Mr Orr, On the product J m (x) J n (x). 99 



If n — ^ is a positive integer equation (18) is to be replaced by 



/-»' -In" = 2 _ 2w(n ( _ n) y F(^-n- 1-n, l-2n; a-)* ...(26), 



wherein the series on the right is not continued beyond the zero 

 terms ; this equation and (20) are now identical. 



In the general case the equations connecting the two divergent 

 series with the four convergent are equivalent to 



(TO —n)7T j j j j . (TO + 7l) 7T . , T T T , 



COS - (J m J n + J — m J —n) — COS o \y m" -n + " — m" n) 



_ 2 sin TO7T sin nir 



1TX 



and 



1+to + w < +m — n 1—m + n 1—m — n 1 — V 



(27), 



^V 2 ' 2 ' 2 ' 2 ' 2' a ; 



(ro-n)7r , _ . . (ro + n^ , j j _r j, 



mil ^ \y m<J n ° —m<J —n) bm n \° m u —n ° —m° n) 



(m 2 — ri 2 ) sin TO7r sin nir 

 irx 2 



„ /2 + to + n 2 + m — n 2 — to + n 2 — m — n 3 — 1 

 V 2 ' 2 ' ~~ ~~2 ' 2 ' 2' ~a? 



(28), 



wherein the argument of x 2 lies between — ir and + ir. 

 We also have as relations equivalent to the above 



<f> U + m, \-m\ —J . <f> U + n, \ - n ; -^-J 



f l + to + n 1 + to — n I —7ii + n 1—m — n 1 1 



= F 



2 ' " 2 ' 2 ' 2 ' 2' « 5 



(to- 2 — w 2 ) „ /2 + ??i + n 2 + m — n 

 ^ « * 



2x 2 ' 2 



2—m+n 2—m—n 3 1 



(29), 



2 ' 2 ' 2' a 8 , 



and another analogous obtainable by changing the sign of x. 

 * This result in a different form is given by Lommel, Math. Annal., iv. 



