40 Proceedings of the Royal Society of Edinburgh. [Sess. 
and y. These having been determined, the frequency can be obtained 
as before. 
If the loading is symmetrical with respect to the ends, i.e. 
if \ = Z 4 , l 2 = Z 3 , R = R, and r 1 = r B , 
if the phase at C is the same as the phase at E, the motion at these two 
points being opposite to the motion at D, nodes are formed at F and G 
equidistant from D. If the phase at C is not the same as the phase at 
E, the vibration is such that a single central node is formed at D. 
For more than three loads the motion is somewhat complex, and the 
equations become difficult to handle. In practice, however, it will usually 
be found possible to group the loads in such a manner as to allow of 
solution by one or other of the above methods. Such grouping will not 
in general lead to errors of large magnitude. Thus in a long beam where 
two or more of the loads are near one end, these may be assumed to have 
the same dynamical effect as a single load having the same moment of 
inertia. 
VII. Beam Loaded Uniformly. 
When the beam supports a uniformly distributed eccentric load, ex- 
pressions for the frequency of vibration may be readily obtained. Thus 
in the case of a “fixed-free” beam of length L supporting a uniformly 
distributed load of magnitude w per unit of length and of constant 
eccentricity r, the angle of twist at the free end due to the static appli- 
cation of an element of load distant l from the fixed end is given by 
wrldl 
UT’ 
and hence the torque at the free end due to the static application of the 
whole load is 
wrL 2 
= 2CJ' 
1 wr . L 
= 2~W 
1 TL 
where w is the total load 
2 CJ" 
i.e. the effect of a uniformly distributed load applied statically is the same 
as that of a load of half the magnitude, and having the same eccentricity, 
concentrated at the free end. If such a system vibrates, the frequency 
