MR. R. C. ROWE ON ABEL’S THEOREM. 
737 
The condition expressed by this equation is then necessary if the inequality (A) is 
to hold. 
It is also sufficient, as is easily shown. 
For from it we obtain 
if V>S [p,7\ — [p, s ] = Sp s . T s + §/L+ r Tj + i -f . . . +8 p r _ v T r -x 
> (Sp,+ Sp 
s+l + . . . +Sp r _ 1 )r ,_ 1 
> {p S — pi)T r -i 
> ( p,—pr)<r r 
if r<s 
$pr-Tr — tyr+l-Tr+i— ■ ■ • “8 
>—(8p,-H-Sp r+1 + . . . -fS Ps~])t,- 
>{p s — p,)(T r . 
The relation (B) (in which [pj is entirely arbitrary and the r’s are only subject to 
the necessity of lying between consecutive <x’s) expresses the necessary and sufficient 
condition for the satisfaction of (a). 
18. Let us next examine (8). 
The condition is expressed by the inequality 
\_P J fh pm ^ \_®- ] ffi ® O 'm 
where a is any term of the series 0, 1, . . . (n —1) which is not one of the p’s. 
Let a belong to the X th set so that 
[a] + acr A < [p A ] + p A cr A 
and let 
[«] + ao- A = [p A ]+p A cr A — A a . 
. . . (C) 
A 0 being 
a positive quantity. 
We have then to make 
[p*] + /o»o-*>Lp A J+p A a- A +a(o-* — cr A ) — A a . . . 
. . . (D) 
Now this inequality clearly holds when m = X. Again it holds when X-j-1 
provided that 
lp\+ 1] — W > — Pa+1 ^ +1 + Px<r A -h «(o'a+ 1 — cr A ) — A a 
i.e., if 
(p A — Pa+i)t a > — pA+l°'A+l + PAO'A + a (o'A+l — O-a) — A b . 
But this will always be possible if 
i.e., if 
(Pa — Pa+i)u"a > — pA+i^A+l +PaO-a+ “(o’a+i — O-a) — A. a 
(a — Pa+i)(o-a — o-a +1 ) > — A a , 
a relation which is always true since a—p A+1 and cr A — <x A+1 are both positive. 
5 c 
MDCCCLXXXI. 
