300 Mr. J. H. Cotterill on an Extension of the 



conservation of energy being implicitly satisfied, we have simply 

 to make the work done a minimum, subject to the statical con- 

 ditions of equilibrium. 



The principle in question is (theoretically) sufficient to deter- 

 mine the law of distribution of the stress within an elastic body 

 exposed to given forces ; but if we attempt to apply it by ex- 

 pressing the work done in an element of the body in terms of 

 the elastic forces by the known formula given by Clapeyron, 

 and then make the integral of the result a minimum by La- 

 grange's method, subject to the well-known differential equations 

 expressing the equilibrium of that element, we shall simply fall 

 upon the general equations given by Lame; the method, there- 

 fore, being of no practical value, I shall not dwell upon it. 



There are, however, a number of questions of practical im- 

 portance, in which it is required to find the stresses on the 

 several parts of a structure composed of beams, chains, pillars, 

 and the like. In these cases the work done in the structure can, 

 on hypotheses more or less perfectly realized in practice, be 

 expressed in terms of the stresses at certain parts of the struc- 

 ture j the ordinary method of maxima and minima then furnishes 

 the values of the stresses. And here the application of the prin- 

 ciple seems to me to possess some advantage. Mr. Moseley has 

 given some formulae for the work done in a beam acted on by 

 given forces, but they are not so convenient for the present 

 purpose as one which I shall presently give ; I first, however, 

 premise the following demonstration of a slight modification of 

 the formula (510) in his work on Engineering and Architecture 

 (first edition) : — 



Let a couple M turn through an angle i, then the energy 



expended is Mi if M be constant throughout the angle ; but if 



M be producing a gradual deformation of a perfectly elastic 



body, so that M varies as i, then will the work done be J Mi. 



Now consider the condition of a transverse slice of a beam 



made by planes originally parallel at a distance dx, but now, in 



consequence of a bending moment, M acting on it, inclined at 



an angle di, which, as is shown in all works on applied me- 



M 

 chanics, is equal to ^ dx, where E is the modulus of elasticity, 



and I the moment of inertia of the sectional area ; the energy 

 expended in distorting it is J Mdi, or ^-r dx, and therefore, by 



M 2 

 the law of conservation, the work done in the slice is ^y dx, 



C M 2 

 and the work done in the whole beam \ ^j dx, a formula which is 



