g. 50 shows the curve of bending moment under one of a series of
travelling loads at fixed distances. Let W_1, W_2, W_3 traverse the girder
from the left at fixed distances a, b. For the position shown the
distribution of bending moment due to W_1 is given by ordinates of the
triangle A'CB'; that due to W_2 by ordinates of A'DB'; and that due to W_3
by ordinates A'EB'. The total moment at W_1, due to three loads, is the sum
mC+mn+mo of the intercepts which the triangle sides cut off from the
vertical under W_1. As the loads move over the girder, the points C, D, E
describe the parabolas M_1, M_2, M_3, the middle ordinates of which are
1/4W_1l, 1/4W_2l, and 1/4W_3l. If these are first drawn it is easy, for any
position of the loads, to draw the lines B'C, B'D, B'E, and to find the sum
of the intercepts which is the total bending moment under a load. The lower
portion of the figure is the curve of bending moments under the leading
load. Till W_1 has advanced a distance a only one load is on the girder,
and the curve A"F gives bending moments due to W_1 only; as W_1 advances to
a distance a+b, two loads are on the girder, and the curve FG gives moments
due to W_1 and W_2. GB" is the curve of moments for all three loads
W_1+W_2+W_3.

[Illustration: FIG. 51.]

Fig. 51 shows maximum bending moment curves for an extreme case of a short
bridge with very unequal loads. The three lightly dotted parabolas are the
curves of maximum moment for each of the loads taken separately. The three
heavily dotted curves are curves of maximum moment under each of the loads,
for the three loads passing over the bridge, at the given distances, from
left to right. As might be expected, the moments are greatest in this case
at the sections under the 15-ton load. The heavy continuous line gives the
last-mentioned curve for the reverse direction of passage of the loads.

With short bridges it is best to draw the curve of maximum bending moments
for some assumed typical set of loads in the way just described, and to
design the girder accordingly. For longer bridges the funicular polygon
affords a method of determining maximum bending moments which is perhaps
more convenient. But very great accuracy in drawing this curve is
unnecessary, because the rolling stock of railways varies so much that the
precise magnitude and distribution of the loads which will pass over a
bridge cannot be known. All that can be done is to assume a set of loads
likely to p