871 

 potential energy foi- ;■ ^ oo being chosen as zero. Further 

 /t = — ;, /■ being Planck's constant. Finally the attraction is sup- 

 posed to decrease sufficiently rapidly with increasing ;•, for tiie integral 

 in (5) to be convergent. 



§ 3. For tlie proof of the first of the theorems mentioned in § 1 

 we develop /-" according to ascending powers of A. The first term 

 becomes : 



00 It ;i27i2;i2;r 



- 8^ '' ff ff / f'"-!''' ''"' '^> «'" ^= drdd/W^dy_,dy^d'i . . (6) 

 1) (I 

 The integration according to 8^, y,^ and ff, the coordinates /■,, é', 

 and Xi being kept constant, must necessarily give 0, if the condi- 

 tions {A) are fulfilled. In fact the result of this integration can be 

 represented as the potential energy of a molecule 1 relative to a 

 great number of superposed molecules 2, ail with the same centre, 

 but further as regards their orientations uniforiuly distributed over 

 all the possible positions. By this supei'position at the limit a sphere 

 is obtained in which the agent is uniforuily distributed over con- 

 centric shells. According to a well known theorem of tiie theory of 

 potential, the potential outside such a sphere is constant if the total 

 quantity of the agent acting according to Coulomb's law of the 

 inverse square of the distance equals 0; from this, together with 

 the assumption mentioned above about un becoming for r = od, 

 follows the above result; the theorem in question is hereby proved. 



§ 4. The odd powers of h in the development of /•" (§ 3) occur 

 in the following form : 



" 8^ ■ (2^1)/ • ''"'^'JjJSjJ'""'"'^'''' "'■" ^^' '"' ^.drdS^dOJy^dy.dr, (7) 



7 



q is here a whole positive number. 



If the conditions {B) are fulfilled, the integration of this integral 

 according to 0^, ■/_, and </,, the coordinates r, 0^ and Xi heing kept 

 constant, will again necessarily give 0. This results froni the fact 

 that each contribution to the integral, obtained from positions of the 

 second molecule indicated by definite values of 0„, '/^^ diwiS. (f , with 

 the ranges dO^, dl,. d/f, is neutralized by the contribution obtained 

 from positions, which can be derived from the first by a revolution 

 through ail angle of 2.t/^' about one of the axes of inverse symmetry. 

 With this the second theorem mentioned in § 1 is proved also. 



56 



Proceedings Royal Acad. Amsterdam, Vol. XVIII. 



