74 
Proceedings of the Royal Society of Edinburgh. [Sess. 
These functions satisfy partial differential equations, and can be 
expressed as definite integrals. Appell has given some applications of 
them to certain problems of celestial mechanics, and expressed in terms 
of them the polynomials of Hermite and Didon and some more general 
polynomials. 
An interesting advance in the theory has been made recently by Appell, 
who has shown that the polynomials V m n of Hermite, which are particular 
cases of the function F 2 , are solutions of the potential equation in hyper- 
spherical co-ordinates, and can be considered as hyperspherical harmonic 
functions on the hypersphere 
x 2 + y 2 + z 2 + 1 2 = 1. 
PART I. 
DEFINITION AND PROPERTIES OF THE FUNCTIONS. 
Chapter I. 
FORMATION OF THE CONFLUENT HYPERGEOMETRIC SERIES. 
The confluent hypergeometric functions of two variables may be formed 
by confluence from Appell’s functions in the following way : — 
First, in Appell’s function F 3 (a; ft, ft ' ; y , x, y), make ft' oo , at the 
same time dividing y by ft': we thus obtain the first of our confluent 
functions, 
^i(a ; ft; y; x, 
Jy y ( a, m + n)(ft , m) x m y n . 
^ ^ (y, m + n) m \ n\ 
A second function can be obtained from F t by dividing x and y by a, and 
causing a to tend to infinity : we are thus led to the function 
%(ft> ft' ; y; ^ y ) = 2 2 
(ft, m)(ft', n) x m y n 
(y , m + n) mini 
A third new function can be derived from F 1 by making the two para- 
meters a and ft' infinite, after replacing x and y by x and this gives 
a aft 
the function 
i / / , \ v’' (B, Tfi) x m y n 
(y, m + n) m ! n ! 
Taking next Appell’s series F 2 , we apply to it the same process, and 
obtain two new functions, the first one by dividing y by ft', and making 
ft' infinite ; and the other one by dividing x by ft, y by ft', and making 
