450 Mr. S. Batterworth on the Coefficients of 



Since 

 Q» W =iPnlo gl — - - ~Y~n~ n ~ l+ 3(^T) ?l " 3 



272-9 



P«-5+... 



P» log *(! + *) 



and 5(n — 2) 



+ 3(^-2) (Pn ~ Pn - ,) ~ '-'J 



3(2/2-2) 



the logarithmic term in (18) is 



V n (iv)\og\^/TVv 2 . 



Also, if s takes the values n, n — 2, n—4z, ... and 



a _i iio/ 2 ™ -1 272-3 2n-5 "> 



A n - 10g ¥ + 2 | y-^ - 2 (2n-l) + 3(272-2) "" • " J 



A _ ^~3 . 272-7 



(19) 



271-1' ""* 272-3' 



the value of V along the axis of symmetry becomes 



P,(^)log v / l+^: 



Therefore to obtain the solution sought it only remains to 

 expand (l + x 2 ) m in a series involving V n {ix) and to apply 

 the results (17) and (19) to each term of this series. 

 The series in question is 



(l + a?)*=(-)'"2*|m ^ ••• ^ + ^|(4m + l)P 2m (^) 



. 772 4772+1-p ., . 



, , . _.w(m — 1) (4772 + 1) (4m -1) \ r9m 



+ (^-~ 7 ) L |g ' (2m-l)(27n-3) F2 -^ ) - '">' (20) 

 from which 



l + a*=-|p,(i*) + fp o (M>), 

 (l + * 2 ) 2 = ggP*^)- gjPi(w)+ ^?o(«). 



