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



where D denotes d/dx. Thus, if I n (x) = i~ n J n (ix), xl m (x) I n (x) 

 satisfies the equation 



[x i D i + 2x 3 D 3 - 2x 2 (2x 2 + a) D 2 - (8x 3 - 2ax) D + b 2 -2a}y = 



(2), 



where a = m 2 + n 2 — 1/2, b = m 2 — n 2 . 



If we seek for a solution of this in the form of a series 



%a r x r 

 we obtain the relation 

 (r + 1 +m + n)(r+l +m — n)(r + 1 — m + n) 



(r + l—m — n) a r+2 = 4<r(r + l)a r (3). 



This is satisfied by four hypergeometric series, always convergent, 

 of which one is 



>^( ffl+ 2 n+1 , ™ + *+-2 . m + n + 1} m+1> n + 1 . ^ 



(4), 



and the three others may be obtained by changing the sign of m, 

 or n, or both ; it is also satisfied by two series, always divergent, 

 unless they terminate, viz. 



jpfl + m+n \+m—n 1 — m+n 

 { 2 ' 2 ' 2 ' 



1 — m — n 1 1 \ 



~2 ' 2' x~ 2 ) {) ' 



1 _ iS t/2 + w + w 2 + m—n 2 — m +n 

 and x Ji ^ — -, — - , - - , 



2 — m — n 3 1 \ , n . 



— «- ; 2 ; it) (6) - 



An interpretation of such divergent series and relations con- 

 necting them with the convergent series are given in the Camb. 

 Phil. Trans., Vol. xvn., Part III. 



Equation (2) is also satisfied by two other functions which for 

 infinite values of the modulus of x tend to equality respectively 

 with* 



e~ 2X , e +2x (7), (8). 



* Loc. cit. , Art. 13. 



