384 
Proceedings of the Royal Society of Edinburgh. [Sess. 
convergent for certain positive values of v, and that the function defined 
only for these positive values of v is continuous from t> = 0 to oo . We, 
moreover, suppose that constants A and l can be found such that 
f n (v) < AE q (lv ) , (cf. Bromwich, p. 340, T.I.S . ) 
where n denotes the index of any ^-difference. Writing f\v) to denote 
the difference — ^ and putting v— - , we see that 
qv — v x 
which is 
A,/(^) = l/(l) 
J(*) = §E(-gO/(-W) 
0 \ X / 
--[E( -<)/(!)]% !§!!(-„<)/( !>(,<) 
1 00 
=%+ — S> etc * 
X 0 
Repeating, we obtain after n integrations 
“ 0+ *" + ^ + ' ‘ ' ' + ¥ + ^ S - iS e (- 20/” +1 (|)%0 
Now supposing, as stated above, 
f n+1 ( t -)< AV q (U/x) 
the integral on the right side of (7) 
< ^iS E ( - qt)V t Wx)d(qt) 
<A JL 
af +1 x + l 
A 
< 
x n (x 4- /) ’ 
and the asymptotic nature of the expression is established for 
Jjmit | J(x) - a 0 - ajx - ajx 2 - . . ajx* 1 J- = 0 . 
• (7) 
At this point I conclude, for my object is not to work out in great detail 
the sums of special series. There are many points which require elabora- 
tion, such as the convergency of the infinite ^-integrals. In the case of 
