128 Railway Curves. 



Sin. A" : sin. B : : d : A"D = d sin. B cosec. A" 



Sin. A" : sin. D : : d : A"B = d sin. D* cosec. A!'=d cos. a cosec. A". 



But A"=A or DAB and BA"T"=supt. A- 5 (fig. M) 



2 



and the cosect. of an [_ =cosect. of its Supt. 

 . •. A!'D=d sin. (A+D) cosec. A ; A!'B=d cosec. a cosec. A 



A"B 



and by rt. |_d. trigy. r— cosec. A. 



The choice of Curves. 



If the arc of contact be set out merely to the available extent, 

 then an assistant may be directed to move along it, and be signalled 

 from the D or B, to fix an eligible point of osculation. It will then 

 be absolutely necessary to measure only the angle D or B where you 

 stand of the A, DAB, of which the base DB and its opposite [_ A, 

 are already known ; we then have all the angles and the base of 

 A, DAB, and the difference between a and D = either [_ at the base 

 of the isosceles, whose base is the chd. AD, and equal sides OA, AD 

 s and the difference between /3 and B = either of the corresponding 

 angles of the A, whose base is AB, and sides the radii AP, PB of the 

 other arc. 



The chords are the two remaining sides of ADB — ■ 



Sin. A : Sin. D : : BD or d : Chd. AD = d. Sin. D. Cosec. A 

 Sin. A : Sin. B : : d : Chd. AB = d. Sin. B. Cosec. A 



Then r'=^L Sec. ADO* and r" = ^5. Sec. ABPt 



2 2 



It is thus seen that after the arc of contact is set out the rest is 

 exceedingly simple, and in obtaining the elements for that arc no 

 difficulty presents itself, since the angle A of the segment, which is 

 the key to the ■whole, is plainly enough stated at pages 126-7, and in 

 terms of the known angles a and /3 ; and the |_ at base of each 

 isosceles on this page. 



N.B. — For distinction the arc of contact, and the position of its 

 tangent, may be marked by arrows tipped with red cloth. 



One radius given, to find the other radius, in Ogee and 

 Compound Curves. 



It will be useful to run through the process by which, with the 

 aid of the arc of contact, we may find a radius for the second arc, 

 the radius of the first being given. 



* Cos. a - sin. D = sin. a -90° 

 t ADO and ABP are the [_es at bases respectively of the isosceles Aes, 



