568 PROCEEDINGS OF SECTION H. 



given to all transition curves ; in others the super-elevation of the 

 outer rail is run out by means of the transition curve on a fixed grade. 

 In South Australia it is usually run out in this way on a grade of 1 in 

 360, so that the length of the transition curve is made 360 times the 

 super-elevation. If we adopt the latter method, which seems to be the 

 more rational of the two, the constant a' then becomes the same for all 

 the curves on a given line provided the super-elevation of the outer rail 

 is in all cases that given by the ordinary formula. 



For, if e denotes the super-elevation in inches, g the gauge in feet, 

 V the maximum train velocity in miles per hour, s the cotangent of the 

 angle of grade at which the super-elevation is to he run out, It the 

 radius of the circular curve in feet, 



i-25/«; 



and <9C(in feet) = llf, 



.•, a'z=3. <9C.i2=:-^r-> a constant depending only on the grade, 

 6 



gauge, and train velocity, and independent of the radius of the circular 

 curve. 



If we make s = 360, g = 5^, v = 50, then a^ = 787,500. As, how- 

 ever, it is convenient to have a simple square number for a^, and there 

 is no especial reason for adopting the particular grade of 1 in 360, it 

 seems reasonable to choose the value of « a little move than tliis so as to 

 make a' a convenient square. In this case 810,000 or 1,000,000 would 

 be suitable values to adopt for a^, giving a = 900 or 1,000 feet, or we 

 might make it 1,500 links. 



It thus appears to be correct practice and one that lends itself to 

 oreat simplicity in the preparation of tables and in the necessary 

 calculations to adopt the same value for a throughout; any particular 

 line. Whether this is done or not, in any case, as all the necessary 

 elements can be expressed in terms of a, it is possible to compute a 

 simple set of tabes of the necessary elements for setting out that shall 

 be of perfectly general application, the various quantities having only 

 to be multiplied by a. This alone gives the lemniscate a great 

 advantage over the cubic parabola. 



As a' = 3E.0C, it follows that if ^ = 20 chains, the value 

 a = 1,500 links will make the length of the transition curve about 

 3| chains. If a = 2,000 links its length becomes 6f chains. 



