914 



DH JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



It is easy to show that, with the definition of (7) and (9), 



2 "r-(2-2')' 



Tv-o _i ] / /M r + (z-z') 

 D3 -r 1 = (2 - 2 ) log ) ^ - r, 



L r - \z-z) 



(10) 



and there would be no difficulty about extending this method of representing a 

 Zonal Spherical Harmonic of the second kind to the general case ; but all we need to 

 note here is that (7) and (9) are first and second z-integrals of r"\ that they are 

 harmonic functions without singularity in the region E, > 0, and that they are functions 

 of a; — a?', y — y', z — 2'. 



The last property allows a convenient modification of (6), so that we may write 

 finally for a Z force of A-kixv units at {x,' y' , z') 



X or Y force of ^tthv units at {x' , y', z'). 

 For such an X force 



(11) 



u,=^ - {x - x)^^ ^ {v - \)r \ u,= ~{y-y)^^ , ^^^=-{z-z)~ 



dx ' ^ ' ' " ""' " ' 3a; 



These displacements may be exhibited in the forms 



(12) 



u^= — X 





, 32 



'^dxi ^3x-2 ^^ ^dx ^3x3//)^' 3^1 "dxdy 

 \ 3a;3z di^dz dz/dx " oz ( ex- dx oxdy ) ^ 



I 



dy 



d:v-> 



(13) 



which are obviously equivalent to 



3 



dx 



\rh-\ 



'=i-"3.2-^''-l)3-.-^9-^P^^^ ' 

 , f ,32 , ,,3 ,32 ) ., , 



(14) 



that is, to 



ex ^ 



^=fe-3-:^'-^7r^"'''"' 



_3_ ^ 



>)d:V-. 



(15) 



