Browne — On an Integral Equation proposed ly Abel. 81 



through a multiple point of those curves, and also not to pass through any 

 of the roots of the equation 1 - k{&) = 0. B and \ being thus chosen, the 

 function 



F(Vl ' " j) " l-Ha)-m)-K(°,P) 

 will have only isolated singularities of the first order in the plane of r^, »)2 ; 

 and the integral 



|(| J" (*,,*) I *h Ah 



will have a meaning when taken over any finite area in this plane. This 

 guarantees the convergence of the integral 



x a y& a-i-" v l -P \p (it, v) d/iij dv 



J J (I 



Un (x, y) 



4tt- 



D„ 



da d$, 



jAn l-k(a)-K(H)-K(a,P) 



for any l) n , .!„ (which have the same meaning as B n in former cases). 

 10. We will now prove that if 



V(x,y) - Y 1 U n (x,y) 



Vn, An = X 



exists, then U (x, y) is a solution of our equation. For we should have 

 <p( x >y)- I k(t)<j>(tx,</)dt-\ K(r)<p(x,Ti/)dT - '< K{t,T)<h(tx,Ti/)dtdr 



Jo J o J °J o 



U„(x, y) - [ l h(t) U„(tx, y)dt - 1 * k(t) U n (x, ry)dr - j [ ' K(t, r) ^(Ar, r^ rfr 



Lt 







-Dn, An = CC 





U-(x,y) 



-\: 



i 





Lt 





I>„ 



A„ = 



00 





Lt 





Tin 



A„ = 



CO 



- ^— r ! a; B wfl it-i-«n)-i-&\L(v,,v)diidv 



in- J D„ J A n ' L J J o 



i r r rr += ° r +c ° 



4tt Ji3„JAnLJlo S f J loe^ 



rfa rf/3 



log -•'log-' 



dadfi, 



by use of the substitutions « = xe~ e i, v = ye -9 ?. This last integral (when we 

 integrate first with regard to a and j3) is equal to 



m,v°-=<* * J lo 



J log ! ' ^ d - 



Lt 1 |"+o>r + =o sin^,))! srn0 2 7j 2 



og? log 

 a 



which, by the theory of the double Fourier integral, is equal to 

 [e e i s i + fa s 2\p(xe- a ', ye- 02 \ _ „ „ = ip(x, //), 



since the function 



I efliSi + fl 2 5 2 tp (xe-h, t,e-te) 

 remains integrable for 2 ->- + 00 and 0~ -^- + 00 . 



