732 Mr. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



The numerical results contained in Table III. are represented graphically in figs. 2 and 3 of the 

 accompanying plate, where however some of the curves are left out, in order to prevent confusion in 



the figures. In these figures the numbers written against the several curves are the values of — 



to which the curves respectively belong, the symbol co being written against the equilibrium 

 curves. Fig. 2 represents the trajectory of the body for different values of q, and will be understood 

 without further explanation. In the curves of fig. 3, the ordinate represents the deflection of the 

 centre of the bridge when the moving body has travelled over a distance represented by the abscissa. 

 Fig. 1, which represents the trajectories described when the mass of the bridge is neglected, 

 is here given for the sake of comparison with fig. 2. The numbers in fig. 1, refer to the 

 values of /3. The equilibrium curve represented in this figure is the true equilibrium trajectory 

 expressed bv equation (65), whereas the equilibrium curve represented in fig. 2 is the approximate 

 equilibrium trajectory expressed by equation (64). In fig. 1, the body is represented as flying 

 off near the second extremity of the bridge, which is in fact the case. The numerous small 

 oscillations which would take place if the body were held down to the bridge could not be properlv 

 represented in the figure without using a much larger scale. The reader is however requested 

 to bear in mind the existence of these oscillations, as indicated by the analysis, because, if the ratio 

 of M to il/' altered continuously from co to 0, they wo iH probably pass continuously into the 

 oscillations which are so conspicuous in the case of the h ger values of q in fig. 2. Thus the 

 consideration of these insignificant oscillations which, strictly speaking, belong to fig. 1, aids us in 

 mentally filling up the gap which corresponds to the cases in which the ratio of M to M' is neither 

 very small nor very large. 



As everything depends on the value of q, in the approximate investigation in which the inertia 

 of the bridge is taken into account, it will be proper to consider further the meaning of this constant. 

 In the first place it is to be observed that although ilf appears in equation (6l), q is really indepen- 

 dent of the mass of the travelling body. For, when M alone varies, /3 varies inversely as S, and .9 

 varies directly as M, so that q remains constant. To get rid of the apparent dependance of q on M, 

 let iSi be the central statical deflection produced by a mass equal to that of the bridge, and at the 

 same time restore the general unit of length. It' a) continue to denote the ratio of the abscissa of 

 the body to the length of the bridge, q will be numerical, and therefore, to restore the general unit 

 of length, it will be sufficient to take the general expression (5) for /3. Let moreover t be the 

 time the body takes to travel over the bridge, so that 2c = Vt ; then we get 



'-S^; ••••; '«) 



If we suppose t expressed in seconds, and Si in inches, we must put g — 32.2 x 12 = 386, 

 nearly, and we get, 



28 T 



'=V^ '''' 



Conceive the mass M removed ; suppose the bridge depressed through a small space, and then 

 left to itself. The equation of motion will be got from (59) by putting M = 0, where M is not 



divided by S, and replacing — , by — , and V -— by — . We thus get 



o S\ dw dt 



d- M 63^ 



-TV + ^ /u = ; 



dt- 31 St 



and therefore, if P be the period of the motion, or twice the time of oscillation from rest to rest, 



P= 2^ '^^' '^^ -""¥ ^^"^ 



Hence the numbers 1, 2, 3, &c., written at the head of Table III. and against the curves of 



