320 REPORT—1905. 
gives y='236 as upper limit, it being otherwise known that the reversed series 
is convergent up to y=1. 
75) Je 
The series Sah Soe 
ae hae 
gives y=‘236 as upper limit; the reversed series is actually convergent for all 
values of y. 
It is to be observed that no result is given for the case when any of the d’s 
become indefinitely great, since 8 ee [vas 28)? + B,?—(8, + 28) | = 0, provided 
8, £o, a case which does not arise. 
The case excepted may be dealt with as follows :— 
Suppose y=O,.t . . 1 +dna" +... : . . : F ay 
where 6 =, and of course the series on the right of (1) is convergent for values 
of x within a certain region, say |'|<p. 
Then, if be any quantity such that |oj>p we have L he = 0, and none of the 
n.,2 O 
quantities jon other than finite. 
Oo 
Transform (1) by the relations ov =é 
wee He arbi + Sngn y he, 
oC o 
and we have a series to which the previous reasoning is applicable, and if € is 
expansible in a series of powers of y, then a = E is also so expansible, 
oe 
The condition is y< ./(8, + 28)” + By? —(B, + 28) 
Dy | 
lo 
where B,= , 
“and B is the greatest of the set 
o 
OY 0 eg ase RO pS os be 
putQr=£ y=} E42. B+ EB... 
Bi=38=3 y<"lb. 
The same principle can also be applied to the first case considered; thus the 
reversed series of (1) will certainly be convergent for values of 2 less than the 
greatest of the quantities [/(B,|o + 28,|o")? + B,20? — (B,|o + 2B,|0”) | 
where o is any quantity whatever and B,,|o” is the greatest of the set 
Bylo; Ba|O2s Ba OF ce 
I am unable to find any general expression for the maximum value of this 
limit, owing to the difficulty arising from the fact that different values of o require 
different A’s to be selected to give the greatest coefficient, It is to be observed 
that the values of the above limit for o=0 and for c= are both zero, 
As an example of the limits obtained we have for the case of 
ae ae 
eee eo. * . gis see 
o= 1} 2 4 1 2 P3} 
y<'315, 369, 324, -236, 152, “111. 
