164 
RADELFINGER, 
therefore during the past year of two more volumes* of his 
estimable series of treatises on modern analysis is noted with 
special pleasure. These volumes deal with the theory of the 
ordinary non-linear differential equations. 
The second volume is devoted almost exclusively to the 
equation of the first order, and includes a very full and satis¬ 
factory account of Briot and Bouquet’s researches, before 
mentioned, together with many amplifications due to others. 
A good representation of Painleve’s theorem relative to the 
movable singular points and the methods of Fuchs and 
Poincare is given. The beauty and symmetry of Poincare’s 
analysis is somewhat marred by transforming it to render it 
more elementary in form. Painleve’s crowned memoir is 
dismissed as being beyond the scope of the treatise. 
The third volume is devoted to equations of the second 
and higher orders, and contains chapters on the expansion 
of the integrals of an equation of the second order in the 
neighborhood of its fixed singular points, following Horn, 
Konigsberger, and others. The subject of singular solutions 
is treated quite fully, after the methods of Darboux, Goursat, 
and Dixon. Picard’s crowned memoir is merely touched 
upon, but everything relating to birational transformations 
is ignored, the author making use only of an inconclusive 
investigation on integrals with “ apparence uniform ,” termed 
subuniform by Forsyth, to which he devotes a chapter. 
The volume ends with a long chapter on Bruns’s f theorem 
respecting algebraic integrals in the problem of n bodies. 
The demonstration of this theorem placed the problem of 
the algebraic solution of the problem of three bodies in the 
same category as the quadrature of the circle, the algebraic 
solution of the equation of the fifth degree, etc. The analysis 
used in the original demonstration is long, loose, and com- 
* Theory of Differential Equations, part II, vols. II and III; ordinary 
equations non-linear. By Andrew Russell Forsyth. Cambridge Uni¬ 
versity Press, 1900. 
f“Ueber die Integrale des Vielkdrperproblems,” Acta Math. t. XI 
(1887). 
