os 
2 
1903.] MERRIMAN—LEAST WORK IN MECHANICS. 163 
energy init. The amount of shortening is, moreover, proportional 
to the stress, if the elastic limit of the material be not exceeded. 
Accordingly, the stress energy in the four legs due to the given load 
is proportional to the sum of the squares of the four stresses, and 
this sum is to be made a minimum. This condition, in connection 
with the three statical ones, enables the four stresses due to the 
load to be readily determined for any given position of that load, 
and that these stresses actually occur is easily verified by experi- 
ment. 
A close analysis of the principle of least work as applied to any 
framed structure will show that its applicability and its validity depend 
upon the fact that the longitudinal deformation of any member is 
proportional to the stress uponit. ‘This law of elasticity, commonly 
known as Hooke’s law, is closely true for the materials used in 
engineering structures, provided the elastic limit be not exceeded. 
In all cases of the design of structures it is intended that this limit 
shall not be surpassed, and hence the principle of least work may be 
used with confidence and success in computations of stresses in 
statically indeterminate trusses. 
It is sometimes asserted that the principle of least work is a state- 
ment of a general law of nature which is obeyed not only by 
materials under stress but by.animate beings. While it may be true 
that men and animals endeavor to perform their tasks in the way 
most economical of effort, this analogy has no bearing upon the 
demonstration of the principle of least work. For this demonstra- 
tion rests upon the theorem of virtual velocities, the formula for 
the stored stress energy being the integral of that of virtual veloci- 
ties. On analyzing this proof it is seen that the integration is 
rendered possible by the fact that the deformation of each member 
is assumed to be proportional to the stress upon it. This assump- 
tion indeed is the same as that of the superposition of forces, for it 
supposes each stress to produce its effects independently of the 
existence of other stresses. The theorem of virtual velocities applies 
to all cases of equilibrium, but its integral form does not give the 
principle of least work unless Hooke’s law of elasticity is fulfilled. 
This principle, therefore, is of limited application in mechanics, 
and it states no general law of nature. 
In the method of least squares the conditions and rules for find- 
ing the most probable values of observed quantities are derived 
from the principle that the sum of the squares of the residual errors 
