108 REroRT— 1882. 



Before proceeding further I shall prove a very remarkable relation 

 between the slope of the surface of an elastic horizontal plane and the 

 deflection of the plumb-line caused by the direct attraction of the weight 

 producing that slope. This relation was pointed out to me by Sir William 

 Thomson, when I told him of the investigation on which I was engaged ; 

 but I am alone responsible for the proof as here given. He writes that 

 he finds that it is not confined simply to the case where the solid is 

 incompressible, but in this paper it will only be proved for that case. 



Let there be positive and negative matter distributed over the horizontal 

 plane according to the law wh cos {zjh) ; this forms, in fact, harmonic 

 mountains and valleys on the infinite plane. We require to find the 

 potential and attraction of such a distribution of matter. 



Now the potential of an infinite straight line, of line-density p, at a 

 point distant d from it, is well known to be — 2/(p log d, where ji is the 

 attraction between unit masses at unit distance apart. Hence the potential 

 V of the supposed distribution of matter at the point x, z, is given by 



F = - 2nivhr" cos ^ log s/ [x^ + (4 - 2)2} di: 



= — f^ivlib ■ 



'sm^~\og{x' + (i'-,y} 





It is not hard to show that the first term vanishes when taken between 

 the limits. 



Now put t = , so that sin - = sin — cos -+ cos — sin -, and we 



X b b b 



have 



V = luwhb sm — cos - + cos — sm - ) 



^ J_«V b b b bJl + f' 



But it is known ^ that 



{+" t sin cf, di _. f+^t cos ct., ^ 

 — ^ ^7-^ = 7re '^j dt = 0. 



Therefore V=2iruw'kbe cos-. 



b 



o 



If g be gravity, a earth's radius, and c earth's mean density, 27r/t = -'^~. 



Zao 



And TT Sqwhj ""' * z . ,. 



V=-~rbe cosr- . . . (4) 



zac b 



The deflection of the plumb-line at any point on the surface denoted by 

 a; := 0, and z, is clearly dVjgdz, when a; = 0. Therefore, 



,,,„,. 1 3n7ijh . z /„v 



the deflection = X -^r-^ sm - . . . (o) 



g zac b 



But from (2) the slope (or—-, when z is zero), is — ^-— ' sin -'. 



dz 2v b 



Therefore deflection bears to slope the same ratio as v/g to ^ aS. This 

 ' See Todhunters Int. Calc. ; Chapter on ' Definite Integrals.' 



