404 THE REV. F. H. JACKSON ON 
This expression, when expanded in a LaureEnr series of ascending and descending 3 
powers of ¢, takes the form (7rans. R.S.E., vol. xli. p. 117 (#)), 
lol 4) 4 > aaNet -\p™In( 5) : . - (34) q 
In case vx=1, the product II (Sep (a ps ; 
m=1 
may be expressed as 
Peat Se , : | 
cee 1—p(@+i)+ptt+i-4) - 2... | : : . (35) | 
(Cf. Cayiey’s Elliptic Functions, p. 297, ed. 1876.) 
We see incidentally that 
1 
Fa-pm ga) - 0 
for all positive integral values of n. Denoting the nature of the base by an index, we write 
1 1 1 
rgia)= tly) =Algba) =o — 
1 1 1 
Hot) mata)-elt) 
ae Ti lp) ? ota l-p * Jeon ; 38 4 
which is the expression in generalized Bessel-function-notation of the well-known result 
1 gute 1 
gee Wl eng a pp Se IN He. 
aq ge rao * dd -a) m=1(1 — g?”) 
On replacing ¢ by ce”, the equation (31) becomes 
tel te-t6 
dole )to(iee—1)_ 
P gS AO i, (1 — 2a? cos 262" * +- atp4m—2) © ; . 63) 
. 10 5 nt -10 m= 
Son ie ee Wes ig Pe =) : 
p-l/ p-| 
Using now Jacosr’s notation, and writing wat P pix NP 9-2 eat (1s zm) 
G p it 1 b) p re 1 170 ee qg “4 . 
gq =p, we obtain 
ono, a) (40) 
titans Wl (Re) | | 
SC orm co wc aee ae ang 
tenn lE ee Ce) 
Pn on ee 
