976 DR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF THE 



where 



J\+^fi + \ 



T.^l + ..,3_,4+p'^.^ + log^-^^P^, (7) 



and € is finite, and tends to zero for m infinite, for all values of t. 

 Hence, by the First Theorem of the Mean, (5) and (6) give 



■P 

 where e has the same properties as c. 



Now, from (7), 



'-^'^'n-a) iAtt^^)'-"'''*^ w 



so that T is an increasing function of t, and the asymptotic value of the integral in 

 (8) obviously comes from very small values of t. 



Following a well-known method, we change the variable from t to T. 



From (7) and (9), near t = 0, 



(IT 1 , ., o\. \ 

 dt 2 ^ ^ ' 



T = l(a2-p-2)<^ 



(10) 



dT Ja'-p^' 

 From these and (8) we find 



\ay Jo Ja^-p^ 



Similarly 

 and 



Consider again 



\ K^^r— ^F^^- f/K = n M—^ 2^'(a^ - p2) 2 ,71 2 ''f: (13) 



h dp" ,J,Ma \ 2 J ^ ' ' a™ ^ ' 



jo J„.^'<« 



^„j r JJmp'fJJmpiGJmat ^^ /^gv 



jo J,Jmat 



in which we will suppose p<.a, p'<.a. 

 From (1), 



^^ ^ "'^ ' 2m Jl + aH"^ 

 so that 



t' = jL/ 1 ■],„imp't J,„impt _.. 



2 7" \/l + a2^2 ,]^jmat J,Jmat " 



" YV ?7i N/2a2 - p2 _ p'2 (^^ j ' • • • • • ' ^ ' 



by the same process as was used for I. 



