Outline of a Theory of Functions of an Abstract Variable 85 
where y’(x) is obtained from the given equation, where the mul- 
tiplication indicated is made in order from left to right. The 
successive derivatives are calculated in the same manner. From 
the correspondence we may conclude that the power series thus 
obtained converges uniformly for 
| a] < pi(1 — €7%2/2H01) | 
where wu is the least upper bound of II F(x, y)|| within the (p1, ps) 
neighborhood of (xo, yo). 
In a similar manner it is possible to prove an existence theo- 
rem for the homogeneous linear differential equation of order , 
whose coefficients are regular functions with derivatives appro- 
priately multiplicable. 
Note: Abstract polynomials have been defined by Gateaux* 
and Frechet,} but since the definition of Frechet has a restricted 
applicability, not being valid for the complex number system, 
we shall discuss only that of Gateaux. It is as follows: 
P(x) is said to be a polynomial of degree 7 if 
(1) It is continuous for every x. 
(2) P(Ax—py) is a polynomial of degree m the complex num- 
bers and yu for all distinct pairs x, y and X, yu. - 
In addition, P(x) is said to be a homogeneous polynomial of 
degree n if P(Ax) = \"P(x) for every pair \, x. Now it was 
shown by Gateaux that every polynomial may be decomposed 
into a finite sum of homogeneous polynomials, and it may also 
be shown that the polar form of a homogeneous polynomial 
P,.(x) of degree n is a continuous, homogeneous, linear function 
Q(x1, %2,--- , x,) of the m increments %1, %2,° °°» Xn such that 
P,,() one O(x, Rye 8 x). 
Moreover, the polar form is symmetric in each pair of its 
arguments, and hence, by T.9, may be expressed in a form 
QinjXiX2 - - - x, where An} is some symmetric element which be- 
longs to Etnj. It follows that 
P,(x) = @[n1*" 
for every x. 
* Fonctions d’une infinite de variables independantes; avec note de Paul 
Levy, Soc. Math. de France Bull., v. XLVI, 1919. 
t Les polynomes abstraits, Journal de Math., 9e serie, v. II, 1929. 
