352 



PROCEEDINGS OF THE AMERICAN ACADEMY 



We found that P„ = 1, hence jP^^dfi = 1 — ^m ; also P^^ = p.; 

 therefore jfidij. equals J (1 — ■ m"). In the same way, we might find 

 values for all the values of P, but it has been found that all the inte- 

 grals of the form i Pidjx are satisfied by the equation, 



J} 



P,da = 



1 — 



dPi 



[IX. B.] 



i (J -]- 1) d fi 

 Substituting this expressiou in equations VIII. and IX. we have 



1 — fi^ zi d Pi 



V=2: 



1 _ ^ _|_ &c. + 



zi d Pi I 



ai dfx ) 



iii + l) 



1 — /ii2 ai d Pi 



[X.] 



V' = 2."-\l-f. + &c. + ^-^^'^/-fi]. [XL] 



These last two equations are the expressions 

 for the potential at any point Z upon the axis. 

 To make these equations general, or for the po- 

 tential at any point R distant r, it is found that 

 we can substitute r for z, provided each term of 

 equation [X.] and [XI.] be multiplied by the 

 Pof the order corresponding to the power of r; 

 the P's being functions of the angle 6 which 

 the line CH makes with the axis ZC We have, 

 then, for the potential at H, 



a r dpCa) ^^_ 



a d cos a 



l-cos2a r« dPi{a) | 



T/f o "^ f 1 I 1 — cos^a a dP,(a) „ .^. , „ 



F' = 2.-|l-cosa+— ^— -^-iA-2p^(^)+&c. 



1 -C^ Oj <IPUa) I , 



' I (I -{- 1) ^ d cos a ^ ' ) ^ ■' 



We can readily see that these expressions are true ; for if we make 

 ^ = and r = c, we shall have equations [X.] and [XI.]. More- 

 over we find that each of these expressions satisfies Laplace's 

 equation. 



