542 Vibrations of Rotating Shafts. 
and ec. It vanishes, it is true, when ¢ vanishes, but the two 
loads then coincide in position. As R is positive, the appli- 
cation of Dunkerley’s hypothesis gives a larger value for — 
1/@?, and so a smaller value for w?, than does (28). 
This does not entirely disprove Dunkerley’s hypothesis, 
because we are not entitled to assume that (25) accords 
absolutely with the true type of displacement, and we know 
from Rayleigh’s general theorem that, unless this is the case, 
the value given by (26) for 4? must be somewhat in excess, and 
consequently the value (28) for 1/w? somewhat too low. We 
may however expect, in accordance with Rayleigh’s general 
reasoning, that (26) is a very close approach to the truth; 
and whilst R is usually much smaller than 1/#?+ 1/w2, it is 
by no means negligible, unless one of the loads be much 
less than the other, or one of the three lengths, a, b, and ¢ 
be small. 
Considering, however, the various sources of uncertainty, 
it must be allowed that in the present instance Dunkerley’s 
hypothesis gives at least a fair first approximation. Taking, 
for example, the fairly representative case presented when the 
two loads are equal, and a, 6, ¢ all equal, we find 
1 /m? = (15/16) (1/w? + 1/2). 
§ 46. In carrying out investigations in cases where there 
are two, three, or more loads, the physical significance of 
the processes is more easily seen by adopting a generalized 
notation. In the above case, for instance, it will be found 
_ that the displacements at the points where the loads occur 
are really of the types . ; 
1 (Myyut Moyy2) and 7’(Myy12+ Moy»), 
where y;; and yj, are the displacements at the point where M, 
occurs, due respectively to unit loads at this point and at the 
point where M, occurs. (By a well-known general theorem 
Yio and yg, are equal.) ‘The kinetic and the potential energies 
vary respectively as 
My (M yyy, + Moye)? + Mo(Myyyo + Moyoe)? and 
(M2y11 + 2My Moyo + M2 yop), 
and the function which appears in the expression for 1/w? 
really varies as 
My + Maz22— 
(YinY22— Yn) My Mo( Mi + Mz) = { Mig + 2M Moyo + Mo y22f- 
The sign of R in (26) and (28) really turns on the sign of 
(YawYo2—Yi2)- 
