734 Mr. STOKES'S DISCUSSION Of A DIFFERENTIAL EQUATION 



the actual effect is insignificant. Now we have seen that if we take only the most important terms, 



1 25 



the increase of deflection is measured by the fractions — and — of S- It is only when these fractions 



are both small that we are at liberty to neglect all but the most important terms, but in practical 

 cases they are actually small. The magnitude of these fractions will enable us to judge of the 

 amount of the actual effect. 



To take a numerical example lying within practical limits, let the span of a given bridge be a 

 feet, and suppose a weight equal to ^ of the weight of the bridge to cause a deflection of 1 inch. 

 These are nearly the circumstances of the Ewell bridge, mentioned in the report of the com- 

 missioners. In this case, Si = ^ x .2 = .15; and if the velocity be a feet in a second, or 30 miles an 

 hour, we have t = 1, and therefore from the second of the formula; (72), 



25 TT 



— = .0434, q = 72.1 = 45.9 x - • 

 Sq 4 



The travelling load being supposed to produce a deflection of .2 inch, we have j8 = 127, - = .0079. 



Hence in this case the deflection due to the inertia of the bridge is between 5 and 6 times as great as 

 that obtained by considering the bridge as infinitely light, but in neither case is the deflection 

 important. With a velocity of 60 miles an hour the increase of deflection .0434^ would be 

 doubled. 



In the case of one of the long tubes of the Britannia bridge ^ must be extremely large ; but on 

 account of the enormous mass of the tube it might be feared that the eff'ect of the inertia of the tube 

 itself would be of importance. To make a supposition every way disadvantageous, regard the tube 

 as unconnected with the rest of the structure, and suppose the weight of the whole train collected at 

 one point. The clear span of one of the great tubes is 460 feet, and the weight of the tube 1400 

 tons. When the platform on which the tube had been built was removed, the centre sank 10 

 inches, which was very nearly what had been calculated, so that the bottom became very nearly 

 straight, since, in anticipation of the deflection which would be produced by the weight of the tube 

 itself, it had been originally built curved upwards. Since a uniformly distributed weight produces 

 the same deflection as ^ ths of the same weight placed at the centre, we have in this case 6"! = # x 10 



= 16 ; and supposing the train to be going at the rate of 30 miles an hour, we have t = = 10.5, 



44 



25 

 nearly. Hence in this case — = .043, or Jg- nearly, so that the increase of deflection due to the 



Sq ''■' 



inertia of the bridge is unimportant. 



In conclusion, it will be proper to state that this "Addition" has been written on two or three 

 different occasions, as the reader will probably have perceived. It was not until a few days after 

 the reading of the paper itself that I perceived that the equation (l6) was integrable in finite terms, 

 and consequently that the variables were separable in (4). I was led to try whether this might not 

 be the case in consequence of a remarkable numerical coincidence. This circumstance occasioned 

 the complete remodelling of the paper after the first six articles. I had previously obtained for the 

 calculation of z for values of x approaching 1, in which case the series (g) becomes inconvenient, series 

 proceeding according to ascending powers of 1 — x, and involving two arbitrary constants. The 

 determination of these constants, which at first appeared to require the numerical calculation of five 

 series, had been made to depend on that of three only, which were ultimately geometric series with a 

 ratio equal to ^. 



