402 Prof. Challis's Theoretical Considerations respecting 



second term of the value of V, gives rise to a repulsive action of 

 each atom on neighbouring atoms varying inversely as the /owr/A 

 power of the distance, and that the velocity expressed by the 

 first term, and accompanied by the condensation a, also causes 

 a repulsion between neighbouring atoms, but of incomparably 

 less magnitude than the repulsion of the first kind. Since, how- 

 ever, this part of the velocity, as well as the condensation, varies 

 simply as the inverse of the distance, it appeared that the dy- 

 namic effect of an aggregation of atoms on any single atom 

 might be of sensible magnitude, and be either repulsive or 

 attractive according to the distribution of the condensation of 

 the incident waves about the surface of the atom. 



Now quite apart from the above theory of molecular action, a 

 general expression for the resultant of the action of sun'ounding 

 atoms on a given atom may be obtained as follows, on the hypo- 

 theses that each atoni is insulated, and that the forces urging it are 

 sensible only at insensible distances. Suppose the collective atoms 

 to constitute a medium, either solid, liquid, or aeriform, and 

 conceive a surface of equal pressure and density .of the medium 

 to pass through the position A of a given atom. Then the result- 

 ant of the molecular action on the atom is plainly in the direc- 

 tion of a normal to this surface, and tends from the denser to 

 the rarer part of the medium, or from the rarer to the denser, 

 according as the component action is repulsive or attractive. 

 Within the sphere of molecular activity the change of density 

 A/3 corresponding to a given change A^ of the distance from 

 the tangent-plane at A may be considered constant : or D being 

 the density at A, the density p at any distance z from the tan- 

 gent plaue is D -f Q,z, Q being constant. Hence if r be drawn 

 from the atom in any direction making an angle 6 with the 

 normal, and ■>/r(r) express the law of the aggregate action of the 

 atoms included in the small space 



A?* X rA^ X r sin ^At;, 



the number of which is proportional to p and is supposed to be 

 very large, then the resulting action in the direction of the 

 normal is proportional to 



2 . Ar . rM . r sin ^A??(D + Qr cos 6) cos d-<^{r). 



If this quantity be treated as an integral, and the integration be 

 taken from 77 = to r] = 27r, and from ^=0to ^ = 7r, the result is 



3 



Or, integrating from ? =0 to ?•= infinity, because by hypothesis 

 the values of ■\lr{r) are insensible for all but very small values of r, 



