OF UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
445 
geometric functions which satisfy the differential equation, at least half of which are 
convergent at any given point of the p-plane. 
In order that the relations between the various particular integrals in the form of 
series may be exhibited, it appears to be most convenient to start from integral 
expressions which satisfy the differential equation ; this is the course adopted in the 
present memoir. A definition of the two functions P/' (p), Q n m (p) by means of 
integrals taken along complex paths, which shall be valid for unrestricted values of 
the degree and order, has been rendered possible by the introduction independently 
by Jordan* and PocHHAMMERt of the use of integrals with double circuits ; the use 
of such integrals has the great advantage over the employment of integrals taken 
between limits, that the constants have to satisfy no convergency conditions, and 
thus that the functions may be defined by means of expressions which have a definite 
meaning for all values of the constants. 
In the special case m = 0, the zonal functions V n (p), Q„ (p) can be completely 
defined by means of integrals with single circuits ; this has been done by Schlafli,| 
who bases his theory of the series which represent these functions upon such 
definitions. 
In the first part of the present memoir the two functions P n m (p), Q„ m (p) are 
defined by means of integrals in such a manner that the functions are uniform over 
the whole p-plane, which, however, has a cross-cut extending along the real axis 
from the point p = 1 to p = — oo ; these definitions are so chosen that in the 
ordinary case of real integral values of n and m, the functions coincide with the well- 
known functions used in ordinary Spherical Harmonic Analysis; from these defini¬ 
tions various series are obtained which represent the functions in various domains of 
the p-plane. Special conventions are made as to the meaning to be attached to the 
functions at points in the cross-cut. Various other integral expressions are obtained 
which would serve as alternative definitions of the functions. It is shown that all 
the known definite integral expressions for the functions in restricted cases due to 
Laplace, Dirichlet, Heine, and Mehler are special cases of the more general 
formulae. In the latter part of the memoir various definite integral formulae are 
deduced for cases in which the degree and order are subject to special restrictions. 
In conclusion, the forms of the functions required for the potential problems connected 
with the ring, the cone, and the bowl are deduced from the general formulae ; in 
particular, convergent series are obtained for the tesseral toroidal functions. 
As much confusion is caused by the variety of notation used by different writers, 
it is convenient to state here for purposes of comparison the relations between the 
symbols used in the most important works on the subject; for this purpose the 
* See ‘ Cours d’Analyse,’ vol. 3. 
f See various papers in volumes 35 and 36 of the ‘ Mathematische Annalen.’ 
$ See a tract “Ueber die beiden Heine’schen Kugelfunctionen.” Bern, 1881. 
