78 Proceedings of the Boyal Society of Edinburgh. [Sess. 
that for is 
fx( 1 - x)r - xys + [y - (a + /3 + 1 )x\p - (3yq - a/3z - 0 
yyt + (y - y)q -v x - ^ = o, 
and similarly for the other functions. 
Each of the systems is of the type 
* fr = a x s + a 2 p + a s q + a^z 
V= b x s + b 2 p + b^q + b±z 
(the as and b’ s being functions of x and y), of which a general theory has 
been given by Appell, with the aid of certain propositions established by 
Bouquet. When the expression 
1 - a l b 1 
is different from zero (which is the case for the ' V E and 3? systems), the general 
solution of the system is a linear function of four independent solutions 
and if 
z=C 1 z l + C 2 z 2 + C 3 z 3 + C 4 z 4 , 
1 - a l b l = 0 
(which occurs for the systems), it is a linear function of three independent 
solutions 
2 — CjZj + C 2 z 2 + C 3 z 3 . 
We may observe that the system satisfied by the function 
^i( a ; P ; y ’> x > v) 
admits also the independent solutions 
x 1 ~ y ^ 1 (a+ l-y;/3 + l- y;2-y , y ; x, y) 
y l ~ y ^ 1 (a+l-y' ; [3 ; y, 2-y ; X, y) 
x 1 ~ y y 1 ~ y '$r 1 (a +2 -y-y ; fi+l -y ; 2 - 7 , 2 - y ; x, y). 
A similar result may be obtained for the function so that the general 
solution of the two T' systems may readily be expressed in terms of the 
\E functions themselves. 
Chapter IV. 
SOME SPECIAL PROPERTIES OF THE AND H FUNCTIONS. 
We shall next give a few formulae illustrative of the properties of the 
<f> and H functions : the function which has a special importance, will 
be considered in the next chapter. 
The dq function admits recurrence formulae analogous to the well- 
