THE INTERNAL CONSTITUTION OF BODIES. 455 



The double integral / / is to be extended only to the points in 

 respect to which the radius u from the surface of the molecule is /_ r, 

 and the integral / / is to be extended to the points in respect to 

 which uy r. 



By means of a beautiful theorem which M. Poisson has demonstrated 

 in the Memoir already quoted, and in the additions to the Cotmais- 

 satice des Temps for the year 1831, the functions given by the integrals 



/r f* a 



gwpn + 'idp, J ^ gvypn+idp, 



may be represented by series of integer and rational functions of the 

 spherical co-ordinates. Let 2 Hn, 2 H'n, 2 H"n, be these series ; if 

 the functions H'li, H"n shall be found, so that they may be discon- 

 tinued, and such that they are reduced to zero, the first for all the 

 values oi u Z_r, and the second for the values of m Z r, we shall be able 

 by means of the known theorems to reduce the expression for G to the 

 form 



o 2« + l \r"+l /' + ! ^ 



Such are the expressions for F and G which should be introduced 

 into the equation (HI). We might directly employ those which give 

 the values of G, because they are always determinable when the po- 

 sition, figure, and density of the molecules are known ; but the same 

 tiling cannot be done with the expression for F. This integral includes 

 the quantity q, which is still unknown ; and we should not be able to 

 determine it by the condition that it would render the equation (III) 

 identical without previously performing the integrations, an operation 

 which would require the same function to be known. In order to 

 avoid this difficulty, we are about to employ for the pur-pose of deter- 

 mining q a differential equation corresponding with that marked (III), 

 but in which the density q is no longer included under the signs of 

 integration. 



The sum of these equations (I)', when they are differentiated in re- 

 ference to X, y, z, respectively, gives 



