MR. R. C. ROWE OR ABEL’S THEOREM. 
735 
and if we choose k so that j‘ is the greatest of the series ~ , we have, deter¬ 
mined as a unique value, what we will provisionally call cq. 
Next put 
(n—k)cr- f- tik =(n—s)cr J r a s 
whence <j =——— 
S — K 
Now this value is to make 
{n — k'j<T -f - QjJc ^ (n ~ t^jcr(hi 
or {t—k)a s —(s—k)a t >(t—s)ai,; 
and since by interchanging s and t we get the contradictory of this inequality, it is 
impossible that by putting 
(n—k)cr-\-ai c —{n—t)(j-\-at 
each of these could be made > (n— s)o--{-a s . 
Therefore the second step is also unique ; and 
a s — Ok Ojc 
s-k < k 
Ci s Ctjc 
since — < 
s K 
so that the second a is less than the first and may be called oq. 
Now, resuming the process of art. 13, divide the terms of the expression 
• • • +2i2/d-2o 
into sets : calling the first /q of them the first set , the next k 2 — k l the second set, and 
so on, the last ki — ki_ x constituting the I th set. 
Also call that term of the first set in which when y 1 is written for y the highest 
resulting index of x is the largest the major term of the first set, call that term of the 
second set in which on the substitution of y. 2 the same happens the major term of 
the second set, and so on. 
Then I proceed 
(i) to show that by a proper choice of the quantities y,_ l( (/ n _ 3 , . . . q x , q 0 , which 
are at our disposal, we can make the major term of the first set an absolute major (for 
the substitution yj, i.e., furnish a higher index of x than is furnished by any other 
term ; the major term of the second set an absolute major (for the substitution yj), and 
so on, 
(ii) to show that the condition of (i) is necessary in order that p —a may have the 
-smallest value of which it is susceptible. 
