160 
RADELFINGER. 
Painleve persevered and undertook to determine all equa¬ 
tions of form (A) whose integrals take only a finite number, 
n, of values around the movable singular points and have 
no lines of essentially singular points in other parts of the 
plane. By a laborious and extensive analysis, based on the 
theory of birational transformations, he solved the problem 
and presented his results in a voluminous memoir, crowned 
by the French Academy, and published in 1891-1892.* By 
the solution of this problem the lower limit is raised and the 
gap between it and the upper limit partially closed. Unfor¬ 
tunately, however, no new functions were added to those 
already known. The resulting equations can be integrated 
by rational functions, by quadratures, or they degenerate 
into linear differential equations. Thus stands the problem 
today. 
EQUATIONS OF THE SECOND ORDER. 
Since the appearance of the memoirs of Fuchs and Poin¬ 
care several attempts have been made to solve Fuchs’s prob¬ 
lem for equations of the second order of the form 
f(xyy'y") = 0, (C) 
with rational coefficients. 
Picard .—Picard tried to use Poincare’s analysis for the pur¬ 
pose, but found that it would not, in general, apply, because 
the resulting transformations were no longer birational, but 
biuniform. However, he worked out a special case, in which 
the transformations were limited to being birational. No new 
functions resulted. His work, together with much more per¬ 
taining to functions of two variables, was embodied in his 
crowned memoir, which appeared in 1889.f Subsequently % 
* “ Memoire sur les equations differentielles due premier order” par 
Paul Painleve, Annales de l’Ec. Norm. 1891-1892. 
f Memoire sur les fonctions algibriques de deux variable Louville’s 
Journal (1889), pp. 135-319. 
t Remarks sur les equations differentielles; extract d’une letter adres- 
sesse a’ M. Mittag-Leffler, Acta Mathematica (1893). 
