158 
RADELFINGER. 
Cauchy’s theorem, which establishes the existence of in¬ 
tegrals of differential equations within circular domains 
excluding all singular points, applies to these equations. 
Briot and Bouquet simplified and extended Cauchy’s demon¬ 
strations and set themselves the task of determining the form 
of the expansions representing the integrals of the equations 
in the neighborhood of the excluded singular points. These 
expansions were determined in the most important cases, but 
they yielded little information as to the nature of the func¬ 
tions defined in general by these equations. 
To obtain more definite properties inverse methods were 
resorted to and applied to more special equations of the 
form 
f(yy') = Q, (B) 
with rational coefficients free from x. They first determined 
the necessary and sufficient conditions which must be ful¬ 
filled in order that the equations of form (B) should possess 
only uniform integrals. Next they found the reduced equa¬ 
tions, and lastly they showed that all these equations could 
be integrated by means of functions already long known. 
Then they solved this problem, viz., to determine all equa¬ 
tions of form (B) whose integrals are limited to assuming a 
finite number n of values at every point in the plane; but 
again the results obtained agreed with the previous ones in 
that no new functions were needed to complete the integra¬ 
tion. Thus their industry and genius yielded little but the 
negative results that a solution did not lie in those direc¬ 
tions. 
Movable Singular Points .—For some time after the publi¬ 
cation of Briot and Bouquet’s memoirs no material progress 
was made, from the function theory point of view, in the 
solution of the ordinary differential equations. During the 
interval, however, the nature of functions in general was 
extensively investigated and one important fact discovered. 
This was that the principal obstacle which stood in the way 
of the direct application of the principles of the function 
