PROGRESS OF PURE MATHEMATICS IN 1900. 159 
theory, as then developed, to the ordinary non-linear equa¬ 
tions was the fact that all of the singular points of the inte¬ 
grals of these equations are not revealed by the form of the 
equations themselves, but that other singularities are intro¬ 
duced during the process of integration by means of the 
constants of integration. These newly discovered singular 
points were named 'parametric or movable to distinguish them 
from the fixed singularities determined directly from the 
coefficients of the equation. 
Fuchs 'and Poincare. —Strengthened with this new knowl¬ 
edge, Fuchs, of Berlin, attempted to determine all the equa¬ 
tions of form (A) whose integrals would possess no movable 
points but poles. His results, published in 1884,* were not 
conclusive, but were completed by Poincare, of Paris, in 
1885,f in a short and very remarkable paper. Poincare 
obtained his results by an application of the analysis of 
birational transformations borrowed from the theory of 
curves. But, even the great generality of the problem solved 
by Fuchs and Poincare revealed no new functions, as in 
every case they found that the equations, satisfying the im¬ 
posed conditions, degenerated into types integrable by ordi¬ 
nary algebraic methods, by simple quadratures, or they can 
be transformed into an equation of Riccati. It will be seen 
that the above results establish a lower limit for the func¬ 
tions defined by equations of the first order. 
Painleve. —The next step in advance was made by Pain¬ 
leve, who in 1887 J established an upper limit by proving 
that all movable points of equations of form (A) must 
be either poles or algebraic critical points. This theo¬ 
rem excludes points essentially singular from the class of 
movable singular points in equations of the first order. 
* “ Ueber Differentialgleichungen deren Integrate feste Verzweigungs- 
punkte besitzen.” Berlin Sitzungsberichte (1884), pp. 699-710. 
f “ Sur nn tlteoreme de M. Fuchs,” Acta Mathematica t. YII (1885), 
pp. 1-32. 
t Sur les lignes singulieres des fonctions analytiques, Ann. de Faculte 
de Toulouse (1888). 
