76 
Proceedings of tlie Royal Society of Edinburgh. [Sess. 
so 
F 9 = 
( - l)- a r(y)r(y) f 1 ^ )y ^ _ ux) m du 
T(P)T(P')T(y-p)T(y'-p)^^ mini J { ' { ’ 
jfV _1 ( 1 - 1 - vy) n dv 
=(-ir22 < i^brf ) F( ~ m ’ p ’ y ’ *) F ( - ». p ; y ; 2 /), 
a formula to which we can apply the ordinary process of confluence, 
obtaining the interesting expansions for the ' V F functions : 
*i(<h P ; y> y ; *. y) = ( - F ( _ m > P : v> *) $ ( - *> y’> s') 
%(a ; y,Y; a, j /) = ( - l)"‘22l^r ®( “ TO > T> *)*( ~ »> y’< 2/)- 
By reasoning of a similar type we find 
r.(. : ; ft ft ;r 2 . ! ■ - »* F,(. . - i ft +« ; ft + - ; 
y-p' + m, y + m; x , y) 
or 
F, = y, ( - w) y F 2 (a+m ; p + m, ft + m ; y + m,y - p + m ; x, y). 
" (y - P, m)(y, m) m'. 
From these we obtain by confluence 
*,(«;ft;y; *, y) - 2 (- 1)” m) s ^ 
" (y - P, m)(y, m) m ! 
(ft m) ?/ 
^(a + m, p + m; y + m, y - p + m; x, y) 
Pi= 
<f 3 (A y ; y) = 2 ( - 1 ) m (y _^’m)(y, m) m T ^ +m ’y +m ’ “W? ~ P + m > v)> 
which may be transformed into 
$ 3 (ft y; X , y) = e x ^(-l) m 7 $(y-a, y + m, -x)B(y- p + m, y) ; 
(y — ft m )(y, m ) m • 
i 
lastly, from F 3 we obtain in a similar way 
Ei(a, a ; P ; y; x, y)|| ^ M 1 ) m ^ m ^, a ’ f F(a + m, P + m, y - a + m, x) 
^ (y,m)(y-a,m) mi 
3>(a' +m, y + m, y) 
$ 2 (a, a ; y; X, ( “ 1)™ ( a > ™ ) ( a » m \ $( a + m, y - a + m, x) 
(y, m)(y- a , m) ml 
<J>(a +m, y + m, y) 
a„(a, ft y, x, y) = 2 ( - 1 )“ 7 TT — 7 -C F(o + m, /? ; y + m ; x) 
" (y, m)(y - a, m) ml 
B(y — a + m, y). 
Another type of expansion may be obtained as follows : in the formula 
r (y) 
npw 
)F(y -ji- W) fofy ~ ^ ~ \ 1 -ty-fi-P-'i 1 -vyP+l - tz-vy + vtx)-‘dtdv 
