132 THE LAWS OP * 



where 



f 1 r 8 



U a = jdd'd'& sin tr Q 8 1 r' 3 ~ n dr f (1 r' 2 )^ g < (r' 2 ), 



J r V 



and the part of U B due to the general term By* in <f> (/ 2 ), 

 will be 



r Jdffdvr sin ^' (2 8 5 t f r '^~ n+2t ~ g dr (1 - r' 2 )^ (b) ; 



v' r 



which, by employing the formula (3') Art. 1, and rejecting the 

 inadmissible terms, gives for truncated formula 



' ( + 1} 



By continuing to follow exactly the same process as was be- 

 fore employed in finding the value of F^, we shall see that s 

 must always be an even number, say 2s' ; and thus the expres- 

 sion immediately preceding will become 



-2^ 



\ 2 / 



Moreover, the value of V 2 will be 



*%> U i> U *> U s> &c - bein ^ what U > U i> U *> &c - Become when 

 we use the formula (b'} instead of the complete one (b). 



The value of F answering to the density 



by adding together the two parts into which it was originally 

 divided, therefore, becomes 



