337 
1911-12.] The Railway Transition Curve. 
curves are then set out, and exact compounding is obtained by this use of 
the cubical parabola. The only restriction on the generality of this method 
is that when Q is given the position of A is fixed. The more general case 
will be examined later. 
If neither the radius nor the degree of curve is given, but the point 
B is fixed, the calculations would be as follows : — BO = (R + /3) is calculated, 
the approximate value of £ selected as before, and the exact value of \fs 
at the point of compounding determined. Hence y 1 is calculated and the 
radius of the circular curve deduced. The shift is then known and the 
intermediate pegs may be set out as before, the compounding being exact. 
Y = IX 3 . Method of Exact Compounding, Radius or Degree of 
Curve being given. 
If x and y , the co-ordinates of the P.C.C., be treated as unknowns, there 
are three unknown quantities, x, y, and l , to be determined at the point of 
compounding in terms of the known quantities \fs and R. 
The equations are 
y = lx 3 . 
y' = tan if/ = Six 2 . 
„ = (1+^)4 = secV = 6 
y R R 
Hence x, y , and l may be found in terms of R and \fs : — 
x = 2R sin if/ cos 2 if/. 
I = 1/(1 2R 2 sin if/ cos 5 i f/). 
y = JR sin 2 if/ cos if/. 
These formulae are suitable for logarithmic computation, as are also 
the required values of £ and y, viz. ^=Rsin \fs, ^ = 2Rsin 2 ^. Hence the 
values of a and /3 are both known, viz. a — x— £, /3 = y — y. The point B 
is now known, since SB = (R + /3) cot co, where 2 w is the known angle 
between the tangents. The circular curve is then set out in the usual 
way from the tangent DS r , and the transition curve from the tangent 
AS, either by offsets, using the equation Y = £X 3 , or by vectorial angles. 
Before the exact position of the point of compounding is chosen, it is 
desirable to know the approximate values of the shift, and of the length 
of the transition curve. Since £ = R sin and x 1 = 2R sin \fs cos 2 \Js, ex- 
panding in ascending powers of \[r, since the deflection angle is small, 
we have 
^ =Pv (^“ 6 ~ + ' ' •)> ^i = 2 r | xf/-~^+ • • • y 
VOL. XXXII. 
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