1920-21.] Hypergeometric Functions of Two Variables. 
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PART II. 
THE CONFLUENT HYPERGEOMETRIC FUNCTIONS 
AND THE POTENTIAL. 
Very important connections exist between the confluent hypergeometric 
functions of two variables and the theory of potential in hyperspace, a 
fact which generalises in an interesting way the well-known relations 
between the hypergeometric functions of one variable, or their confluent 
forms, and the potential in three-dimensional space. We shall now 
develop some of these propositions. 
Chapter I. 
THE FUNCTIONS OF THE PARABOLIC HYPERCYLINDER. 
Let us consider a four-dimensional space, where the Cartesian co-ordinates 
will be denoted by x, y, z and t, Laplace’s equation in this System being 
AU = — + 0 5 J +!^ + — = 0. 
lu'i pf 
Let us now make the change of variables 
X = UV COS (j) 
y.= uv sin <f) 
t — t. 
The hypersurfaces thus introduced are rather simple ones : it is at once 
obvious that t — const, and 0 = const, are hyperplanes; as for u = const., we 
obtain the corresponding hypersurface by eliminating v and <p between the 
first three equations, obtaining 
x 2 + y 2 = u 2 (u 2 -2z) (A) 
so that it is an hypercylinder, with its generatrices parallel to the £-axis, 
its basis in the xyz space being the quadric (A), which is a paraboloid of 
revolution. We shall say, therefore, that this hypersurface u = const, is 
a parabolic-hypercylinder. The hypersurface v — const, is of the same 
nature. 
If now we transform Laplace’s equation, we obtain 
