PROGRESS OF PURE MATHEMATICS IN 1900. 
BY 
Frank Gustave Radelfinger. 
[Read before the Society March 2, 1901. J* 
Introduction .—It is not intended to attempt a review of the 
progress made in the whole field of pure mathematics during 
the past year, but I shall mainly confine myself to an ac¬ 
count of one important research completed within that period 
by Painleve, of Paris, and forming a fitting conclusion to the 
work of the past twenty years. The first half of the paper 
is a historical introduction to Painleve’s work, while two 
new publications are reviewed in the latter part and some 
general remarks made in conclusion. 
EQUATIONS OF THE FIRST ORDER. 
Briot and Bouquet .—The problems presented by the ordi¬ 
nary differential equations are not equaled, either in extent 
or difficulty, by those in any other branch of mathematics. 
Soon after the successful application of Cauchy’s methods to 
the theory of algebraic integrals attempts were made to deal 
similarly with the ordinary differential equations. The first 
attempt was made by Briot and Bouquet, of Paris, in their 
classic memoirs of 1856.* Their researches were restricted 
to differential equations of the first order with rational co¬ 
efficients represented by the general form 
f(xyy') = 0. (A) 
* Journ. de 1’Ecole Polytechnique, t. XXI (1856), pp. 134-198, 199- 
254. 
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