746 
MR. R. C. ROWE OR ABEL’S THEOREM. 
means of algebraic and logarithmic functions; so that Abel’s theorem becomes 
unnecessary. 
Except then in these two cases it is always possible by satisfying the conditions 
(A) and (B) to render the sum of the series of functions equal to a constant.* 
The number of arbitrary constants, being equal to the number of relations con¬ 
necting the variables of the functions which we sum, will by art. 20 (G) be 
t n r m r n s p s -\-^tn z mp—^2 l nm—^Zn—^n-{- 1 . 
s~>r 
It is not necessary that we should assume F 0 («r) = l for the correctness of the 
processes of the last two pages. 
Our equations will be the same if for any other reason F 0 (x) disappears from the 
general formula, and reduces it to the case of art. 9. 
But this will happen if in the denominator of —— % 
/si®) 
also occurring in F 0 (x) ; and this will be so if F 0 (.r) and 
the same value of x. 
JMy) i 
wr Iog 
xf) 
ff x > y) 
Qtj there is no factor 
do not vanish for any 
If this condition hold the results just arrived at will remain true. 
APPENDIX. 
Lemma. 
To find the values (i) of the integral parts, (ii) of the fractional parts, (iii) of the 
complements to the fractional parts of the series of terms 
a a + i a+2b a-\-(n—l)b 
n n 5 n ’ n 
where n is a positive integer, and a and h are integers positive or negative. 
By the integral part of a term we mean the integer next less than or equal to 
it; by the fractional part that positive fraction (zero included) which added to the 
integral part gives the number ; by the complement of the fractional part that fraction 
which added to the given number produces the next higher integer. 
Let these functions of the term be denoted by the symbols E e e'. 
* A notable particular case is that in which ffv, y ) consists of a single term, x Jc y 1 ’ 1 ; where m is so 
chosen as to satisfy the condition (B) above, and k so as to satisfy the equation (i) of the last page. 
