1889.] which multiply Semr-convergent Developments. 365 
Hence in considering the ultimate value of the series we may 
restrict ourselves to the portion of it mentioned above. 
Now when m is very great we have ultimately by a known 
theorem 
T (m +a) =J2a (m+ a-—1) {(m+a—1)/ey""* 
— % a~1\™ - = 
or /2arm.m** (14) é Cay 1) 
or 2am. m** m™e"™. 
Meee. =a) =. =s, and lett besthe excess of the 
number of I'-functions in the denominator of u,, over the number 
‘n the numerator; then the expression for uv, becomes ultimately 
(Qarm)—? . m= ***. (e/a) ps... cere eeee sore -(B). 
The ratio of consecutive terms, which may be obtained from 
this expression, or more readily directly from (A), is since m is 
supposed very large m~‘p, and hence for the greatest term we may 
take 
Strictly speaking m, would be the integer next over the (in 
general) fractional value of m, which satisfies the above equation, 
but it is easy to see that in passing to the limit we may suppose 
the equation satisfied exactly. Within the specified limits of that 
portion of our series which it suffices to consider, we see at once 
that m,may be written for m when we are dealing with any 
finite power of m, since a and 8 may be supposed to vanish after 
p has been made infinite. We need therefore only attend to the 
last portion (v) of u,, where 
y= (e /: mm fay't — en WT gE 
Now treating m as continuous, i.e. not necessarily integral, and 
putting w for log v, we have 
w=t(1+log m, —log m) m, = tm, when m =m, 
= =t (log m,—logm), =0 when m=™m,, 
Piedolsw ction te when m=m 
dm mm’ | ™ OG a? 
1 
whence putting m =m, + » we have by Taylor’s theorem 
2 
t 
w= tm,—5— +... 
1 
