450 Archdeacon Pratt on the Undulation of an 



m and b are the same to the degree of approximation to which it 

 is necessary to go. 



3. Having thus found the elements of the two variable cate- 

 naries, I will calculate the curve into which the roadway is 

 thrown. 



Fig. 2. 



Let D N D r be the undisturbed roadway, D S T W the undula- 

 tion into which it is thrown when the train is on the road; 

 DN = Y, NS = X the coordinates to any point in the part of the 

 curved roadway corresponding with, the thicker catenary. The 

 depth of any point of that catenary below the fixed horizontal 

 line A A' (fig. 1), the horizontal ordinate from A, being Y, and 

 therefore from the lowest point of the catenary being k— Y, is 



fc-Y 



When /3=1 and there is no train on the roadway, c=C, k=b ; 

 let h=H; then this expression becomes 



H- (*- Y > 2 



2C 



The excess of the first of these above the second is the depression 

 of the roadway at the corresponding point, and therefore equals 

 NS". Hence the equation to the curve of the roadway is 



or 



X 

 H 



G)*(-t)-t v ('-^)= 



(since k 2 = 2ch and 5 2 = 2CH by the equation to the catenary, 

 neglecting extremely small quantities), or 



£C_ 



e?-0' 





c c c 



This is the equation to a parabola with the vertex at the lowest 

 point : let N, D be the horizontal and downward vertical coordi- 



