Page:American Journal of Mathematics Vol. 2 (1879).pdf/23

Rh The equations of condition for equilibrium in these cases, from equations 2, will be the three following:

Three other equations of condition result from the conditions that $$N_2,$$ $$N_3$$ and $$T_1$$ each equal zero. These give in connection with equations (1)

These equations, as it will afterwards be seen, aid in the determination of the displacements $$u,$$ $$v$$ and $$w.$$ The last two of equations (3) may be integrated at once, and will give

In which $$f$$ and $$F$$ signify any arbitrary functions of $$y$$ and $$z$$ whatever; they correspond to the "constants" of integration and must be written because the intensities of the internal stresses are, in general, each functions of $$x,$$ $$y$$ and $$z.$$

Denoting by $$f'_y\,(y,z)$$ and $$F'_z(y,z)$$ the partial derivatives of $$T_3$$ and $$T_2,$$ respectively, in respect to the variables indicated, the first of equations (3) may be integrated, and will give

The quantity $$\Psi\,(y,z)$$ is any arbitrary function of $$y$$ and $$z,$$ and it will now be shown that in general it is independent of $$y$$ and $$z,$$ as well as of $$x,$$ and that many of the cases of pure flexure it may be put equal to zero. Rh