Railway Curves. 133 



and the angles deflected out of the direction of tangents are evidently 

 the compliments severally of these angles at the bases aforesaid. 

 In a compound curve, the radii approximate equality as the angles 

 a and B do so ; when a= 8 then the two arcs merge into one simple 

 arc. 



Note. — The extraordinary figure S is drawn to show an S curve 

 DGB of equal radii, to unite an arched unto a straight tangent, the 

 arc of contact being only a small portion of a very small circle, on 

 which osculating points for two arcs of differing radii might, however, 

 be chosen between D and A" on the arc of contact ; and figure T, 

 by way of showing the general applicability of the formulae, is drawn 

 with two symmetrical $ curves of equal radii, to which the above 

 formulae are applicable ; and the arc of contact of each is also drawn, 

 to show that the principle of that also applies to all sorts of sym- 

 metrical figures formed by the ogee curve. 



Intermediate Straight. 



Investigation of a rule for the direct calculation of a common radius, 

 to admit a given length of intermediate tangent. 



Fig. V. 



CP= Vrt+h? .-. OP=2 V^+A 2 



PN = r (sin. a+sin. 8) ; VW = NO = a/OP 2 -NP 2 



Let in be put for sin. a + sin. 8 = 2 sin. ' cos. ~^ 



Then PN = rm 

 (Note. — If tangents are parallel m = 2 sin. a because a = /3) 



VW 2 =OP 2 -PN 2 



■ VW 2 - 4 h 2 



and VW 2 -4A ! =4r-m-r . •. r- = 



4-wi 2 

 But (see fig.) VW 2 = d-r (cos. a+cos. ft) = 

 Let n be put for cos. «+cos. /3 = 2 cos. "*"" cos " 



2 2 



Then YW = d-rn 



(Note. — If tangents are parallel, then n = 2 cos. a.) 



