Page:American Journal of Mathematics Vol. 2 (1879).pdf/52

Rh To illustrate the effect of resisting the lateral distortion of the fibres the following procedure may be employed. In equation (1), as is sometimes done, suppose $$\frac{dv}{dy} = \frac{dw}{dz} = \frac{1}{4}\frac{du}{dx}$$ and $$\lambda = 2\mu$$; then there results $$N_1 = 3\mu\frac{du}{dx}$$. If there is no lateral contraction then $$\frac{dv}{dy} = \frac{dw}{dz} = 0$$ and $$N_1 = 4\mu\frac{du}{dx}$$, giving an increase of $$\frac{1}{3}$$ over the result obtained with lateral contraction.

It is not by any means an insignificant fact that the same increase in the example taken would almost entirely make up the discrepancy observed.

Now in regard to the method by which N was established in equation (32) and those following. The principles there applied are perfectly general not being restricted to any assumptions or kind of material; they may be applied in absolutely all cases.

The restriction in the application lies in making N a function of z and y only, for any given section, and in the present case, as has been shown, that does not affect the generality of the results.

It is believed that the principle of least resistance has not heretofore been applied in the discussion of this problem.

It is also believed that the determinateness of the problems of elastic equilibrium has not before been so generally stated. Clebsch in his admirable work on the theory of elasticity gives a demonstration of the principle, which, however, appears to the writer to be somewhat unsatisfactory.