344 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Example . — 
R = 2000, £=80', a=220'. 
Assuming 
6 = — — = ®0000 =1*9 approximately. Take R as T9'. 
H 24R 48000 J H 
.*. sin i//- == yV • i^ = 2° 17' 33". 
77 = 1*6007 
aq = 300 
yi = f3 + r) = 3*5007 
. \ h = Vit x i = -000,000, 12965. 
Thus 
Y = *000,000, 12965X 3 
is the equation of this transition curve. Hence the gradient of the T.C. 
at the point of compounding, a? = 300, is tan 2° O' 18", in which the angle 
differs by 17' from the required angle. By calculating the value of y" it 
may be shown that the radius of curvature differs by nearly 2290' from 
the required value. 
This illustrates the importance of the correct use of the transition curve, 
so as to obtain exact compounding. 
The Sine Curve instead of Circular and Transition Curves. 
The use of a sine curve would get rid of all trouble of fitting transition 
curves to the circular curve. This was suggested long ago by Mr Gravatt, 
but the method proposed was by offsets from the long chord. Such a 
method would usually be unsatisfactory, owing to the length of the offsets, 
and perhaps this accounts for the fact that the sine curve has been so little 
used in railway work. In the case of an excessively “ flat ” curve in open 
country such a method might, however, be employed. 
The use of the sine curve, if it could be set out easily, would obviate 
all trouble with regard to calculation of shift and compounding, and it 
would only be necessary to see that the radius of curvature at the vertex 
was not less than a standard value. The curve might be set out easily, 
at least on flat ground, by offsets from three standard tangents — those at 
the two points of springing and at the vertex. A scheme for the deter- 
mination of these offsets is shown, from which their values may be obtained 
rapidly by calculation or even by slide rule. This method might be useful 
also in the case of setting out tramway curves. 
Let 2(o be the angle between the two pieces of straight track which are to 
he joined. AB = L, the long chord ; Y the vertex ; VC = H ; CAS the angle a. 
PR is drawn perpendicular to AS. Let A be the origin, AB the o?-axis, 
