u = 2 [J„+^,+i (.r) — /„+/,+3 (.^) + /«+/,+5 (.^0 — ...]• 

 thus in all cases 



ƒ 



/„ (ii) /^, (.i--p) c^^i = 2 [i„+,+i (.■) - /„+,.+3(aO H- i„+,.-H.. (.f) - ...] (8) 

 Applying this result, the equation (7) takes the form 



00 X 



and comparing the two members we have 



«/<+! = 26„ 



c,+3 = - 26„ + 26, 

 .^+4 = - 26, + 2/>3 

 etc. 

 thus 



iLi U £i ^ 



and 



The solution of the integral eqnation (1) in the form (2) may be 

 easily deduced from ihis eqnation. For putting [) ^ 0, we obtain 



In this case 



ƒ (.f) = c, ;, (.1') -f c, /, (.r) + 0-3 7-3 (.t') + . . . 



thus 



and 



dx 2 2 ^ ^ ^ ^ 2 " ^ -* ' 2 3 V / 1 



ax 



which, according to (5) may be written 



(/.r J x—^ 



3. We shall next show that the solution (9) may be written in 

 the same manner as the solution (2). It is however convenient to 

 examine tirst the special cases p =z: 1, 2, 3. 



