[HARKNESS] THE ALGEBRAIC BASIS OF CERTAIN BESSEL SERIES 111 
Thus we have a very elementary proof for the elegant formula 
2 
Another proof will ue eae in Nee a S56, 
It is instructive to show that the formula (23) can be connected 
with two others, due to Lommel, namely 
S, (2x) = RE aan AL (Ne p.203) (24) 
where SS; (x) is the integral sine function (= dx, and 
2x ‘ ts é 
= Js 1 8/3 gt EUR st. (25) 
(see N. p. 295 res the A eee form of this equation (25).). 
The formula (24), on differentiation, gives 
sin 2x 
25, JT TROT pI p+... =; 
2 3 D 22 

TX 
hence 
FE J,=29,) +7, e Fy-2I,)+Jy (2), oh os: 
2 26 9) 2 5 
_ sin 2x 
age 
2 sin 2x 
1 
But— (SR -- os. 4...) — = and ———- = J, J 4; therefore 
x 2 2 vy T TX 2 2 
we have J_y Jy +2Jy Ji +27, Jat. = 5, 
as above in (23). 
Without performing the analysis it may be worth while to point 
out that (24) can be proved at once by combining (2) with the identity 
GC) ©), à 
The following is an instructive proof of (25): 
DATE 
er res ie J, (2x cos ¢) d¢, 
2 {5 
Joes = och (2x cos D) db, 
2 [= 
En = É J; (2x cos ¢) dd, etc. 
Burn) +37: G) Lo ye. = 5 (G. and M. p. 19); 
therefore 
TiHBP ++... [x cos à dé = =: 
