89 
1920-21.] Hypergeometric Functions of Two Variables. 
Laplace’s equation, written in this system, is easily solved by the change 
of variables 
u + v 
u — 
V2 
u! - v' 
V— . , 
x/2 
which reduces it to the equation for the hyperparaboloid of revolution. A 
solution is then 
W 
multiplied by a convenient factor, or 
W k " + i[(^ + v ) 2 > ~ ( u V Y]‘ 
A similar remark may be made regarding the change of variables 
u 1 - v 1 
y = cos 0 
j 2 r 
z = uv 
t = t, 
where Laplace’s product is readily reduced to the parabolic hypercylinder 
function. 
3. We shall show presently that there exists, between the hyperspherical 
zonal functions and the functions of the hyperparaboloid of revolution, a 
connection similar to that between Legendre polynomials and the parabolic 
cylinder functions. 
Let us first investigate the relation between the one- variable functions, 
and for that purpose give a preliminary definition. 
If between two differential, linear and homogeneous, equations, E and 
D, there exists a relation such that, deriving equation D n times with 
respect to the independent variable x, we obtain equation E, we shall say 
that D is the equation of Didon of E for the order n. Similarly, if between 
two linear and homogeneous two- variable systems, S and A, there exists a 
relation such that, deriving both equations of A m times with respect 
to x and n times with respect to y, we find the system S, A will be called 
the system of Didon of S for orders m and n. This definition, which we 
proposed some time ago,* has its origin in the fact that F. Didon made 
an extensive and successful use of the transformation in question to solve 
differential equations occurring in the theory of two- variable polynomials. 
* Nouv. Ann. de Math., decembre 1919. 
