164 MERRIMAN—LEAST WORK IN MECHANICS. [April 2, 
shall be a minimum. In theoretical mechanics the condition for 
finding the centre of mass of a system of bodies may be expressed 
by saying that the moment of inertia of the system shall be a mini- 
mum. In the mechanics of elastic bodies the principle of least 
work is analogous to these, for the conditions which must be 
fulfilled are those found by making the stored energy of the system 
a minimum. In all these cases the algebraic conditions are 
expressed by linear equations while the laws from which they are 
derived are in quadratic form, and these laws are only true when 
each elementary error or particle produces its effects independently 
of others. 
Solid beams and tubes, as well as framed trusses, are subject to 
the principle of least work, provided the materials of which they 
are made conforms to Hooke’s law of elasticity. For instance, the 
thick hollow cylinder of a gun tube is stressed under the pressure 
arising from the explosion of the powder, and the stress at any 
point varies inversely as the square of the distance between that 
point and the centre of the tube. It is easy to show that this law 
of variation is the one which makes the stored energy in the tube a 
minimum. So in a hollow sphere subject to interior pressure, the 
stresses throughout the spherical annulas vary inversely as the cubes 
of their distances from the centre, and this law of variation is the 
one which renders the stored stress energy a minimum. 
The ether that fills space and transmits the force of gravitation 
from every particle of matter to all others has been regarded by 
many physicists as an elastic solid which obeys Hooke’s law. Ifso 
it must be subject to the principle of least work. Any portion of 
matter may be supposed to exert upon the ether a compressive 
force, due to the fact that its molecules have displaced the ether 
and crowded it outwards. Then the stresses in the ether due to 
this displacement must be so distributed that the stored energy in 
the infinite sphere may be a minimum. Stating the algebraic 
expression for this energy due to a spherical body, it is found that 
its minimum value occurs when the stress at any point in the ether 
varies inversely as the cube of the distance of that point to the 
centre of the body. If gravitation be a differential effect, due to 
the difference of the stresses upon opposite sides of a body, the 
force of attraction between two spheres should vary inversely as 
the fourth power of the distance between their centres. From no 
point of view does it seem possible to deduce the actual law of 
