722 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION. 



To take a numerical example, suppose that we wished, by means of a model bridge five feet 

 long and weighing 100 ounces, to investigate the greatest central deflection produced by an engine 

 weighing 20 tons, which passes with the successive velocities of 30, 40, and 50 miles an hour over 

 a bridge 50 feet long weighing 100 tons, the central statical deflection produced by the engine 

 being one inch. We must give to our model carriage a weight of 20 ounces, and make the small 

 bridge of such a stiffness that a weight of 20 ounces placed on the centre shall cause a deflection 

 of Jyth of an inch; and then we must give to the carriage the successive velocities of 3-s/lO, 

 4\/l0, Sy/lO, or 9.49, 12.65, 15.81 miles per hour, or 13.91, 18.55, 23.19 feet per second. If 

 we suppose the observed central deflections in the model to be .12, .16, .18 of an inch, we may 

 conclude that the central deflections in the large bridge corresponding to the velocities of 30, 40, and 

 50 miles per hour would be 1.2, 1.6, and 1.8 inch. 



G. G. STOKES. 



Addition to the preceding Pajjer. 



Since the above was written. Professor Willis has informed me that the values of ji are much 

 larger in practice than those which are contained in Table I, on which account it would be interesting 

 to calculate the numerical values of the functions for a few larger values of /3. I have accordingly 

 performed the calculations for the values 3, 5, 8, 12, and 20. The results are contained in Table II. 

 In calculating s from « = to a; = .5, I employed the formula (12), with the assistance occasionally 

 of (15). I worked with 4 places of decimals, of which 3 only are retained. The values of z for 

 a = .5, in which case the series are least convergent, have been verified by the formula (42) given 

 below : the results agreed within two or three units in the fourth place of decimals. The remaining 



values of z were calculated from the expression for {x - x") '^(p{x). The values of T and - were 



deduced from those of z by merely multiplying twice in succession by ix (l - ,r). Professor Willis 

 has laid down in curves the numbers contained in the last five columns. In laying down these 

 curves several errors were detected in the latter half of the Table, that is, from x = .55 to x .95. 

 These errors were corrected by re-examining the calculation ; so that I feel pretty confident that the 

 table as it now stands contains no errors of importance. 



