526 Royal Society : — 



crepant measures or estimates of the earth's mean density one more 

 discordant than ever ; so that we have now vahies ranging from 4-7 

 to nearly ()"(3 ; a range, which, in the ahsence of any sutficient ground 

 for selecting any one as true to the exclusion of the rest, would seem 

 to deprive us of all confidence in their correctness as measures, and 

 leave them rather to he classed as estimates of a very rough de- 

 scription. 



But it will he my endeavour to show, that, while none of the 

 methods employed are capable of giving strictly accurate results, the 

 Cavendish experiment is the one which may be relied on as giving a 

 good approximation to the truth, within limits of error (when con- 

 ducted with proper precaution) far less than those to which either of 

 the other methods are liable. 



The three principal methods which have been tried are, — 1st, the 

 SchehaUien or Huttonian, which consists in comparing the total 

 attraction of the earth with that of a mountain mass, by measuring 

 astronomically the inclination of the normals at a given distance in 

 the meridian-plane on each side of the mass ; and then inferring the 

 attraction of the mass from the difference of this inclination from 

 what it would be on an exact spheroid ; 2nd, the Cavendish experi- 

 ment, in which the earth's total attraction is compared, by means of 

 the torsion-balance, with that of a small mass of known dense material ; 

 and 3rd, the pendulum, or Airy^s experiment, in which the total 

 attraction is compared with that at some distance below the surface, 

 or by means of differences, with that of the outer spheroidal shell, 

 whose density may be supposed, approximately at least, to be 

 kirown. 



Now none of these methods give the mean density as a direct 

 result ; for the result obtained, the earth's total attraction, is=^ X the 

 sum of (all the particles divided respectively by the squares of their 

 distances) instead of ^ x (the total mass divided by the square of the 

 radius or mean distance) : and to assume the equality of these, is to 

 assmne the earth to be a sphere, and to have its matter arranged in 

 concentric shells or layers of equal density throughout each layer, 

 both of which we know to be untrue. Mr. Airy has indeed shown 

 that, in the case of his experiment, it is sufficient if we know, as 

 regards the upper shell, the form and density of that portion which 

 is in the immediate neighbourhood of the place of observation, with- 

 out attending to irregularities of distant parts ; but he has not shown 

 that variations of density heloiv and near to his lowest station would 

 not sensibly vitiate his results. 



In order to show the nature and amount of error that might 

 thus be introduced, let AB be a section of the earth through the 

 centre, AC an inscribed sphere of half the diameter ; then it is 

 e^adent that on the supposition of a uniform density throughout, the 

 attraction of the small sphere on the point A would be just half of 

 the total attraction of the earth, although its mass would amount to 

 but i ; and if this small sphere were to have its density doubled, the 

 total attraction at A would be increased by one-half, while the mean 

 density would be altered by only \. 



Now it is quite true that we need not fear any great deviation of 



