26 
RADELFINGER. 
By transposing the origin to the point x B , the fundamental 
integrals pertaining to the domain surrounding this point 
take the form: 
K 4- <h ® + <h + a s & ..)• (G) 
When x turns around the origin, now a singular point, the 
integrals are each multiplied.by a constant e~ 2nim . 
This is a characteristic property of equations with regular 
integrals, equations generally called Fuchsian, from their dis¬ 
coverer. In general these constants are different for each of 
the fundamental integrals pertaining to the singular point. 
In case they are equal the equation still belongs to the 
Fuchsian class and the integrals are still termed regular, 
though they are of a more complicated form and involve 
logarithms. 
The Fuchsian class of equations includes many important 
special cases, long known to mathematicians from their con¬ 
nection with investigations in physics and astronomy. Some 
of the most celebrated are Bessel’s, Legendre’s, and Gauss’s 
well-known equation defining the hypergeometric series. 
The theory of each of these equations has been worked out 
in great detail from the modern point of view. Owing to 
the relative simplicity of these equations, most of the results 
had been anticipated, having been arrived at by methods of 
transformation. 
We have seen that Fuchs’s expressions for the fundamental 
integrals consist of the product of a uniform power series 
and a factor which becomes infinite at the singular point. 
The integral can be said to have been rendered uniform at 
the singular point by the multiplier. In the Fuchsian equa¬ 
tions this multiplier is always algebraic; but this method of 
rendering integrals uniform admits of an obvious extension 
by making use of transcendental factors. This has been done 
by several analysts, and they have succeeded in constructing 
fundamental systems for differential equations with irregular 
integrals—that is, differential equations the strength of some 
or all of whose singular points are infinite. Before going 
