116 THE REV. F. H. JACKSON ON 
9. 
In this article certain results will be obtained which will be required in subsequent 
work. We define the function H,(a) as 
2 
E,(a)* = 1 + ant pt Bache 
If we invert the base p 
S(s— as 
Bi (a)=1+ 5 . +p" mont eee 
ayy 
without difficulty we have 
B,(a)E(b) = isonet et Reh ae 
Changing p to p” 
(a+ b) aa b)\(a+ we 2 4 
E,3(@) Ep-2() = Ooh oe io p? ae pte 
Pa el 
[2](a +b) [2 (a + b)(a + bp?) 
age cr 2 ae 
Se @ DNs @ b) (a+ b)(at bp? 
Bs( Brel) = 1+ TD af aa ay P ae My Rae teens Jat: Ye ; . (y) 
In part (1) we have established 
Jp-m(@) = (-1)Spu(a) ‘ ‘ : : (4) 
Inverting the base » we obtain also from this 
dem) = (-1)"am() . : (x) 
10. 
A consideration of the product of the two absolutely convergent series 
at\ at Wits a 
Mee Cy ae 
at at} eee 2 no , 
ela) <a tanny  Oat 
shows us that 
E(B - fo) = Dlr + 2l- Dn 5! 
= oF ine" : ; ‘ ; (A) 
In precisely the same manner, if we consider the product of 
bt cls, ye Ore Amey Oe 
Bm) = py ayy? tat 
: THOUS Se a, ae Oe 1)spnn— pro” 2 ea Galina 
ay) Sty Pet 
* Proc. kdim. Math. Soc., vol. xxii. 
