230 Cambridge Philosophical Society. 



Bernouilli, and demonstrated by Euler, to this effect : that if from 

 any rectangular curve a string be unwrapped, and from the curve so 

 described again a string unwrapped, and so on perpetually and alter- 

 nately in opposite directions, the curves constantly tend to the form 

 of the common cycloid. The extension is to this effect : that if the 

 original curve be not rectangular, the curves perpetually tend to the 

 form of an epicycloid or hypocycloid, according as the angle is greater 

 or less than a right angle. 



May 21. — Discussion of a Differential Equation relating to the 

 breaking of Railway Bridges. By G. G. Stokes, M.A., Fellow of 

 Pembroke College. 



In August 1847 a Royal Commission was appointed •" for the 

 purpose of inquiring into the conditions to be observed by engineers 

 in the application of iron in structures exposed to violent concussions 

 and vibration." Among other branches of inquiry, the members of 

 the Commission have lately been making experiments on the motion 

 of a carriage, variously loaded in different experiments, which passed 

 with different velocities over a slight iron bridge ; the object of the 

 experiments being to examine the effect of the velocity of a train in 

 increasing or decreasing the tendency of a bridge over which the 

 train is passing to break under its weight. The remarkable result 

 was obtained, that the deflection is in some cases much greater than 

 the central statical deflection, and that the greatest deflection takes 

 place after the body has passed the centre of the bridge. In in- 

 vestigating the theory of the motion, reducing the problem to the 

 utmost degree of simplicity by regarding the moving carriage as a 

 heavy particle, and neglecting the inertia of the bridge, Professor 

 Willis, who is a member of the Commission, was led to a differentieil 

 equation of the form 



d^y by 



dx^~^ {2cx—x^y 



where x, y are the horizontal and vertical co-ordinates of the moving 

 body, 2c is the length of the bridge, and a, b are certain constants. 

 Professor Willis requested the author's consideration of this equation, 

 with a view to obtain numerical results, and to determine, if possible, 

 the velocity which produces a maximum deflection. 



The author has expressed y in a series according to ascending 

 powers of x, which is convergent when x < 2e. The convergency, 

 however, becomes very slow when x approaches the limit 2c ; and 

 the series does not point out the law according to which f{x) or y 

 approaches its extreme value as a? approaches 1c. When the con- 

 stant term in the second member of the preceding equation is omit- 

 ted, the equation may be integrated in finite terms ; and consequently 

 the variables can be separated in the actual equation, so thaty(a:) 

 can be expressed explicitly by means of definite integrals. In this 

 way the author has obtained y(2c — x)—f{x) in finite terms, so that 

 the numerical value of /(a?) may readily be obtained from a?=c to 

 it;=2c, after it has been calculated from the series from x=^0 to a;=c : 

 and between these limits the series is very convergent, being ulti- 



