34 



since f^(l) = 1. Using this result in Equation [84], the relations [82] follow. Hence, by 

 induction, these relations hold for all n. 



By Equations [78] and [82], a^ and 6^ may be easily evaluated. Since a° = 4>(y ^ and 

 b° = 0, a^ and 6| can be found by repeated use of Equation [78]. Then Equation [82] can be 

 used to find at and 6f . 



< 



1 5V 



n\ dx" 





^: 



= 





< 



2 





K 



2 

 (n + 1)! 



5"-l 



(n + 1)! 



dx"-'^ 



< 



2 





bl 



2 

 in + 2)! 



a"-2 



{n + 2)! 



,9a;''-2 



< 



2 

 (71+3)! 



^^„_3 (.,,,- 3.,.) 



^i 



2 



^n-3 



(n + 3)! 



dx"-^ 



< 



2 



^^„_4^'^— -""^ry-^ 



K 



2 



^n-4 



(n + 4)! 



(71 + 4)! 



aa;"-'* 



(<?ij 



(2<i_) 



[87a] 



[87b] 



[87c] 



(4<^yyy. -V,,..) [87e] 



SOURCES AND DOUBLETS 



When the singularity at r . is a source or doublet, the expressions for the force and mo- 

 ment take particularly simple forms. Using the values for the coefficients given by Equation 

 [87] we have the source 



F^^{i) = inpa° 



dcf) 



dx 



ley . 



36 



dz 



[88a] 



[88b] 



[88c] 



F,ri;=-4;rp<q,(r.) 



[88'] 



