130 Railway Curves. 



Next having found 



The chd. AD and its position 

 Let x be the [_ at base AD of 



isosceles ADO (compt. of AOQ) 

 Then in A BAD, D,(=a — a?) and 



A being known, we have given 



two sides AD, DB and all the 



angles to find AB. 

 Lastly in the isosceles APB we 



have AB and L ABP, or y= 



/3=L:ABD* and _ Sect. y=r" 



2 



In a compd. curve y = /3 + ABD 



Or, Chd. AB and its position. 

 Let y be the [_ at base AB of 



isosceles ABP (compt. of APQ) 

 Then in A BAD, B=/3 02 y; (i.e.) 



the difference between ft and y 



and A being known, we have 



given two sides AB, BD and all 



the angles to find AD. 

 Lastly in the isosceles AOD we 



have AD and |_ ADO or x = 



a — ADB and Sect. x=r' 



In a compd. curve x = a + ADB 



Moreover when x or y is known, then y=BAT> — x ) in a compound 



or x= BAD — y j . curve. 



It has thus been attempted to simplify and generalise the various 

 possible cases of § an( i compound curves, and to j>resent the whole 

 at one view, in order, as has been premised, to anticipate practical 

 difficulties in the field by such general theory, and provide a 

 starting point for their solution. 



It may be excusable here to point out the importance of keeping 

 these few essential points steadily in view, viz., the distance BD 

 and the angles a and ft (alpha and beta) are of course given, x (and 

 y for distinction when two arcs are concerned) is the angle at the 

 base of each isosceles A (see fig. X), and this angle is the compliment 

 of the several [es marked, in the figure, G (the capital theta) viz.: 1st, 

 of the |_ of deflection of the chord from the tangent of the arc. 2nd, 

 of any angle in the alternate segment. And 3rd, of half the |_ at the 

 centre (i.e.) it is the complement of the angle of a right-angled A, 

 whose hypothenuse is the radius of the arc, and whose opposite side 

 is half the chord. In short, x= 9 - TAB= 9 0° - ASB= 9 0°— APR. 



The L BAD is easily made out (see page 127). Also, AOQ and 

 PAQ, when one radius is given (see page 129) ; and thence to find 

 AOQ, when the radius OA is given, and APV when the radius PA 

 is given, and their compts. severally x and y at bases of isosceles. 



* In a compound curve, and in the ordinary figure M of an $ curve, 

 y=ft + ABD, the minus sign occurs when (as in fig. N, O) the two 

 centres of an $ curve are on the same side of DB. 



In fig. P, an $ curve might be adopted up to A" from D. Let the 

 blue Hues be the radii of such ; then the |_es OA'Q, PA"Q, the re- 

 quired |_ AOQ or APQ, and chord AD or AC, will be as in fig. MNO. 

 The (angle) DAB being the same in either case. 



