. OBA 



The chords may be deflected from the tangents at the counpts. 

 respectively of these angles (EDP and OBA). 



Note. — If the tangents are parallel, then 



. 9 PN 2 r sin. a . _ 



Sin. 8 = = = sin. a cos. 



OP 2r sec. 

 Tangent 9 being as before _ 



