446 



Archdeacon Pratt on the Undulation of an 



be perfectly flexible, in the present calculation. The effect of 

 the train when in any position on the bridge will be found by 

 considering the problem, To determine the position of equili- 

 brium of a chain of given length suspended from two points in 

 the same horizontal line, the chain consisting of two parts, of 

 different thickness, but both uniform. 



A P C A' is such a chain, of which A P is the thicker part and 

 is the portion of a common catenary A P C R, and P A' is the 

 thinner part, and is a portion of the common catenary R/P C'A': 

 € and C are the lowest points of these two catenaries ; P Q, C N, 

 C'N' are vertical, PMM' is horizontal. Let c and c 1 be the 

 lengths of chain which measure by their weight the tensions at 

 the two points C and C' respectively, and j3 the ratio of the 

 thickness of A P to that of P A'. Let 



CM=#, MP=y, CN=A, NA=£, CP=s, CA=/, 



C'M'W, M'P=V, C'N'=#, N'A=#, CT=s', C'A'=Z', 



AA'=2&, APA'=2w, AP = *. 



By the equations to the catenary we have 



S 2 + C 2 = ( a? + C )2 # . (i) 



Z 2 +c 2 =(A + c) 2 . . (2) 



Jc k 



c(^+"e"c)=2(A + c). (3) 

 y = clog e -Z Z (4) 



s' 2 + c'*=(*' + c') 2 . . . (5) 



i ! *+(j*=(h'+(/)% . . (6) 



c '( e ? + e -7<)=2(A' + c'). (7) 



y ^ H f_±_^±£\ (8) 



The conditions of equilibrium require that the two curves 

 should have a common point and tangent at P, and that the ten- 

 sions there should be the same. These lead to the following 

 equations : 



h-sc^ti-at, (9) 



H (io) 



{*-+c)ff=J+J; (11) 



also 



s—l—t, 



(12) 

 (13) 



