TRANSACTIONS OF SECTION A. 319 
This must now be substituted in equation (2) and a second approximation 
obtained. The second approximation is now substituted in (2) to obtain a third. 
If the series thus obtained is convergent it is the reversed series wanted. 
Suppose, first, that b, is positive. 
Then the coefficients of the series obtained by successive approximation are 
clearly not greater than would be got if all the coefficients of powers of x in (2) 
were positive, 7.e., not greater than the coefficients of reversed series obtained from 
wae 3 o0%4 809...) . . . (8) 
b, b, 
where 8, =|0,|. 
If 8 is the greatest of the set 8,8, ... the series from (3) has coefficients certainly 
not greater than those from 
rat 1 a 3 
x a a aerate 
If 6, is negative, the numerical value of the coefficients is clearly not diminished 
if we make every sum on the right-hand side of (2) positive. Combining the two 
cases, we see that the coefficients of the reversed series are certainly not greater 
than those obtained from 
wa Ya 2180+ Bao +. Stic 
The coefficients in this last case can be calculated. 
Wack Pt rae 2 oe 
e have a B.* a, —a) 
whence -  2°(B+B,)—a(y+B)) +y=0; 
: : ‘ 
®= 58 4p) (it 9— VB 2G) +28) +97}, 
the solution with the negative sign before the square root being evidently the 
appropriate one. 
Now the expression for will be expansible in a series of powers of y provided 
[1—oy #28, 9° 7h 
L fel Bs 
is so expansible. 
Values of y for which this is certainly true are given by 
i.e, y? + 2y(B, + 28) —B,2<0. 
The range of positive values of y for which this is true is 
¥< /(B;, x 28)? +.B,’ — (8, + 28), . 
and for these values of y the series obtained from (1) by reversion will also be 
convergent, 
It is assumed throughout that the value of y is positive; a negative value 
_ would be met by changing both sides of (1) and obtaining one or other 
(0,+ or 6, —) of the cases discussed. 
2 3 
The series eg ee 
uy, 91 31 
* Chrystal II. chap. xxvii. § 7. 
