we obtain 

 Lt 



£=oo 



Jo 



Browne — On an Integral liquation proposed by Abel 

 ~s,mZ,(v p - o m ) 



67 



dv p dv m P(v p - v p - x )dv p .i 



fp.i 



P {v p _t - v p _ 2 ) dv p _. 



P{Vi-Vi)dvi 



P x '» m -v m - l )dv m _ l . . . 



P{v p+ ,-v p+l )dv pU 



= Lt 



sin £{v p - v m ) 



S {v p ) T (v m ) dv p dv„. 



o J o u p u ™ 



This, by reasoning similar to that already employed, is equal to 



7T S(u)T(u)du, 



- 



if this integral exists. 



Now it is easy to see that if two functions Y(t), Z(t) are integrable in 

 an interval (a, b), the function 



W(v) = 



Y{v -u)Z(ic)dit 



is finite in this interval, unless perhaps for such values as v - v, = u = a, 

 i.e. v = 2a, when Y{t), Z(t) are both infinite for t = a. 



1 , „ 1 



If Y{t) becomes infinite like 



\t 



, and Z(t) like 



\t 



(0 < A < 1, < fi < 1), 



it can be easily proved that W (t) becomes infinite for t = 2a like 



1 



\t-2a\ A+M ~ 1 



We name \, /n, etc., orders of infinity. 



By successive applications of this theorem we see that if the highest order 



of infinity of P (w) or of K(t) is A, then the highest orders of S(u) and T (u) 



are p\-(p-l) and q\-(q-l) respectively, q being equal to m-p. 



Hence the highest order of infinity of S(u) T (u) is m\ - (to - 2), and this 



function is integrable from to oo if 



toA - (to - 2) < 1, 

 that is to say, if 



1 



to > 



1- A 



Hence we have only to take m = d, the first even number satisfying this 

 condition. It is then easy to show that the integral 



Q= [ I Vt*K(t)dt I 'Ida 



J D I J o 



has a meaning for any m > d, even or odd. 



K.I.A. PROC, VOL. XXXII., SECT. A. 



[10] 



