740 
MR. R. C. ROWE ON ABEL’S THEOREM. 
20. To find the value of y — a we must investigate the fractional parts. 
Considering any set (say the X th ), they are, with the notation of the Lemma (p. 746), 
of the form 
(/A — 
/**. 
where a takes each value from n — k A-1 — 1 to n—k A ; and k K —= ?y. A p, A . 
But m K and y K are prime to each other. 
Therefore, by the result of the Lemma, the sum 
= n k 
1 
So then finally, giving to 2A its least value, we have 
S6i/—%q={[n— !) + • • • -\-(n—k 1 )}o- 1 -\-{(n—k 1 —1) + . . , 
-\-(n—k 2 )}cr. 2 -\-. . .+ NI»(/x — 1) 
This expression 
=Pi°'i(2^—^’i—1) -{--^k^crf^n—k^ — k% —!)-(-. . .-\-%i>n(y — 1). 
Now Zq — -J-Wo/Xg j &c. = &c. n = ki—■ n^y^ -J- n k y.i-\- . . . 
Substituting we obtain 
n i m i(~Y~ 1 +n.2Pz+ • • • +my}j 
+ %h^3^3+ • ’ • J T n . 
+ • • • 
+ %i(/r-l) 
= tn r m r n s y s + ffrdmy-\-f%ny —— ffnm. 
sf-r 
Now, returning' to the values of art. 13 and inserting the numbers A and B for the 
correction there explained and writing instead of %ny its equivalent n, we have the 
result following. 
The least number of functions in terms of which the sum of any number may be 
expressed is independent of everything but the form of the function considered (he., the 
form of y given as a function of x by the equation y(//) = 0), and if this equation has 
