236 Royal Society :— ~ 



Clairaut's Theorem follows independently of the adoption of the hy- 

 pothesis of original fluidity, or even of that of an internal arrange- 

 ment in nearly spherical strata of equal density. 



But though the law of the variation of gravity which was ori- 

 ginally obtained as a consequence of the hypothesis of primitive flui- 

 dity, and was afterwards found by Laplace to hold good, on the 

 condition that the surface be an ellipsoid of revolution as well as a 

 surface of equilibrium, provided only the mass be arranged in nearly 

 spherical strata of equal density, be thus proved to be true whatever 

 be the internal distribution, the question may naturally be asked, Does 

 not the condition that the potential at the surface shall have its actual 

 value require that the internal distribution shall be compatible with 

 that of a fluid mass, or at any rate shall be such that the whole mass 

 shall be arranged in nearly spherical strata of equal density ? Such a 

 question was in fact asked me by an eminent mathematician at the 

 time to which I have alluded. I replied by referring to the well-known 

 property of a sphere, according to which a central mass may be dis- 

 tributed uniformly over its surface without affecting the external at- 

 traction, by applying which proposition to a mass such as the Earth 

 we may evidently, without affecting the external attraction, leave 

 a large excentrically situated cavity absolutely vacuous, the matter 

 previously within it having been distributed outside it. It is known 

 further that the mass of a particle may be distributed over any 

 surface whatsoever enclosing the particle without affecting the ex- 

 ternal attraction ; and in this way we see at once that we may leave 

 any internal space we please, however excentrically situated, wholly 

 vacuous ; nor is it necessary in doing so to introduce an infinite 

 density, by distributing the whole mass previously within that space 

 over its surface, since that mass may be conceived to be divided into 

 an infinite number of infinitely small parts, which are respectively 

 distributed over an infinite number of surfaces surrounding the space 

 in question. These considerations, however, though they readily 

 show that the internal distribution may be widely different from 

 any that is compatible with the hypothesis of primitive fluidity, do 

 not lead to the general expression for the internal density. Circum- 

 stances have recently recalled my attention to the subject, and I can 

 now indicate the mode of obtaining the general expression required 

 in the case of any given surface. 



Let the mass be referred to the rectangular axes of #, y, z, and let 

 p be the density, V be the potential of the attraction. Then for any 

 internal point V satisfies, as is well known, the partial differential 

 equation 



ePV , d 2 V , d 2 V 



~d^ + ~df + dl = ~ 4 ^ 0) 



or, as it may be written for brevity, VV=0. This equation may be 

 extended to all space by imagining the body continued infinitely 

 but having a density which is null outside the limits of the actual 

 body ; and by adopting this convention we need not trouble our- 

 selves about those limits. Conversely, if V be a continuously varying 



