114 THE REV. F. H..JACKSON ON 
n+ mn Ossi aa 
Jny(2) = (= (22) a4 BU fe 
If we now define 
Sin, (Xe) as Ne — er x Ii eet ss ae (46) 
then : 
ee Sie on ER + (I eae 
Operating on this with A! which is 
Ls at oe \ qe {aaa} }---}- (47) 
if 
INLD GEN caller 
the first n terms of the series are destroyed, while the term involving «®”"**” is reduced 
to 
enna [Qn+27r][2n+27-2]...... [27+ 2] amor 
i oar iene ce) 
Taking 
a ee or nv 2. 
Toate) = 2 Vereen 
[ntr+4]! = [nt+rt+4][ntr—-3]..... EE 
()ieray = (Bi A) pes acta ee ee Coal): (2), 
and 
(2),0,({1 + $]) < [2}'T2((1 +$)) 
therefore 
Loe aCe = Doe a. [3] x [2] (3) Ot mee: 
So we obtain 
utigin tH F Der ge” Hier] 
JinenWse) = pyr,.qepae'- ) (Qn +21}. 2 (5) (3) = (2)[4]. . . (27) cy) 
and by a change of the variable x 
on He ep eee eee ae Sim, (at) 
NTinan(yertt) = (1) ct ee ee ; 3 . (51) 
A™ denoting the operator 
ee | 
For further properties of Sin,, Cos,, and their connection with symbolical solutions 
of certain differential equations, reference may be made to a paper on “ Basic Sines and 
Cosines” (Proc. Edin. Math. Soc., 1904). 
