738 
MR. R. C. ROWE ON ABEL’S THEOREM. 
Once more, it holds when m=X—1 if 
i.e., if 
i.e., if 
[/h- l] — M > — pK +1 cr A + 1 + p A o- A + CL (©*_! — o- A ) — A a 
{pK — p\-\)T\-\> — px+iCT A+1 +p A o- A + a(o- A _ 1 — cr A ) — A a 
(p\-i — /3 a )t a _ 1 < /3 A _ 1 cr A _ ] — p \ — a( cr A _ 1 — cr A ) + A a 
and this can, as in the last case, be shown to be always possible. 
Now if the inequality (D) holds, m being greater than X, it will hold when for m 
we write m -\-1 provided that 
I P®*+J [pjd - Pm+L<r m +1 PmCm ^ »i+\ CT »t) 
i.e., if 
Pm(o'm+\ ct ri ij cr 
but 
pm ^ Ot, (T,n X"* CT OT _|_2 
therefore this relation does hold. 
But the inequality (D) is true when rn = X +1. It is therefore true for all larger 
values of m* 
It can similarly be shown that if the inequality holds, m being less than X, it will 
hold when for m we write m —1 ; and that, since it holds when m=\ —1, it holds for 
all less values of m. 
It is therefore proved universally. 
We observe that, as was stated at the outset, the consideration of the case (b) has 
only introduced a restriction into the conditions of the case (a) —viz. : that the t s are 
no longer merely subject to the necessity of lying between consecutive cr’s, but must 
satisfy the closer conditions expressed by the inequalities 
, . >p A oX — p A+1 cr A+1 + a(cr A+1 — cr A ) — A a 
\P\ — P\+i) T \ / \ I A 
<Pk<T\— /3 A +iCT A+1 + a(cr A+1 — cr A )H-A a 
where in the first line a denotes any one of the numbers of the X th , in the second any 
one of the (X+l) th set. 
19. We have next to consider the second proposition of page 735, viz. : The condi¬ 
tion of (i) is necessary if p.— a is to have its smallest value. 
* It must be observed that this method of proof could not be used to deduce the case m + 1, \ + 1 from 
the case m, A; for it would not be necessarily true that p m is less than a.. 
