μ

Now according to our hypothesis the bridge must always have the form which it would assume under the action of a uniformly distributed force; and therefore, if u be the mean deflection at the time t, (57) will be the equation to the bridge at that instant. Moreover, since the point (x, y) is a point in the bridge, we must have ny when = x, whence y = μu.

We have also

We get from (55), F=5Mgμ/2S. Making these various substitutions in (58), and replacing d/dt by V.d/dx, we get for the differential equation of motion

Since μ is comparable with S, the several terms of this equation are comparable with

Mg, Mg, MV'S, M'V'S,

respectively. If then VS be small compared with g, and likewise M small compared with M', we may neglect the third term, while we retain the others. This term, it is to be observed, expresses the difference between the pressure on the bridge and the weight of the travelling mass. Since c, we have V2S/g=1/166, which will be small when ẞ is large, or even moderately large. Hence the conditions under which we are at liberty to neglect the difference between the pressure on the bridge and the weight of the travelling mass are, first, that ẞ be large, secondly, that the mass of the travelling body be small compared with the mass of the bridge. If ẞ be large, but M be comparable with M', it is true that the third term in (59) will be small compared with the leading terms; but then it will be comparable with the fourth, and the approximation adopted in neglecting the third term alone would be faulty, in this way, that of two small terms comparable with each other, one would be retained while the other was neglected. Hence, although the absolute error of our results would be but small, it would be comparable with the difference between the results actually obtained and those which would be obtained on the supposition that the travelling mass moved with an infinitely small velocity.

Neglecting the third term in equation (59), and putting for u its value, we get

The linear equation (60) is easily integrated. Integrating, and determining the arbitrary constants by the conditions that μ= 0, and du/dx=0, when x = 0, we get

and we have for the equation to the trajectory

y = 5μ (x − 2x3 + x1) = 5μ (X + X2)..................................(63),

where as before Xx (1-x).

.....

When V=0, q=c, and we get from (62), (63), for the approximate equation to the equilibrium trajectory,

y = 108 (X + X2)3.....

whereas the true equation is

y=168X2

.(64);

..(65).

Since the forms of these equations are very different, it will be proper to verify the assertion that (64) is in fact an approximation to (65). Since the curves represented by these equations are both symmetrical with respect to the centre of the bridge, it will be sufficient to consider values of x from 0 to , to which correspond values of X ranging from 0 to . Denoting the error of the formula (64), that is the excess of the y in (64) over the y in (65), by S8, we have

Equating do/dx to zero, we get X= 0, x = 0, 80, a maximum; X=1787, x=233, 8=-067, nearly, a minimum; and = }, d = '023, nearly, a maximum. Hence the greatest error in the

approximate value of the ordinate of the equilibrium trajectory is equal to about the one-fifteenth of S.

=

Putting μμ1, y=yo+y1, where μ, y, are the values of for q=∞, we have

м, у

The values of μ, and y, may be calculated from these formulæ for different values of q, and they are then to be added to the values of μ, Yo, respectively, which have to be calculated once for all. If instead of the mean deflection μ we wish to employ the central deflection D, we have only got to multiply the second sides of equations (62), (66) by 2%, and those of (63), (67) by 18, and to write D for μ. The following table contains the values of the ratios of D and y to S for ten different values of q, as well as for the limiting value q=∞, which belongs to the equilibrium trajectory.

The numerical results contained in Table III. are represented graphically in figs. 2 and 3 of the woodcut on p. 216, 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 2q/π to which the curves respectively belong, the symbol ∞ 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 B. The equilibrium curve represented in this figure is the true equilibrium trajectory expressed by 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

·001 ⚫001 ·001

⚫001

⚫001

⚫012

⚫050

⚫30

•191

⚫000 ⚫000

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⚫001 ⚫002

⚫003

⚫004

⚫006

⚫025

⚫017 ⚫025 ⚫037

⚫050

⚫075

⚫096

⚫067 •108

•150

properly 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 M' altered continuously from ∞ to 0, they would probably pass continuously into the oscillations which are so conspicuous in the case of the larger 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 M appears in equation (61), q is really independent of the mass of the travelling body. For, when M alone varies, B varies inversely as S, and S varies directly as M, so that q remains constant. To get rid of the apparent dependence of q on M, let S, be the central statical deflection produced by a mass equal to that of the