284 



PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



T , , wi, + in., + 1 



I (m, 1, m,) = - n, 



m. 



m, + m, + 1 

 i,, m z 1) = - - n, 



. (i l + m.,) (in, + m* + ]) 



1 *7 <115 1 J _ 11 



'i ^j wijj - - n, 



MI, (m, 1) 



I (i, ( m.j 2) 

 From (i.) we have 



m,, (i 4 1 ) 



n, 



. (Ixxxvii.). 



wliui 



r o<, 



n i 1.- i , i ^ (loir v) 



It will be needful to find - , - z , 



//;; 



oL; 1 v) , d- (\u<> y) 



^- , and - v* 



//(.] am, dm., 



'"" 7 ) (Jxxxviii.), 



dtii\ 



(hn\ dm'( 



(//i, + m., + 2) >t- T (;/i, + 1 ) 



= e. e, . . . . (Ixxxix.). 



Similarly 



where 



d (M, + ?/(. + 1)- dmi 



cl ~ ( lo g x)/ (/ ' : ^ e - e - ( x c-)' 



c/ J (log x)/dmi dm, =; e 3 (xci.), 



Cl = d 1 T (m, + l)/t/m'i, e, = d 1 Y (m, + l)/t?4 



, =^ (?- r (TO, + ;, + '2)/d (m, + m, + 1)- 



. . . (xcii.). 



e,, e.j, and e 3 can now be readily expressed in semi-convergent series admitting of 

 calculation. 



e _ ' - . i i 



'2iit'i mi 



2in., -J. S (in.,) 

 '* " 



. . . (xciii.), 



in, 



where 

 and 



2(m l + m s + !)' 

 S ( ) = B,/ - B,/j 



" 



S (/<t, 



+~Hi s "+"l?" J 



J 



(xciv.). 



