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



relation connecting the series (5) with the convergent series may 

 be written more conveniently in terms of the J functions 



J n 2 (x) + J- n 2 0) - 2 COS HIT . J n (x) J-n (x) 



2 . 2 _A, jq-n)(i + n) 



= - Sin 2 n7T.iT Ml ^ ; 



7T ( 1 . X 2 



, i.f(i-w)(t-n)(i+n)(f + n) 1 



+ 1.2. a;* J 



= ?sin 2 W7r.«- 1 0(^, |-n, fc + n; -ar 8 ) ...(20), 



7T 



wherein every power x m has its argument between — mir and 

 + mir. The left-hand member may also be written in the form 



sm 2 7i7r{J n 2 (x)+T n 2 (x)}. 

 Thus we see that if P and Q are the divergent series such that 



J n ( X ) = (—) [P cos {(2n + 1) tt/4 - #} + Q sin {(2n + 1) tt/4 - x}], 



\7TX/ 



P 2 + Q 2 = <f>(h k-n, % + n; -or') (21), 



or by writing x 2 instead of — x~ 2 



<f>($ + n, \-n\ 2x) $ (i + n, £ - n ; -2x) 



= <Mi *+*> *-*; +**) ( 22 ); 



the series in the left-hand member being those on the right of 

 (14). Equations (21), (22) are to be interpreted in the sense that 

 the coefficients of the various powers of x are equal, or in senses 

 obvious from the paper referred to. 



From (12), (13), their analogues, and (17), (18), we deduce that 



[F(\ + n; l+2n; 2a;)] 2 



= e +w F(± + n; 1+n, l + 2n; x 2 ) (23), 



FQ + n; l + 2n; 2x)F$+n; l + 2n; - 2x) 



= F(\ + n; 1+n, l+2n; x 2 ) (24), 



and 



Fft + n; l + 2n; ±2x)F(\-n; l-2n; + 2x) 



= F(\; 1 + n, 1-n ; x 2 ) (25). 



