9 
critical points, where the tops of the lines lie within the parabolic 
curve y= + nf, at a distance less than unity from this curve. 
We shall now shew that the upper limit for n= of the second 
sumquantities (41) is equivalent to n°. The sums 
Nk net nid 
m 8m and i Sm 
6 we 
Nk—Nk ne+l 
are arithmetical progressions consisting of terms of equal signs only; 
their total value is easily verified to be exactly equal to + (77). 
Thus we have, but for a certain constant, whose value is of no 
importance for the asymptotic behaviour of these sums: 
nk + nk? 
Sin so (n,°)’ = (n,°)? HE, GOE (nx’)? 
0 
Since the terms of the sum in the right-hand member increase in 
absolute value or remain constant over a certain number of n-values, 
the sum itself is, in absolute value, less than the last and the first 
term together. So we have 
(2) 
8 
nj np 
where e is a positive quantity which tends to zero for k=o. 
Further, for an n-value lyiug in the interval between nj — nf = 
< (ne)? + (2,5)? < (n1°)' (14e) 
=a + Ba and nz sale the value of s®) lies between the two 
values of s® for these two limits, because s, remains of the same 
sign between them, and therefore se) varies monotonely. So we have 
for all n- values of the interval considered 
a2] — (nyt? (1-46) Z mx (1-48). 
For the part of the n-values greater than nz we therefore have 
a fortiori 
ec 0 (1 Trajet. 46 ate eer B) 
For the remaining part we have 
n> nk — NIS. 
Now a 
nye? — (nz — nye? + nj’) = (nk — nj)? (1 + €) 
where, again, e is arbitrarily small for sufficiently great &. Thus for 
the latter n-values, too, we have 
