Browne — On an Integral Equation proposed by Abel. 63 



s and q being positive integers. We have 



* sinr£ 



-?a sin t£ 



'M^p (pe-r)dT 



r^r-, py e b ( S "T) 



161' 



c t5 i// (pe- T )dr 



Mpy 

 2AJ 



-g\ 

 log'' 



«t (S-7)«?T 



that is to say, less than 



q\ 



(S- 7 )logi- -?A(S-7)- 



8 7- 5 



Mp a 



q\ (y - S) 



1 - e 



( y -S)hoe^+q\\- 



iMp a 



7-, 



q\(y-S) 



where t is a quantity which tends towards zero with A. 

 The integral 



r ~ s * sinr g 



- q\ T 



6 T \p(pC~ T ) dr 



breaks up into a number of sums like the following 



-(»•+!) A 



-(r-l)A. 



with the possible exception of an extra term 



Smr£ t5 _ T 



e <p(pe )St, 



-(S+1)\ T 



which is less in absolute value than 



\M 7 4(8-7) 



\i\ P 

 Putting r + A instead of t in the second of the above pair of integrals 

 we see that their sum is equal to 



-(r+l)A 



sinr| 



e T5 i//(pc- T ) 6(t + a)5^ (pe-T-A) 



dr ; 



hence the sum of all the pairs is less in alsolute value than 



log! 



C" 5 ip (pe~ T ) e(-"-+A)5 yp (pe-T-A) 



r + A 



(-6 



dt 



= p7 



,er(S-y) I fl(r) - eM*^) 0(r + A) | dr, 



