304 Royal Society. 



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

 satisfying the equations of internal equilibrium, witluarbitrary 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 formula? thus obtained are employed to compute ideal systems 

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

 sional pressures," which are developed in trigonometric functions of 

 the independent co-ordinates of the faces to which they are conceived 

 to be applied. Should the provisional pressures agree with the 

 actual residual pressures on those faces, the process is complete; 

 should they not so agree, the provisional pressures are to be sub- 

 tracted from the actual residual pressures, leaving systems of re- 

 mainders called " Secondary pressures." 



The series representing the molecular displacements corresponding 

 to the Secondary Pressures on the faces normal to y are to be found 

 in the manner already referred to. The formulae thus obtained are 

 to be employed to compute an ideal system of " Provisional Secondary 

 Pressures" on the faces normal to z, which are to be developed in 

 trigonometric functions of x and y. 



Should the provisional secondary pressures thus found agree with 

 the actual secondary pressures on the faces normal to s, the process 

 is complete. Should they not so agree, the provisional are to be sub- 

 tracted from the actual secondary pressures, leaving a system of re- 

 mainders called " Tertiary Pressures" on the faces normal to z, whose 

 effects are to be computed in the usual manner. 

 Process VII. is not required. 



Process VIII. consists in combining the terms of the molecular 

 displacements due to the constant and linear terms of the internal 

 pressures, the residual pressures on the faces normal to x, the se- 

 condary pressures on the faces normal to y, and the tertiary press- 

 ures on the faces normal to z, and finally determining and adding 

 to the other terms, those depending on the displacements and ro- 

 tations of the prism as a whole. 



The Fourth Section relates to the integrals of the equations of the 

 internal equilibrium of an isotropic elastic solid. 



The constant and linear terms of the internal pressures are to be 

 determined by the methods described in the previous sections, for all 

 solids, whether isotropic or not. 



The transcendental terms of the internal pressures and molecular 

 displacements in an isotropic elastic solid, require a special method 

 for their determination. 



The three projections of the molecular displacement, with all their 

 functions, in an isotropic solid, are deducible from one primitive 

 function or series of primitive functions of the co-ordinates, by cer- 

 tain processes of derivation, distributive, but not necessarily commu- 

 tative. 



