•572 PROCEEDINGS OF SECTION H. 



The double lemniscate curve is thus seen to have its crown always 

 considerably further from the intersection point of the straights than 

 a simple circular curve, and its tangent point is much further along, 

 so that as a rule where its insertion is possible it will also be practicable 

 to insert a simple circular curve of larger radius equal to about five- 

 iourths of R. The double lemniscate curve may be inserted between 

 two straights at any angle with one another ; it may be set out between 

 two parallel lines or between two lines making an angle of nearly 

 180°. 



Instead of a double lemniscate having a minimum radius R, it will 

 in general be possible to set out through the same crown tangent point 

 a circular curve with radius greater than R and having a transition 

 curve at each end. The double lemniscate construction, however, shows 

 in every case a saving in total length of track, and, in some cases, 

 <juite a considerable saving. For instance, if the angle of intersection 

 of the straights is 90°, and the minimum radius 20 chains, the distance 

 /C is 1,098 links, and the total length of the curve 6,168 links. To 

 insert instead a simple circular curve of 20 chains radius will mean an 

 increased length of track of no less than 321 links. But it would be 

 possible without increasing the distance IC to insert a circular curve 

 with a radius of 26 chains, and transition curves at each end. If we 

 take a for the transition curves equal to 1,500, the distance IC works 

 out at 1,083 links, and the total length of track is longer than the 

 double lemniscate by 61 links. Taking a similar curve with a radius of 

 26*5 chains, the distance 10 becomes 1,103 links, that is to say the 

 crown tangent point is further away from the point of intersection 

 than in the case of the double lemniscate, but the double lemniscate still 

 has the advantage in length of 40 links. Making a similar calculation, 

 with the angle of intersection 60^ and the same minimum radius, the 

 total length of the double lemniscate is 8,105 links ; but the saving in 

 total length of track over a circular curve of radius 26'2 chains with 

 transition curves at each end, which will have a crown tangent point 

 at about the same distance from the intersection point, is 121 links. 

 If the angle of intersection is 120^, still, with the same minimum 

 radius, the total length of the double lemniscate is 4454 links, and the 

 track is shorter than that of a circular curve of radius 26"5 chains, 

 with transitions at each end, by 13 links. 



Calculation of the Length of Curve. 



The length of the arc OP (Fig. 1) is expressible as the difference 

 of two elliptic integrals. From the properties of these functions it 

 follows that the lemniscate is one of the curves whose arcs, like those 



