MR. R. C. ROWE ON ABEL’S THEOREM. 
747 
Then, by the theory of numbers, if b and n are prime, the integers 
a ct-\-b ct-\-n —1.5 
ne-, ne -, . . . ne - 
n n n 
will he (in some order) the series 0, 1 , 2 . . . , n —1 ; while if b and n are divisible by 
c, c being their greatest common measure, the integers 
cc a + 5 a+ n — 1.5 
ne~, ne —,. . . , ne - 
n n n 
form an arithmetical progression whose common difference is c, repeated c times; and 
the smallest term of this progression is the remainder when c is divided into a. 
If this remainder be called d we have 
whence 
and 
t e^±^=y{d+(d+c)-h(d+2c)-|- ... to ^ termsj 
1 = 0 
n 
ii l 
= d- 
n — c 
1 ,a + lb ,_ h _1 ct + lb 
t e- —n— £ e- 
i=o i=o n 
= -d 
n + c 
ct + lb ^ a + lb 
i=o n n 
-d- 
ii+ c 
' y 
Corollary i. 
If c the greatest common factor of b and n also divides a, then d— 0, and we have 
the simpler forms 
. / n-\-c n — c 
2e Se=——. 
Corollary ii. 
The sum of the fractional parts of any n terms of the series (repetit ions being allowed ) 
differs from the sum of the fractional parts of the values of the same terms when a is 
put equal to zero, by an integer. 
For, if the sum of the coefficients of b in the numerators of the n terms be X, then 
5 d 2 
