272 THE REV. F. H. JACKSON 



We see that 



/ x \ (l + a X q-i)(l + au:g-*)*(l+axq-sy . . . . . (&Q) 



"\q-\J (\+b.rq-%l+bxq-y(l+bxq- s ) s ' V ' 



and that 



* ( ? v V (* + aM f)( l + azq^a + a*q 6 ) 3 



*""Vl-flV (1 - bjq*)(l + krq*)\l + bxq^f ' 



also 



. ( ,,,■ v E ! ^f$M5 ! ) 



and 





? y . . . (61) 



r 2 ' 



,2 



The expression on the left side of this equation is by (47) and (49) 



[1 +axq*(t* + t*) + q»aVl 1 [ l+aa»j 4 (< 8 + f 8 ) + o^g 8 ] 8 (62) 



= / , ( a: ) + / x 1 (^ 2 + r 2 ) + / x 2 ( a; )(^ + r'')+ 



The expression on the right side of equation (61) may be given in various forms, owing 

 to the following properties of the function E g (x), 



E»xE rl (-^)=l, (63) 



W_ E / a v xE (Ml, ("-&)| ( fl -fr)( a -g 6 ) + . 



We shall first transform the expression on the right side of (61) into 



V/-I/ Vg 2 -1 / , . . . (64) 



Now I have shown * that 



= J [ o,(A) + (^ + nJ [1 ,(A)+ +(f m + t-*»)J w (X)+ , 



Jt " ](X) = f§ {2w + 2r}!{2r} ! ' 



Replacing in (64) -^ : by .=-= , for convenience of writing, and -^- by ..— = , 

 we^have finally 



/ , (; C )+ / x 1 (*)(* 2 + n+ _J| 0) (^) + (< 2 + nJw(^)+ (65) 



*>(-) + /*i(-)(< 2 + '- 2 ) J ^) + (t 2 + ^)J «+ 



* /Vw. L.M.&., series 2, vol. ii. p. 196. 



