| FIELDS] EXISTENCE OF THE POWER-SERIES 87 
Choosing k sufficiently great to ensure that the order of the 
function F(z, #%i+....+u,) is > 2d we write the equation (29.) in the 
form 
c® co c™ 
31. ve +2 Cn- cars) Do Pope US +. 43° £0(Z) =0. 
The A olituée of this nation a 1 we eee 
k k k k k k 
ont, Pade BaP) 
Ais 2 3 n 
We therefore have c® — c® <sc® and consequently c®+sc® >c®) 
for s=2,3,....n. Fractional exponents may or may not present 
themselves in the coefficients of the equation. For the moment we 
shall assume that they do not present themselves. 
In (31.) make the substitution v, = 2°”w, bearing in mind the 
equality c+4+c®=c® and the inequalities c® + sc™>c® for s>1. 
(k) 
Division of all the terms of the resulting equation by z ° evidently 
gives us an equation which may be written in the form 
32. 2% we 4 g¥n-lyn-14 | 42%s 9, (z)ws +... + m(z)w +20 (2) =0 
where the exponents yx, Yn-1,.... Y2 all have positive integral values. 
Again on dividing through by g, (z) we can write this equation in the 
form 
33. W= So (2) +2 { Bo (G) wet... .. +2 (2) wS +....+8n (2) w } 
where the functions £ (z) are power-series in 3 involving no negative 
exponents. Furthermore where b is the constant term in & (2), we 
obtain, on writing w=y7+0, an equation which can evidently be 
written in the form 
34. 9=2{ ho (3) +helz)y?+.... ths (2) n° +....+hn (2)y” } 
where the coefficients h(z) are power-series in z which contain no 
terms with negative exponents. The power-series here involved may 
or may not converge for the value z=1. In any case they will all 
converge for some value 7 and on substituting z=7€ in (34) we 
obtain an equation of the same type in the variables (¢, 7) in which 
all the power-series in € which present themselves converge for the 
value £=1 and the proof of the existence of a power-series in & 
which substituted for 7 satisfies this HRA OE brings with it the 
proof of the existence of a power-series in 3 for » which satisfies 
equation (34). We wish to prove the enr of a power-series 
35. = 2422 +....+AS5S + ..... 
which satisfies the equation (34), and from what we have just seen 
it will suffice for this purpose to prove the existence of such series 
in the case where the power-series h(z) which present themselves 
in the equation all converge for the value z=1. Assuming this to be 
the case then and substituting for y in (34.) the series (35.) we obtain 
for the determination of the coefficients À, a succession of equations 
which express each A, , by aid of additions and multiplications only, 
