8 
becomes of an order which is by less than one unity higher than 
the first sumquantity (35), i.e. of an order less than n't. We 
proceed in the following way. For a certain value n =p we take 
8) =p, 
where by the notation p’ the same thing is meant as Z(p’). Now 
we make the following sums decrease as soon as possible by taking 
a certain number of coefficients a, for n >> p, all equal to — 1. 
After p? terms the sum s, has in this way become equal to zero. 
And after p? terms more s, has attained the value — p’; the value 
of n is then equal to p + 2p%. It may happen that for this value 
of n the quantity n° is still equal to p’*). In this case we may say 
that |s,| has again reached the required upper limit and we assume 
a certain number of next coefficients a, all equal to +1. If, on 
the contrary, nf for n=p+2p° is greater than pf, we go on taking 
a, equal to —1 as long as s, has diminished to such an effect that 
sa is again equal to 7’. Such an n-value must be reached, if 
O6<1, for |s,| would become of order n, if we never stopped 
taking a, ——1. Just as in the former case we assume, after this 
n-value, the next coefficients a, equal to + 1, until s, has attained 
the value „? again. And so on. The upper limit for » =o of the 
sums s, is then equivalent to »? and that of the coefficients a, is 
equivalent to n°. We call the values of n for which s, is equal to 
ni critical values and we denote them by nz for k—=1l, 2,.... If 
we assume arbitrarily a certain value n=p as a critical value, 
then it follows from the preceding construction that the following 
critical values are uniquely determined. We might continue the 
same construction towards the side of the smaller n-values, but this 
is of no importance, since the asymptotic behaviour of the quantities 
in question is solely effective. The graphical representation of the 
coefficient @p, considered as a function of n, consists of a number 
of horizontal line-segments, which are alternately above and beneath 
the n-axis, and are at a distance 1, the change taking place in the 
critical points. The graphical representation of the sums s, consists 
Jt 
of joined line-segments including alternately angles of Dn and 
JT 
ae with the n-axis; and the alternation again takes place in the 
1) We may write (p + 2p 4/9 = pi+26p24—1 (1—e), where lim «=O. From this 
poe 
it follows that for 6<1 and large p-values, E (p+ 2p)? is in general equal to 
E (p'). 
