Royal Society. 803 



tracted from the nine actual reduced external pressures, there re- 

 main nine resMual reduced external pressures for each such element, 

 which form three systems, each suitable for development in series 

 of trigonometrical functions of a different pair of independent co- 

 ordinates. 



V. The parts of the three projections of the molecular displacement, 

 which correspond to each system of residues of the reduced external 

 pressures, are to be expressed by infinite series in terms of the sines 

 and cosines of linear functions of the proper pair of independent co- 

 ordinates, each order of terms containing (except in some special 

 cases) four kinds of trigonometric functions, multiplied by six expo- 

 nential functions of the third co-ordinate, whose parameters are the 

 roots of an equation of the sixth order, and by twenty-four arbi- 

 trary constants. 



From the expressions thus formed are to be computed symbolical 

 expressions for the values of the system of residues or transcendental 

 parts of the reduced external pressures, for each pair of independent 

 co-ordinates, which, by the aid of the equation of fiie form of the 

 surface of the body, are to be transformed into series containing 

 terms in trigonometric functions of the independent co-ordinates 

 only, multiplied by linear functions of the arbitrary constants, which 

 are (in general) twenty-four times as numerous as the orders of 

 terms. 



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- 

 I adent 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. 



