60 



Proceedings of the Royal Irish Academy. 



I have discovered the following formula for the solution of this 

 equation : — 



1 r « a (V 1 ~ a i/'(2/) ( fy 

 <f> (.*■) = / (x) = v-. — ^ 



da. 



- f ^ K{t) dt 



To explain this formula, which holds good f or < x < a : we suppose that 

 | \p (.r) | is less than Mxy, M being a constant and y > - 1, and that K(t) is 

 integrable for o < t < 1. Putting a = ij + i0, we suppose that D is an infinite 

 straight line in the n-plane parallel to the 0-axis, not passing through any 

 root of the equation 



1 - 



t°-K{t)dt - 0, 



and lying between the lines »j = - 1 and >) = y. It can be shown, with 

 certain very general assumptions about the nature of Kit), that the roots of 

 the equation 



1 - \* PK(t)dt = 



are finite in number, and that the integral 



f K(t) dt 



tends towards zero with : — -, so long as a remains to the right of the line 



I a I 

 i) = - 1.* The integral along D is not to be taken arbitrarily, but equal 



portions of D must be measured above and below the ij-axis, and the limit 



found when these equal lengths tend to infinity. 



2. Supposing that this limit exists, we will prove that /(a;) satisfies the 



equation 



<f> (x) = i// (x) + Kit) <p (tx) dt. 



. 



Take the length of the equal portions of D to be £, and call their united 

 length DL Then 



I(x) 



We find easily 



Lt Tir e-x Lt J_ 



da. 



^ l-\\t«Kit)dt 

 I(x,£)-\ Kit)I(tx,£)dt 



2«L*r 



-l 



y ^iy)dy 



da. 



* See These de Doctoral, pp. 74-79. 



