40 8 Note 19: determination of law of electric attraction 



In this case we find for the potential of the inner shell when tested by the 

 electrometer M . 



" 



. 

 } 



Let us now assume with Cavendish, that the law of force is some inverse 

 power of the distance, not differing much from the inverse square, that is to 



sa y >let #(r)-r-'^. ...... (21) 



then /(,) = __,!-,. ...... (22) 



If we suppose q to be a small numerical quantity, we may expand / (r) by 

 the exponential theorem in the form 



~ q log r + (q log r) ~ ~ - ....... (23) 



and if we neglect terms involving q z , equations (19) and (20) become 

 I a r [~a . a + b 4<z 2 



[~a . 

 \b 



2 ^~b b a-b ~ *- 



. + & 4a 2 "I 



g log g - log fl -^ 2 J - ...... (25) 



Laplace [Mec. Cel. I. 2] gave the first direct demonstration that no function 

 of the distance except the inverse square can satisfy the condition that a uniform 



spherical shell exerts no force on a particle within it. 







If we suppose that /3, the charge of the inner sphere, is always accurately 

 zero, or, what comes to the same thing, if we suppose B 1 or B 2 to be zero, then 



bf (20) - of (a + b) - of (a - b) = o. 

 Differentiating twice with respect to b, a being constant, and dividing by a, 



'*- = C ' f"(c + 2b)=f"(c), 



which can be true only if f , (f) _ Q & ^^ 



Hence, /' (r) = C r + C lf 

 and 



whence, <f> (r) = C l ^ . 



We may notice, however, that though the assumption of Cavendish, that 

 the force varies as some inverse power of the distance, appears less general 

 than that of Laplace, who supposes it to be any function of the distance, it 

 is the most general assumption which makes the ratio of the force at two different 

 distances a function of the ratio of those distances. 



If the law of force is not a power of the distance, the ratio of the forces at 

 two different distances is not a function of the ratio of the distances alone, but 



