83 



distribution of sea and land throughout the whole surface of the 

 earth, the result would agree far better with gravity calculated on 

 the hypothesis of original fluidity. 



May 7, 1849. 



Additional Note to a Memoir on the Intrinsic Equation of Curves. 

 By Dr. Whewell. 



This note contained an extension of a theorem discovered by John 

 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, 1849. 



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 differential 

 equation of the form 



d q y by 



dx* (2cr— x a -)*' 



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 



