174 Compressible and Non-Homogeneous Masses [OH. vn 



quantities. With this simplification, equation (447) gives as the value of E, 



E=7rabc 



g 



Jo Jo & 



(!-)= (463), 



.'o A 



in which q is given, now that small quantities of the first order may be 

 neglected, by 



2 _ 

 q ~~ 



a? 



* 



a 2 + X 6 2 + X c 2 + \ ' 



Substituting this value for q 2 in equation (463), we obtain, to our present 

 order of approximation, 



E= 2 7rabc[J-Z(u?J AA + WfJ AB )\ ............ (464), 



where J AA , etc. denote integrals defined by equation (56) of p. 36. 



In calculating Vf(l) we must of course retain small quantities of the 

 first order. The whole potential of a solid of unit density whose boundary is 

 determined by equation (459) is 



r<a)+Ar<a)--*oT [/++(i)j 7 



Jo * 



so that 



o 

 in which, neglecting small quantities of the second order, we may put 



* (1) = e [P - i/DP + i (If)" P - . . .]. 

 Collecting terms, equation (462) becomes 



f *<'>?- ( & r 5 )[ s "' J "-" + 2 ' < >' '"I 



JO ^ \ O ' 



(465) - 



Clearly there will be a solution in which P Q consists solely of terms of 

 degrees 4 and 2 in x, y, z. Let us assume for P the value 



P _ /Po - <r\ [Lot Mtf Nz* 21 fz* 



9 ~~ *'~ """" 



so that 



eP = (?^L\ [L? + Mrf + N? + 2 Vr + 2m? 2 f 2 



4- 2rf a ] . . .(467). 



