708 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



abscissfB and ordinates, the differential equation is reduced to one containing a single constant /3, 

 which is defined by equation (5). The meaning of the letters which appear in this equation will 

 be seen on referring to the beginning of Art. 1. For the present it will be sufficient to observe 

 that j3 varies inversely as the square of the horizontal velocity of the body, so that a small value 

 of ;8 corresponds to a high velocity, and a large value to a small velocity. 



It appears from the solution of the differential equation that the trajectory of the body is 

 unsymmetrical with respect to the centre of the bridge, the maximum depression of the body 

 occurring beyond the centre. The character of the motion depends materially on the numerical 

 value of j3. When /3 is not greater than ^, the tangent to the trajectory becomes more and more 

 inclined to the horizontal beyond the maximum ordinate, till the body gets to the second extremity 

 of the bridge, when the tangent becomes vertical. At the same time the expressions for the central 

 deflection and for the tendency of the bridge to break become infinite. When fi is greater than 

 ^, the analytical expression for the ordinate of the body at last becomes negative, and afterwards 

 changes an infinite number of times from negative to positive, and from positive to negative. 

 The expression for the reaction becomes negative at the same time with the ordinate, so that in 

 fact the body leaps. 



The occurrence of these infinite quantities indicates one of two things: either the deflection 

 really becomes very large, after which of course we are no longer at liberty to neglect its square ; 

 or else the effect of the inertia of the bridge is really important. Since the deflection does not 

 really become very great, as appears from experiment, we are led to conclude that the effect of the 

 inertia is not insignificant, and in fact I have shewn that the value of the expression for the vis 

 viva neglected at last becomes infinite. Hence, however light be the bridge, the mode of approx- 

 imation adopted ceases to be legitimate before the body reaches the second extremity of the bridge, 

 although it may be sufficiently accurate for the greater part of the body's course. 



In consequence of the neglect of the inertia of the bridge, the differential equation here dis- 

 cussed fails to give the velocity for which T, the tendency to break, is a maximum. When j8 is 

 a good deal greater than 1, T" is a maximum at a point not very near the second extremity of the 

 bridge, so that we may apply the result obtained to a light bridge without very material error. 

 Let Tj be this maximum value. Since it is only the inertia of the bridge that keeps the tendency 

 to break from becoming extremely great, it appears that the general effect of that inertia is to 

 preserve the bridge, so that we cannot be far wrong in regarding T, as a superior limit to the 

 actual tendency to break. When /3 is very large, T'l may be calculated to a sufficient degree of 

 accuracy with very little trouble. 



Experiments of the nature of those which have been mentioned may be made with two distinct 

 objects ; the one, to analyze experimentally the laws of some particular phenomenon, the other, to 

 apply practically on a large scale results obtained from experiments made on a small scale. With 

 the former object in view, the experiments would naturally be made so as to render as conspicuous 

 as possible, and isolate as far as might be, the effect which it was desired to investigate ; with the 

 latter, there are certain relations to be observed between the variations of the different quantities 

 which are in any way concerned in the result. These relations, in the case of the particular problem 

 to which the present paper refers, are considered at the end of the paper. 



1. It is required to determine, in a form adapted to numerical computation, the value of y' in 

 terms of x, where y' is a function of w' defined by satisfying the differential equation 



with the particular conditions 



dx" " (2cx' -w''y' ^ ' 



dy 



«' = 0, ~7 = 0, when «■ =0, (2) 



dx 



