110 THE REV. F. H. JACKSON ON 
which gives us 
(lp) ep heer, wel pe) lee pe) ee Lp) é 
a eee EEE ier OTS Tt ge ea o O (25) 
as the coefficient of \”. If r be not integral, the infinite products in HEINE’s trans- 
formation do not reduce to finite products but to expressions in terms of the I’, 
functions, ultimately giving the sum of the series in the form (14), 
Having obtained the coefficient of \*”, we have established, subject to convergence, 
the theorem 
‘ [8 
TA) ain(A) — oH a>) ana(A) +ptih mA)daA)+..... 
pl pt) a. pee) ie (Le py 
(a) {2r}! {2r}! BRC OF Cs (26) 
_ 71 _2(1+p?)r? 
=) — Taya Pras 
which is the extension of 
TONS fa)? = 200A 2 en 
This is a particular case of the addition theorem for Jo. 
To(A + Aq) =ITg(A)T (Ay) — 23,(A)T4(A) given t's 
5. 
Defining * Sin, (A) and Cos, (A) as 
Sin, (A) = Mo BT bode 
Cos, (A) = Tae epee 
we obtain 
Sin, (A) Coss (A,) + Cos, (A) Sing (Ay) = (A ae A.) ns (Xr AP AYA ee \p*) he 
Gin, (A) Cosa (A) + Cos, (A) Sinz (A) = 2A— eee ae 
Dp Pp : 
This suggests that the extension of the addition theorem of J,(A+A,) will be on 
similar lines. 
Consider now the series 
Tio A) Hin(Ay) — bP m(A)da(yy. 1... e. (= ype m(A)da(y)+.. -. . (28) 
~ Proc. Edin. Math. Soc., vol. xxii., 1904. 
