Browne — On an Integral Equation proposed by Abel. 73 



where G lies wholly to the right of the line i| = n, and D between the lines 

 i) = n and »j = n + 1 + y. We get after n + 1 differentiations 



m H a(a-l).. (a-^)^-l,(a) rfa 



•** J c #(«) (i) - I ^<?(«+i) (0 eft 



*h-i l f (-!)"«• fV o+ "0(2/)*/ 



+ 



^"+! 2?ri 



da. 



D G(n)(\)-\ t*G(»+V{t)dt 



Integrating by parts, we find that the denominator is equal to 

 (- l) n o (a - 1) . . . (« - n) [ V»-l Q (t) dt. 



Writing a + n + 1 instead of «, and supposing that G lies to the right of the 

 line t) = - 1 , and D between the lines ti •= - 1 and r\ = y, we find 



/»- S3 (/!=**- 



Jo taG ^ dt 



1 *»+l 



+ 



2wi aV+i 



•« 04 " +1 [ y- a - l g(y)dy 



^ (a + 1) (a + 2) . . . (a + » + 1) I *« <? (*) <ft 



Of course we suppose in all these eases that C encloses all the roots of the 

 equation 



t" G (t) dt = 0, 

 J ° 



and that D does not pass through any of those roots. 



7. The equation we have solved is not so general as that of Abel, namely, 



f[x) (j> (ax) dx = ip (a). 



Supposing 1 1 1 > | k |, by putting Ix instead of x, and - instead of a, we can 



reduce this equation to the form 



/ (x) (j> (ax) dx = ip (a) 

 Jm 



where | ju | < 1. Keeping to our notation, we write it 



G(t)f(tx)dt=g(x). 

 We will further suppose < fi < 1. Integrating by parts as before, and 



