is as follows: Having moved the theodolite forward, look back on the first tangent point and set off the deflection angle for the next chain peg, which is the total deflection angle for that peg, then proceed as before, laying off the angle for each peg in succession, all as originally calculated and entered in the field book opposite the chainage of each peg. This virtually amounts to laying out a tangent to the curve at the point at which the instrument is set up, and from that tangent laying off the deflection angle for the next chain ahead, only of course we do not stop to actually lay out the tangent to the curve, but continue to turn the instrument round to the angle of the next peg; in other words, for the first peg beyond the point to which the theodolite is shifted we have to lay off so much deflection angle behind for the part of the curve behind us plus the deflection angle for I chain forward. Beyond that we have to increase the angle by the deflection angle for 1 chain for each peg in succession, all of which is already calculated and entered in the field book, as originally directed.

When it is necessary to move the theodolite forward a second time we have to look back on the furthest back point of the curve which it is possible to see. Suppose this is 10 chains back, then for the next peg we have to lay off the deflection angle for II chains, i.e., 10 chains behind and I chain in front. We have then to add to this the deflection angle for 1 chain for each peg in succession. It will be seen that in this case fresh calculation of the deflection angles is necessary, these being no longer continuous round the whole curve; but the angles which have to be calculated are merely the deflection angles for even numbers of chains, with the exception of the fraction of a chain from the last chain peg on the curve to the second tangent point.

Curve to Left.-When the curve curves to the right hand the deflection angles are laid off in a right-handed direction round the graduations of the theodolite. When, however, the curve curves to the left hand the angles have to be laid off in a left-handed direction or contrary to the way in which the theodolite is graduated. In this case we must either subtract each deflection angle from 360° and set the vernier to that angle, or we must use the vernier backwards and read off each angle directly. To do this, set the end of the vernier to 360°, not the zero as is usual, the zero being regarded as the end of the vernier for the time being. The degrees

are then counted off in the left-handed direction, for instance 10° will be at 350°, and the minutes are read off the vernier in its reversed position. With a little practice it is almost as easy to use the vernier one way as the other.

As on

Transition Curves.-The object of a transition curve is to make a gradual change from a straight line to a curve. every curve the rails must have a certain amount of "cant" to counteract the centrifugal force of the train, the transition curve also serves to put the cant on the rails gradually. In addition to this there is the "widening of the gauge on curves"; this also may be gradually done on the transition curve. In the absence of transition curves both the cant and the widening of the gauge are usually effected on the straight, so that there is the full cant and extra width of gauge at the commencement and throughout the whole of the curve. It is obviously a very incorrect proceeding to put cant and extra width of gauge on a straight, while equally obviously the change from a straight to a curve ought to be made gradually, the radius of curvature decreasing from infinity at the commencement of the curve of adjustment to the same radius as the circular curve at its termination.

For these reasons it is the universal practice to put in curves of adjustment on the Continent, and the practice is also general in the United States and South America. Of late years they have been adopted to a small extent by English engineers in home work, and their universal use on railways in this country will no doubt not be long delayed.

Various transition curves have been proposed, among them being the cubic parabola and the compound transition curve. The latter is formed of successive circular arcs increasing in curvature a certain amount for each chord.

Froude's Curve of Adjustment.*-The best practical form of transition curve is Froude's curve of adjustment, which approaches nearly a cubic parabola and is readily calculated and laid out on the ground.

To allow of the cant being gradually applied with a gradient of

* Minutes of Proceedings, Institution of Civil Engineers, vol. cxxxiii., "The Field Practice of Laying Out Transition Curves," by Mr John Robinson, M. Inst. C.E.

I in 300, which is the usual gradient and that recommended by Froude, the length of the curve of adjustment will be = 300 x cant.

In Fig. 173, let FI, LI be two straights to be united by a circular curve with transition curves, and let the intersection angle 1= central angle of curve, be measured as usual. Let AB represent the ordinary circular curve, having its tangent points at A and B. The method of introducing transition curves is to "shift" the curve AB to CD, AB and CD being concentric; the ordinary curve AB must be therefore considered as being of a radius = proposed radius of

H

B

Fig. 173.-Transition Curve.

circular curve plus the "shift" AC.

Let R = radius of curve CD,

then radius of curve AB = R+S where S = shift.

The length of the tangents AI, BI is then given by

AI = BI = (R+S) tan 11.

To get s we have length of curve of adjustment = 300 x cant = /

The distances IA and IB are now to be measured along the

tangents, and pegs put in temporarily at C and D, making AC and BD each = s.

The transition curves will bisect AC and BD at E and K.

The points of commencement of the transition curves are now to be marked by three pegs at F and L, the distances AF and BL

being made = -, i.e., half length of transition curve.

2

(This is on supposition that AF and BL measured on the tangents do not appreciably differ from EF and KL measured on the curves, which is in practice the case.)

The terminations of the transition curves and their points of junction with the circular curve are at G and H, EG and KH being

These points are fixed by means of the offsets x and x'H measured at right angles to the tangents FI, LI.

These offsets XG and x'H are each = 48, i.e., 4 times the shift. They are also equal to the offset xa to the circular curve + the shift s or to

LI.

Setting out Transition Curves. The transition curves are set off by means of offsets measured square off the tangents FI, These offsets are proportional to the cubes of their distances from the commencement of the transition curve, and having the offset XG = = 45 at a distance / from F the offset at any other point is easily calculated. Otherwise take the offset AE at a distance from F, and calculate the others from this. The last offset at x should then be = 4s.

S

==

2

2

Having thus set off the transition curves by means of their calculated offsets from the tangents, set ranging rods in the lines F'I, L'I', these lines being parallel to the original tangents and at a distance from them = shift. The circular curve GH may now be set out in the usual manner, by setting up the theodolite at c and working off the tangent F'I'. On laying off the deflection angle for the chord CG the point found should coincide with G as previously fixed by measuring the offset G from FI. Similarly the termination of the curve should coincide with H as fixed by measuring x'H from LI.

R

The length of curve CD = circular measure of angle I × R.
Length of curve GH = CD - l.

Total length of curve from F to L= GH + 2/.

Or total length of curves from F to L= (circular measure

of angle 1 x R) +7.

These lengths of curves are on supposition that EG and CG are equal. In practice they will not differ appreciably.

The portions CG and DH of the circular curve are of course not staked out, and the temporary pegs at C and D are removed, after setting out the curve GH.

The radius of the above transition curve at G and H is equal to R, the radius of the circular curve.

"Cant.”—The length of the transition curve above described will be proportional to the cant, which again depends on the radius of curve and speed of train. The following formula gives the cant:

Cant in inches = gauge in feet

where v = velocity in miles per hour, R

v2

1.25 R

radius of curve in feet. For a 10 chain curve on 4 ft. 8 in. gauge at 40 miles per hour the cant is about 9 in.; at 1 in 300 this gives a length of transition curve of 225 ft. For curves of greater radius the cant is less, and length of transition curve therefore less. The length of transition curve is, however, often made greater for curves of larger radius. When sharp curves follow each other closely there is sometimes little room to get in the transition curves, but in practice transition curves from 180 to 400 ft. long will usually cover all cases. Curves above about 40 chains radius do not usually require transition curves, the small amount of cant and widening of gauge being effected on the straight.

Transition Curves between Reversed Curves.-When there is not room to get in a piece of straight between two reverse curves, the change of cant may be readily effected by uniting the two reversed curves by a transition curve as shown in Fig. 174.

Let ABC be two reverse curves reversing at M. Shift each curve the proper amount to GF and DE. Calculate the length of transition curve, which will be in this case 300 × sum of cants.