460 



Mr. R. Hargreaves on 

 9% 



differentiation ; also the value of ^ when (143) is applied 



has the shape belonging to the original formula of § 30. 



§ 36. It will be observed that the results of the last few 

 pages all attach to the use of the reciprocal form or the 

 second integral in (124). The comparison of a direct integral 

 under an seolotropic law of potential with the general solution 

 has only been made for one position, the centre. A method 

 of derivation well known in isotropic potential is applied to 

 extend the result, and the new forms of the potential integrals 

 have been verified directly. In Part II. of Thomson and 

 Taifs ' Natural Philosophy/ p. 44, § 494 k, a direct integra- 

 tion for the potential at any point within an ellipsoid is given. 

 A similar process for seolotropic potential requires a modifica- 

 tion of the argument of § 33. Since u v has x — x\ . . . for 

 variables, when a fine cone is drawn from x' y' z' in the 

 direction (Imn), u P still becomes ru F (lmn), but r is given by 

 the quadratic 



If instead of using r 2 for one end of the cone and ?' 2 2 f° r the 

 other, we take ^(r 1 2 -\-r 2 2 ) for each, the internal potential is 

 given by 



lep C dco r , / ~du d?/ ~dv \ 2 /0 "I 



* i= t)^?V -«•(»,*«) + (*B7 +y *m+*Tn ) /2 ""'j (U4) 



the dashes now dropped because only one x occurs : also u 

 and u v have Imn for variables except in the term where 

 u a {xyz) is written. Thus the form for the coefficients is 

 dictated by the quadratic determining r. If we set out from 

 the value of (/> implied here, viz.. 



-r~\ i which is equal to I , — =-> 



and obtain L . . . by differentiation as before, we reach the 

 same result. By the two methods we get 



T . 1 Cdco(al + c'm + b'n) 2 . 1 



L = a<f) --\ 



7T 



2 U * 



and 

 L 



,_ , _ 1 Cd<o(c'l+ bm + a'n)(b'l + a'm-\-cn) 



d J 



, , 1 f dco(Om 2 + Bn 2 -2A'mn) 

 + a</> =-J 



> (H5) 



Mo-WpT 



T r , , , 1 Cdco(B'/m + C'lm-Amn-A'/ 2 ) 



Hn'ttnS 



J 



