PROGRESS OF PURE MATHEMATICS IN 1900 . 161 
he expressed the opinion that it was unlikely that any new 
functions would result from the ultimate solution of the 
problem. 
Painleve .—In view of what had been accomplished, the 
mathematical world was hardly prepared for the announce¬ 
ment by Painleve, first made in a series of short notes in the 
Comptes Rendus during 1899 and 1900, that he had com¬ 
pletely solved Fuchs’s problem for equations of the second 
order, and had found three new uniform functions; also 
that part of his method was simple and applied to equations 
of all orders, and enabled Fuchs’s problem to be greatly 
simplified if not ultimately solved in all cases. 
A brief synopsis of Painleve’s principal results has subse¬ 
quently appeared in a short memoir by him, inserted in the 
Bulletin de la Societe Mathematique de France,* which fully 
confirms his previous announcements. 
Painleve’s method consists in forming a series of new neces¬ 
sary conditions which must be satisfied, if the equation has 
only fixed critical points (branch points), and then proving 
their sufficiency. In deducing the necessary conditions a 
system 
^&= H (* 2 /*) 
dz 
dx 
= K(xyz ) 
is taken in which H and K depend analytically on a para- 
mater « and are suppposed holomorphic for « = 0. If the 
general integral of the system (D) is uniform for all values 
of «, excepting a = 0, it is still uniform for a = 0 and the 
serial developments of y ( x ) and z (y) in powers of a must 
have coefficients which are uniform functions of x. By 
making H and K depend on an arbitrary paramater and 
applying the above proposition, which is easily proved, the 
* Memoire sur les equations differentielles dont 1’integrals generale est 
uni forme; par M. Painleve, t. XXVIII, pp. 201-261. 
