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Prof. G. G. Stokes on the Internal [May 16^ 



May 16, 1867. 

 WILLIAM BOWMAN, Esq., V.P., in tlie Chair. 



The Riglit Honourable Lord Chief Justice Bovill was admitted into the 

 Society. 



The following communications were read : — 



I. ^^On the Internal Distribution of Matter which shall produce 

 a given Potential at the Surface of a Gravitating Mass.''-' By 

 G. G. Stokes, M.A., Sec.B.S., Lucasian Professor of Mathematics 

 in the University of Cambridge. Received April 18, 1867. 



It is known that if either the potential of the attraction of a mass 

 attracting according to the law of the inverse square of the distance, 

 or the normal component of the attraction, be given all over the surface 

 of the mass, or any surface enclosing it (which latter case may be 

 included in the former by regarding the internal density as null between 

 the assumed enclosing surface and the actual surface), the potential and 

 consequently the attraction at all points external to the surface and at the 

 surface itself is determinate. This proposition leads to results of particular 

 interest when applied to the Earth, as I showed in two papers published 

 in 1849*, where among other things I proved that if the surface be 

 assumed to be, in accordance with observation, of the form of an ellipsoid 

 of revolution, Clairaut's Theorem follows independently of the adoption of 

 the hypothesis 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 originally 

 obtained as a consequence of the hypothesis of primitive fluidity, 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, pro- 

 vided 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 sur- 

 face shall have its actual value require that the internal distribution shall be 

 compatiblewith 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 projierty of 

 a sphere, according to which a central mass may be distributed uniformly 

 over its surface without affecting the external attraction, by applying which 

 proposition to a mass such as the Earth we may evidently, without 

 aff'ecting the external attraction, leave a large excentrically situated cavity 



^ " On Attractions, and on Clairaut's Theorem," Cambridge and Dublin Mathe- 

 matical Journal, vol. iv. p. 194 ; and " On the Variation of G-ravity at the Surface of 

 the Earth," Cambridge Philosophical Transactions, vol. viii. p. 672. 



