Definite Integrals, with Physical Applications. 379 



Comparing this expression with the series for B„ in the last 

 Article, with which it ought to be identical; we may simplify, by 

 expanding- each term in that series according to the powers of h, 

 and reject all powers higher than If, since they must mutually 

 destroy each other; or we may expand each term in the latter 

 expression, and reject the negative powers of h for the same 

 reason . 



Adopting the latter process, it is obvious that the part of B,, 

 between the brackets, is of the form 



r A + AJi + AJi- + &c. 

 — \ A Q Ai A* 

 l + A !+ ;F + ;F +&C ■ 

 1 ., . .,, . 2.4.6...2w 1 

 and the part without = 13 ^.^.^ • & 



2/i.(2n-l)...(n + l) 2».(2?t-l)...(« + 2) , 2«.(2w-l)...(« + 3) .„ - 

 1.2...M + 1.2...(»-1) 1.2...(»-2) " 



2w.(2m -!)...(» + 2) 1 2«.(2»-l)...(w + 3) J_ „ 

 1.2...(»-1) ~'h + 1.2...(»-2) h* C " 



but since the negative powers of h must disappear, this con- 

 sideration gives 



. _ J_ 2.4.6...2w 2w.(2«-l)...(?t + 2) 

 °~2'"'1.3.5...(2h-1) - 1.2...H-1 



. J_ 2.4.6...2M 2w.(2w-l)...(w + 3) „ 

 2 8h '1.8.5...(2»-l)' 1.2...(»-2) ~ 



substituting these values for A A x &c. in the other terms, and 

 making the proper reductions, we get 



Ji - 1 3m , 5?i .(« — !) ,„ „ 



"~ >7+7 + (» + l).(» + 2) (» + l).(M + f).(» + 8)' ' 



IV. IV. / J «r< III. .') C 



