44.6 TRANSACTIONS OF SECTION A. 
each dg=P=Gr=hr>/; at any point of continuity of @, which is not a point of 
continuity of dp; dn=f>?- 
(N.B.— may have points of continuity other than those of dz; e.g., f=9, 
except at one or more isolated points, where f= 1.) 
Theorem 4.—The only points where both ¢r and ¢p are continuous are the 
points of continuity of . 
5. Similar theorems hold for the lower associated functions, Hence, con- 
sidering the common points of continuity of the two pointwise discontinuous 
functions p and y, there is no distinction of right and left for f, except possibly 
at points of an ordinary outer limiting set of the first category. 
This is not the utmost that can be said; the complete theorem is:— 
Theorem 5.—The points, if any, at which dzF fx are countable. 
6. Thus we have proved that the points, if any, where there is a distinction of 
right and left for f are countable. 
Example of a function having distinction of right and left at a countable set of 
points dense everywhere. Make terminating binary points on 2-axis correspond to 
middle points of black intervals of Cantor’s typical ternary set on the y-axis, and 
add 1 to the limiting values at all the remaining points. 
7, The reasoning proving Theorem 5 serves to prove the following :— 
Theorem 6,—The points, if any, where 
f>dr, or f>gr, or f>h 
are countable. 
II. Non-uniform Convergence and Divergence. 
1. Definition of the right and left peak and chasm functions.—Let f,, fy.» » 
be a series of functions having a definite limiting function f. Let P be any point ; 
take an interval on the right of P, say (PQ), and let M, be the upper limit of 
Ff, in (P,Q). Represent these§ numbers M, on the y-axis, and let Mg be the 
highest point of their first derived set. Now let the interval (P, Q) diminish 
indefinitely ; M never increases, and has therefore a limit rp (P). This is the 
right-hand peak function. 
Similarly define the left-hand peak function, and, making suitable changes 
the right and left chasm functions yp and xz. : 
The omission of the subscripts, or of the words right and left, means that we 
take that one which is in the case of m not less, and in the case of x not greater 
than the other. ‘ 
2. Theorems 1-6 are the same as in Part 1, only using now the peak and chasm 
functions. (N.B.—Hence we get precisely the same relations of right and left as in 
the case of the associated functions.) 
3. Assuming now that the functions f, are in the extended sense continuous 
the condition for uniform convergence is = x; it will then follow that r=x= f. 
This includes what I call uniform divergence when r=x=f=00. It may easily 
be shown that the points of uniform divergence form an inner limiting set, 
The reasoning used by Osgood, and subsequently by myself, in discussing non- 
uniform convergence shows that the points of non-uniform convergence and non- 
uniform divergence together form a set of the first category. Hence it follows 
that if a series of continuous functions diverges at a set of points dense every- 
where in an interval, it diverges uniformly at points which form a set of the 
second category. 
3 Ona Remarkable Periodic Solution of the Restricted Problem of Three 
Bodies. By Dr. W. DE SITTER. 
In the third volume of his ‘ Méthodes Nouvelles’ Poincaré has devoted a short 
chapter to what he calls periodic solutions ‘de seconde espéce.’ The principle of 
these solutions is the following. Let two bodies with infinitely small masses 
