587 



ƒ(,(■) = e--' /i (,(,•) aiul u (.7;) = e--'-- 11^ (x) , 



where (4rO takes at once the tbnii of i\BEL's equation and (4^) as 

 its sohition can easily be reduced to its ordinary form. 



In tlie following paragraphs we shall be led to really new api)li- 

 cations. 



4. Let in the first [)lace 



1— ;. 



1_; r in 4- 



-'— r(i— ;.) o. 1^2; 



^-i-Mo 



where, by application of 



r(«) r(r. + i)=.^.r(2«) (5) 



we find that 



and thus, again because of (5) : 



o 



,2»i 



2V^ w/i1[2m— A-f-1) 



= j{i-^) (-1)'" — ^^^^^ ^ ' -,.+, =0, 



in! ri m hi 



V 2 ^ 





SO that : 



^ ^ m/r(m — - + i ^ /v / 2 



Furthermore we find 



/ r(i— /)rf ^ 



2 



by which, on account of (5) : 



