200 



VI. By equating the constant factor of each term of the symbo- 

 lical developments thus formed, to the constant factor of the corre- 

 sponding term of the arithmetical developments found by the pro- 

 cess IV., there are formed as many linear equations between the 

 arbitrary constants and known quantities as there are constants to 

 be determined, from which equations those constants are found. 



VII. Cases in which one ordinate intersects the surface of the 

 body in two or more pairs of points are to be treated by a special 

 method. 



VIII. The results of the previous processes are to be combined, 

 and the solution of the problem completed by determining and adding 

 to them the displacements and rotations of the body as a whole. 



The Third Section relates to the internal equilibrium of a rectan- 

 gular prismatic body. 



Processes I., II. and III. The determination of the constant and 

 linear terms of the internal pressures, and the corresponding terms 

 of the molecular displacements, consists in the special application of 

 the methods of the preceding section. The axes of figure are taken 

 for axes of co-ordinates. 



IV. The means and differences of the transcendental residues of 

 the reduced external pressures on each pair of faces of the prism are 

 developed in series of trigonometric functions of the pairs of inde- 

 pendent co-ordinates of the respective faces to which they are ap- 

 plied ; the series employed being of such a nature, that for the 

 edges of the body all their terms vanish. 



V. and VI. An order having been fixed for the consideration of 

 the forces acting on the three pairs of faces, let yz, zx, xy be that 

 order. 



Series of functions trigonometric in y and z, exponential in x, and 

 satisfying the equations of internal equilibrium, with arbitrary con- 

 stant coefficients, are taken to represent the molecular displacements 

 produced by the residual pressures on the faces normal to x. From 

 those series are computed series representing symbolically those re- 

 sidual pressures, which series being equated to the series numerically 

 representing those pressures, the arbitrary constants are found by 

 elimination. 



The formulae thus obtained are employed to compute ideal systems 

 of external pressures on the faces normal to y and z, called "provi- 



