Formula for the Hyper geometric Function A 4 (V). 169 

 _ i((I+ 0+r+»+l)/ ,t;_=±tbp!±tlj:^y + i+-+P+l) 



/5 - -|t«+/3 + y + <> + 3/2) -~(a + /3-ry + o + 3/2) 



= V27T . p - 1 



-4p^ 4 ±irt(o+i3 + y+*+3/2) 



x <? e , 



according as <f> is positive or negative. 



If <£ = the integral does not occur. 



Consider next the integrals 2 N X and iN 2 . Critical values 

 are given by the equation 



0=!+(2^ + 1)tt; £-0=!-(2/H-lV. 



There will be one, and one only, for each integral, viz., 



according as <f> is positive or negative, except in the case 

 <f> = 0, when there will be critical values it and — it it: 



Hence, according as <j> is positive or negative, 



x.-lW-rr'Oi*")^^ 



Xe L 4J ^e 



v /27r[p^ 2±7r) ] -i(-+f3+r+^+3/2)^ A0/4±,r/ " ) , 



if <^0. 



4p ? 

 



