162 
RADELFINGER. 
set of necessary conditions is obtained. This procedure can 
be repeated until no more new conditions result. It is ob¬ 
vious that this method can be applied to equations of all 
orders to deduce necessary conditions. The proof of the 
sufficiency of the resulting conditions has only been com¬ 
pletely accomplished for the order two. 
As a result of the above process applied to equations of 
the second order of the form 
y" = R (xyy f )) (E) 
where R is rational in y', algebraic in y , and analytic in x> 
Painleve obtains the three following equations defining new 
uniform functions: 
(1) y" — 6 T/ 2 -f x, 
(2) y" = 2 y* xy 8, 
(F) 
(3) y" = y ll + ^-«y , -^ + m + *• 
X X X 
The first two are in their cononical form. The last was not 
given by Painleve in his memoir, but was extracted from one 
of the notes in the Comptes Rendus, and may be capable of 
a simpler form. The integrals of (1) and (2) are meromor- 
phic, and can be represented by the quotient of two entire 
functions which are essentially new. These entire functions 
satisfy equations of the third order. The integral of (1) is 
represented in the whole plane by the series : 
y 
^ (* — x.Y — -g- O - *„) ! + 
h (x — x o y + (x — *„) 8 + 
for x = x of y — y 0 , y' — y 0 and h is an arbitrary constant. 
Painleve announces that he has undertaken the develop- 
* ment of the properties of his three new functions, and prom- 
