406 THE REV. F. H. JACKSON ON 
In the same way if we take 
£10) 3 a a 
Oe = "ome 2n+ 2r}!{Qm + 27}! (In + Ir} ary 
which is related to J... by inversion of the base p, since 
ae C2) Pi han ee) 
Taking the product of two Jd series we find 
Hn 2t) X Fouj(@t™) = FP (2) + Ss (ip ae eta ae) ; . Cm 
In a previous paper (Z7rans. R.S.H., vol. xli. p. 106) it has been shown that 
: = {2n + 2v+4r}! ton 
Talat) x Solel) = 20 (- Ug aaron eee 
There is a certain similarity of form among the series (50), (51), (58). 
Consider now the product of four J functions | 
Jin (at) - J watt » Iua(at) - Ipy(at-) : . . .-| (oem 
This expression may be written in two other forms. Firstly, by (49) and (52) we 
write it 7 
{ Fan Ge) +3 (— mem teeny) be Lae, Cod D(— Tyrmermrcem e-em Ga) |. (65). 
Secondly, by means of (53) we express (54) as 
ae {2n + Qv + 47}! hoc 
pee ‘Tine Wy + Orff (In+ Ir} ve ror} wy) 
{2n + Qv + 4r\! C —1\n+r+2r ‘ 
{SED "Gas 4B} ons rls ry } —— 
Equating coefticients of powers of ¢ in (55) and (56), we find from the terms 
independent of ¢ 
@ {2n+ 2v+4r}! ) antitr Jp Yo 4.9 oo 2mim-+y) Jp yp 57 
2: oasaye Dry In + Wr} Ww + Wy Ay)” nano 2? mmm — 
The terms in the series on the left side of (57) are the squares of the terms in (53). 
Generally 
SS {Qn + QWw+4r}l{In+ 2+ 4m t 4r}! geet rentar 
S {2m + In + QW + Ww}! Qn + QW + AW} !{Qm + In + Ar} l{ In + BW + QW} !{ Qon + Br} {In + A} !{ QW + Ar} l{ Ir}! 
4 P iD Pp aP 58) 
=a Se | Jaimie se PEO Ae. ; ( ) 
