EQUATION AND ITS TRANSFORMATIONS. 
821 
This formula may he written 
x v 
2 i,+l r>+i) 
2p + 3 i'^x* 2y + 5 i'^x* 0 \ f 
~ lX ^~ 2 v + 2 Y. 2 y + 2 "si &c y * 
, 2z. + 3 i'V , 2v + 5 « , 0 \ 
+ [l + i x + 2v ~ 2] + 2u + 2 -^r+&c.je ■* 
and the expression on the right-hand side therefore corresponds to ^(Qfi-S) where 
Q and S are as defined in art. 3, so that the algebraic theorem to which the two forms 
of Bessel’s Functions (58) and (60) lead is Y = l(Q-J-S). 
46. The formula involving descending series for Bessel’s Function, J v (x) is 
J 'W= \/(£ 
-V 
(4v 3 — l 2 )(4i> —3 3 ) 
1.2.(8#) 2 
—{— &C. | COS ( X —^7r — -|a'7r) 
2 
irx 
4v~ — 1 (4F - l s )(4i/ 2 - 3 3 )(4v 3 - 5 2 ) 
1.8® _ 1.2.3(8r) 3 
&c. I sin (x- 
rlT- 
tI'TT 
); 
the descending series ultimately diverge for all values of v for which they do not 
terminate, but the converging terms may be used for the calculation of •b'(.r); and this 
formula was in fact employed by Hansen in the calculation of his tables of J°(.r) 
and J 1 ^)*. If v—p-\-\, p being an integer, the series terminate and we obtain a 
finite expression for J /J+i (r). 
Replacing the sine and cosine by their exponential values, this formula may be 
written 
n'v+\ 
when 
4a 2 —l 3 1 , (4v“ — l 2 )(4i> 2 —3 2 ) 1 . 
“ =1 - Wx + -12- (W S_&C - 
and /3 differs from a only in having all the terms positive. 
* Rommel’s ‘Studien Tiber die BesseTscben Functionen,’ p. 58. 
