OP ARTS AND SCIENCES. 



351 



The values of P^, P^, P.,, &c. can be readily seen from equation III. 

 Below we write a few values which are determined from [HI.]. 



P, = l. 



P^ = cos a or fJL. 



3 ,,2 1 



a 1 



P. = &c. 



Equations [IV.] and [V.] are general expressions for the reciprocal 

 of the distance, or -^r^. We may therefore express 



ZP 



^•^, = ^(i^ + P,^ + P,|+&c. + i^^ 



Z P 



dS 



=^ 17 Po+ Pr, + P.-. + &<^- + ^i 



[VI.] 



[VII.] 



Equation VI. must be employed if z is greater than a, and fVII.] if z 

 is less. 



We made dS a, small element of the portion of the spherical shell 

 bounded by the coil. 



In polar co-ordinates 



dS = — a^dfi dcj) ; 



5 will evidently be the integral of the above when 

 <l> is integrated between the limits of and 2ir, 

 and n is integrated between cos a and 1, or 



S: 



\7ra'- fd^ 



Fig. 3. 



Hence the potential at Z due to the included shell is F= ^-p, 

 where ZP is the distance to each element of area of the sphere. 



V= 2na I JP.^M + IJK^I^ + &c. + ~,jP^d,. I . [VIII.] 

 V' = 2^j[Jp,df.^ Ifkd^ + &c. + 5j/>,./^ } . [IX.] 



