199 



between certain of these coefficients and the force of gravity, and 

 fifteen equations formed by means of the conditions of equilibrium 

 of portions of the body cut off by the co-ordinate planes, and planes 

 parallel to them. 



III. The six constant terms, and the eighteen linear terms, of the 

 three dilatations or compressions and the three distortions, are com- 

 puted from the corresponding terms of the internal pressures by 

 elimination, or by means of the coefficients of extensibility and com- 

 pressibility, and of pliability. The coefficients of the co-ordinates in 

 those twenty-four terms bear linear relations to the coefficients in 

 the linear and quadratic terms of the three projections of the mole- 

 cular displacement. 



IV. The parts of the nine reduced external pressures correspond- 

 ing to the constant and linear terms of the internal pressures having 

 been determined for each element of the body's surface and sub- 

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

 main nine residual 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 the 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. 



