110 THE ROYAL SOCIETY OF CANADA 
this coefficient can be made to take the form 
CSD ONE Qe ee Mer) 
"Tate? Belgie! 2 GED TEE 
(a: D41+2 921+1 
| 
mn sait ‘Osa 
the coefficient of x?2/+1 in sin 2x. 
Thus we have established Lommel’s formula (N. p. 281) 
D 
sin Qe=x 2 (—1) (2s+1) JF, (x)... (21) 
Lommel’s proof of this equation, as he gives it in vol. 2 of the 
Mathematische Annalen, pp. 632, 3, is made to depend upon the 
equation 
WAS. Ss Gaye. 
Vv x dx Hs Vv Ss 
He appears to have overlooked a still simpler proof, which has the 
advantage of giving the value of the remainder after a finite number 
of terms. Observe that by ee the equations 
Title x 
TEE T. Re 
T Rod eee 
IT aes, 
2 A2: CNT 
by Ji, —J,, J;, etc., and then adding the results, we get 
2 as: 
ip 131 = in ]- fe ce +57 —..] 
$ 10. By equating the coefficients of x‘ in the expansions of 
(1+x)25-1 and in (1+x)?° (1+x) 1, we find that 
EC) ae 
au oes mG wee 
The formula (2) shows that these are all the terms containing x in the 
expansion according to powers of x of 
Ja + WIiS gt 2g Ig. a 
2 2 2002 AC 
There is only one term independent of x, namely the constant term of 
: : 1 2 
J_iJx, and its value is THIS) = = 
