402 THE REV. F. H. JACKSON ON 
Krom this we derive 
(=). (=) =1+ [nja+ ee Seah asenseiers somata a ‘ . 5 
=f (2) . mo ; ; : me) 
B(-). 4, ( )a1tetats sere ge ; . (18) 
Dp 
nit 
= (<1) . ao 
In this expression we notice that inversion of the base p simply interchanges the HE 
functions in the product on the left side of the equation (18). 
nye) =l- (ret ole le ~ Aes eee 
Pp 
=p'(-2). ; - Ql) 
Ro-fes . . . . i 
Jf xfi(-a=f" ). : : > . (23) 
pe (2) x hk (- 2) = h'(2") ; : : : , (24) 
J (ex b2(-a)=1 : : ; : : (25) 
Hence 
The equations 
eve =] 
E,(x)-E,(-2)=1 
are special cases of (25). 
If n be infinite 
Ws (@)-t. (55) (26) 
Fonction I,,(x). 
It is well known in the theory of Brssst’s function that 
e+ 
In+3 ) Qn+5 x i 
e*- L(t) = 5, jn + 3)" news 
ee a Ne 
T(n+1) | 
Te) eetlin(an)s 
In a paper on Basic numbers applied to Busset’s function (Proc. Lond. Math. Soc. 
series 2, vol. 11., 1905), I have extended this theorem in the form 
