An arithmetic treatment of some problems in analysis situs

CO

AN ARITHMETIC TREATMENT OF SOME

PROBLEMS IN ANALYSIS SITUS

A DISSERTATION

SUBMITTED TO THE FACULTY OF ARTS AND SCIENCES OF HARVARD UNIVERSITY IN

SATISFACTION OF THE REQUIREMENT OF A THESIS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

BY

L. I). AMES

BALTIMORE
2:8« fiotb (gaftimotc (prtee

THF. FRIEDENWALD COMPANY
I 90s

M ARITHMETIC TREATMENT OF SOME
PROBLEMS IN ANALYSIS SITUS

A DISSERTATION

SUBMITTED TO THE FACULTY OF ARTS AND SCIENCES OF HARVARD UNIVERSITY IN

SATISFACTION OF THE REQUIREMENT OF A THESIS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

BY

L. D. AMES

I

BALTIMORE
Z^t Boti (Jlaftttnote (prtet

THE FRIEDENWALD COMPANY

IQOS

913088

p

An Arithmetic Treatment of Som^e Problems
in Analysis Situs.*

By L. D. Ames.

CONTENTS.

PAOB

Ihtkoduotiok 343

Past I. In Two Dimensional Space 345

Cbapter I. Fundamental Conceptions 345

Chapter II. The Theorem as to the Division of the Plane 353

Chapter III. Regions, Orientation of Curves, Normals 359

Fast II. In Three Dimensional Space 365

Chapter I. Fundamental Conceptions 365

Chapter II. Solid Angles ' 369

Chapter III. The Theorem as to the Division of Space 377

Introduction.

C. Jordan f has proved that the most general simple closed curve divides
the plane into an interior and an exterior region. But he assumes all needed
facts in regard to polygons without stating clearly just what those assumptions
are. He certainly makes use of more than the special case of the same theorem
for polygons. Later A. Schoenflies J proved the theorem for a more restricted
class of curves including polygons. But his proof is not simple. Complete
arithmetization is at least impracticable; the writer must leave to the reader
the last details, but these details should be only such as the reader can
immediately fill in. Where the line shall be drawn must be left to the judgment

• An abstract of the principal results of this paper was presented to the American Mathematical Society at
Its meeting of December, 1903, and was published in the Bulletin, March, 1904.
t Court cP Analyse, 2d ed.. Vol. I., SS96-103, 1893.
JGottinger Nachrichten, Math.-Phys. Kl., 1896, p. 79.

46

344 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

of the individual writer. Schoenflies has left far more for the reader to do than
have I in the proof that follows.

There is, however, one point in which Schoenflies' work is open to more
serious criticism. He proves that any straight line which joins an interior point
to an exterior point has a point in common with the curve, and then asserts in
the theorem, without further consideration, that it is impossible to pass from an
interior point to an exterior point without passing through a point of the curve.
This does not follow. In fact it is possible to divide the points of the plane into
three assemblages /Si, S2 and B such that a point of Sj cannot be joined to a
point of ^^2 by a straight line having no point in common with B, or by a curve
consisting of a finite or infinite number of straight lines, but such that this can
be done by other simple curves. And the essential difierence between these
assemblages Si and S^, on the one hand, and the interior or exterior of a curve,
on the other, is a property of the interior and exterior which Schoenflies leaves
unmentioned in the theorem or proof, and the omission of which is our final
point of criticism: namely, that each is a continuum, that is, if any point is an
interior (exterior) point all points in its neighborhood are also.

More recently Ch.-J. de la Vallee Poussin * has published an outline of a
proof of the same theorem for the most general simple curve. This work appears
much more simple than either of the proofs already mentioned, but it is not
arithmetic in form, and it is not easy to see how the arithmetization is to be
effected. It can, therefore, be regarded only as a sketch of whatever rigorous
proof may be made following its lines.

Since the publication of the abstract of the present paper G. A. Blissf has
proved the theorem for a somewhat more general class of curves than those for
which Schoenflies proved it. O. Veblen J has recently published a proof for the
most general simple curve. None of these proofs deals with the corresponding
theorem in three dimensions.

The present paper assumes the axioms of arithmetic but not those of geome-
try. It contains a proof of the above mentioned theorem for a class of curves
more restricted than those of Jordan, of Valle6 Poussin, or of Veblen, but more

•Court S'Analytt inflniUtiTnal, Vol. I., (1908), §$300-803.

t The exterior and interior of a plant curve. Bulletin of the American Matbematlcal Society (3), Vol. 10,
(1004), p. 898.

X Theory of plane curvet in non-metrical analytit titui. Transactions of the American Mathematical Society,
Vol. 6, No. 1, Jan. 1905, p. S8.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 345

general than those of Schoenflies or of Bliss. It goes back to fundamental arith-
metic principles, and does not assume the theorem for the polygon. Moreover, it
is extended in Part II to the corresponding theorem in three dimensions, and it
seems highly probable that it could be extended to more than three dimensions.
The proof, both for two and for three dimensions, is based on a conception which
I have called the order of a point with respect to a curve [or surface]. The order
is a point function, uniquely defined and constant in the neighborhood of every
point not on the curve [or surface], and undefined and having a finite discontinuity
at every point of the curve [or surface]. Its value is always a positive or
negative integer or zero.

Part I.
IN TWO DIMENSIONAL SPACE.

I. — Fundamental Conceptions.

1. A. point is a complex of n real numbers {a, I .... ). This number n is
called the number of dimensions in which the point lies. An assemblage is any
collection, finite or infinite, of such points. In two dimensions the assemblage
of all the points is called the plane. In the earlier chapters we confine ourselves
to two dimensions. The numbers x, y are called the coordinates of the point
P {x, y); the point (0, 0) is called the origin; the assemblage of all points of the
type (z, 0) is called the aj-axis, etc. Distance, straight lines, circles, squares, and
other elementary conceptions are assumed to be defined by their usual analytic
expressions without explicit mention.

2. Transformations. K point transformation is a rule by which the points of
an assemblage are individually replaced by the points of an assemblage, in
general different. If, whenever a property belongs to one of these assemblages
it also belongs to the other, it is said to be invariant of the transformation. A
rigid transformation in two dimensions is defined by relations of the type.

jc = a/ cos a — y' sin a + Xq,
y = a/ sin a + ?/' ^^^ ^ + V^-

346 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

We shall assume without explicit mention the simpler facts of invariance. We
shall use the expression change of axes for brevity to denote a rigid transfor-
mation whenever it is desired to emphasize the fact that the essential proper-
ties of the assemblage are unchanged. The reasoning involved can generally be
stated in something like the following form. An assemblage is given concerning
which certain facts are known. The assemblage is transformed into a second
assemblage by a transformation with respect to which the given facts are known
to be invariant. Certain conclusions are reached in regard to the second assem-
blage. This is then transformed into the given assemblage by the inverse of the
first transformation. The conclusions are known to be invariant of this inverse
transformation. They therefore apply to the given assemblage. Unless other-
wise specified a change of axes shall be effected by a rigid transformation.

3. Existence of a Minimum. The following theorem is well known:*

Theorem. If S^ and S^ are two complete f assemblages of points having no
point in common, then the distance of any point of Si from any point of S^ has a
positive minimum.

4. Curves. A simple curve % is an assemblage of points {x, y) which can be
paired in a one to one manner with the points of the one dimensional interval
(^ = ' = <i) in case the curve is not closed, and with the points of the circle

^ = cos/W, >7 = sin>W,

in case the curve is closed; moreover, when t approaches a limiting value (t) the
point (a;, y) shall also approach a limiting point («, y), and this limiting point
shall be the point of the curve which is paired with 7. In the case of the open
curve the points corresponding to t^ and t^ are called end points.

It follows from this definition that a simple curve can be represented analy-
tically by equations of the form

x = 4>(<), y = m^ {k<t<h),

where ^{t) and 4(0 are single valued continuous functions, and

^{t) = <^{t') and 4,(<)=4,(<')

•Of., for example, Jordan, Court cTAnalyte, 2d ed., Vol. I, §30, last paragraph.

t Professor Pierpont snggests complete as the English equivalent of abgeschlossen.

JCf. A. Hnrirltz, Yerhandlungen det ertlen Internationalen itathematiktr-Kongreaus, p. 108.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 347

are not simultaneously satisfied in the case of the open curve when t =f= t', and are
not simultaneously satisfied in the case of the closed curve when t^ t" except

that ^(<o) = <?>(<i) and ^.{to) = ^{ti) .

If the curve is closed, let a = ti — t^. It is then convenient to extend the defi-
nition of the functions <p{t) and ■<i^{t) to all values of t by means of the relations

^{t + no) = ^{t) , ^t + no) = ^{t) ,

where n is an integer and a is defined to be the primitive period of the pair of
functions. Conversely, every assemblage of points defined by the above equa-
tions is a simple curve.

A simple curve is said to be smooth at a point if the parametier can be so
chosen that the first derivatives ^'{t) and ■^/'{t) exist, are continuous, and do not
both vanish at the point. If the point is an end point one sided derivatives are
admitted. A smooth curve is a simple curve which is smooth at every point. A
regular curve consists of a chain of smooth curves. Analytically, it is an assem-
blage which can be defined by the equations

where p{t) and ^-CO are single valued continuous functions whose first deriva-
tives ^'{t) and 4''{i) exist, are continuous and do not vanish simultaneously,
except possibly at a finite number of exceptional points called vertices. Moreover,
these derivatives approach limits as the point t approaches any such exceptional
value if from above, and also when t approaches t' from below, and in each case
the limits approached by 4>'(<) and •4''(0 *i"® '^ot both zero; the forward limits
are not both equal respectively to the backward limits. It follows that one
sided derivatives exist at thq exceptional point and that they are equal to the
respective limits.

A regular curve may admit multiple points, that is, points common to two
or more of the constituent smooth curves, other than the common end points of
two successive smooth curves. Arithmetically such points correspond to distinct
values oft. Two or more of the constituent smooth curves of a regular curve
may coincide along whole arcs. Such curves may be treated arithmetically in
the same way as the Riemann surface is treated. We do not need such curves,

848 AifES: An Arithmetic Treatment of Some Problems in Analysis Sitits.

and shall include all such points without distinction under the term multiple
point. Any point of a regular curve not a multiple point is a simple point. All
of these definitions refer exclusively to assemblages of points and not to the par-
ticular way of representing them.*

The following is a special statement of a well known theorem :

Theorem I. A simple curve is a complete and perfect assemblage of points and
lies in a finite region of the plane. Moreover, if a set of points of the curve has a
limiting point (i, y), then the corresponding values of t have a limit which is a point
of the interval (<o = ^ = ^i)t ow(? (», y) corresponds to i.^

The following theorem is stated without proof:

Theoeem II. A regular curve can be divided into a finite number of parts,
each of which can be represented by an equation of the form

or else by an equation of the form

y=f{^)> ia<x<b),

where f is single valued and continuous throughout the interval of definition.

Most of the results which follow apply to a somewhat more general class of
curves, which by virtue of Theorem II includes all regular curves. We shall
describe such a curve by saying that it satisfies the following condition :

Condition A : A curve which consists of a chain of simple curves (after the
manner of a regular curve) and furthermore is such that each constituent simple
curve can be represented by an equation of the form

y = /(x) or else by an equation of the form x ^f(y) ,

where /is single valued and continuous, is said to satisfy Condition A.

5. Vectors and Angles. We define a vector % to be an object determined
by the two following phenomena :

*For % dltcnssion of change of parameter, orientation of curves, etc., see Sec. III.

f Cf. Jordan, Court cT Analyse, 8d ed.. Vol. 1, §§64, 85.

X This is a special case of an oriented curve, discussed in Art. 12.

Ames : An Arithmetic Treatment of Some Problems in Analysis Situs. 349

(a) A simple curve which can be defined by equations of the form

(x — ait + bi, a/ + a/>0,

\y = a^t+b„ {t,<t<t,);

(b) One of the two possible permutations, Pq Pj or Pi Pq, of the end points.
It follows that the end points are individually invariant of any change of

parameter consistent with the definition, and that a vector is also completely
defined by naming the end points in a particular order, e.g. PqPi- Taking the
four points Po(«o. ^o). Pi{^i< Vi), Po{4, yo)^ and Pi{x[,yi), the two vectors
PjPi and Pj P{ are said to be equal if

Xi — Xo=xi— a;^ and 2/i — yo = 2^i — ^o-

The length of the vector Pq P^ is the positive number

\^ {xi — xof + {yi—yoT '

The angle 6 from the vector Pq Pi to the vector P'^ Py is defined to be any simul-
taneous solution of the equations

X,

a'o Vi — Vo
^ — x'q y'l—yo

sin 0 = ^
where K is the positive number

cos 0 = ^

yi—Vo -(«! — a;o)
x'l — xlt yi—yo

(^)

K= [V(xi-xo)^ + {yi-yof V(xi -x^ f + {yi-y^yr'-

If the vectors are defined by means of their equations

^'>^''-\y = a,t + h,

a? + ai>0,
a[' + ai'>0,

f a; ^aj < -f 6i,
^^^''■\y = ait + b;„ {4<t'<t[),

where the parameter t is so chosen that the value of t at the first named end
point is less than that at the last named end point, then equations {A) are equiv-
alent to the equations

sin 6= X

ai

al

cos 6= X

az

a[

a'

(S)

where K is the positive number

350 Ames: An Arithmetic Treatment of Some Problems in Analysis Sitixs.

We here assume an analytic definition of the trigonometric functions.* Ordinarily
we select a particular solution by some convention. The angle ABC shall mean
the angle from BA to BO.

To justify the above definition it can be shown:

(a) That equations (A), and hence (B), always have solutions differing by
multiples of 27t;

(6) That if the angle from P, Pi to P; P[ is 6, and that from Pl>P[ to P^'P^'
is 6' , then the angle from P^Pi to PqP'i is 0 + 0' + 2mt, where n is a positive
or negative integer or zero ;

(c) That B is invariant of any rigid transformation.

6. Continua. A two-dimensional continuum is an assemblage of points
P(x, y) such that:

(a) If Pq (cco, ^o) is a point of the assemblage, all points in the two dimen-
sional neighborhood :

\x — ocn\<h, \y—yo\<^

of Pq belong to the assemblage ;

(6) Any two points of the assemblage can be joined by a simple curve con-
sisting wholly of points of the assemblage.

The neighborhood of a point Pq may be defined generally to be a continuum
containing Pq and such that the distance of any of its points from Po is less than
h, where A is a positive constant as small as either party to a discussion wishes.
The term near will be used as a technical term to replace the longer and more
familiar expression in the neighborhood of.

Any point of a continuum is an interior point. A boundary point of a con-
tinuum is a point not belonging to the continuum but having points of the con-
tinuum in its neighborhood. Any point Pj not an interior or boundary point is
an exterior point, and all points near Po are exterior points.

The term region is applied both to a continuum, and to a continuum plus its
boundary. A region is said to be finite if it is possible to choose a constant G
80 that if P{z, y) is any point of the region, then

1*1 + \y\<G-

• Of. for insUnce, Godcf roy, TMorit (Itmentaire dtt Sirie$, Chap. 6.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 351

The boundary of a region is a complete assemblage.* If two continua have a
point in common, then the totality of the points of the two continua taken
together form one continuum. The continuum S' is said to be annexed to the
continuum S along B (a part or all of their common boundary) if the points of
S, S', and B form one continuum and are so considered. The definitions of this
section are invariant of any one-to-one and continuous transformation.

There are at least two essentially distinct ways of defining a particular con-
tinuum. One is by defining the points of the continuum explicitly. The more
common way is by defining the boundary explicitly. Section II deals with the-
orems relating to the second method. The following are examples of the first
method.

Example 1. The interior of a circle may be defined by the inequality

a.» + 2^2 — r»<0.

Example 2. The interior of a triangle, or more generally the assemblage S,
defined by relations of the form

Ui{x,y)=A,x + B,y+ a>0, {i=l, 2, 3),

where these equations are satisfied by at least one point, can be proved to be a
continuum as follows:

(a) Let Pq {xq, y^j be one point of S. Since ?<; is continuous and Ui{xQ, y^
> 0 , hence if P (x, y) is any point near Pq, M<(a;, ?/) > 0, and hence P belongs
toS.

{b) Let Po (xo. 2/o) ^^^ P\ (3^1. Vi) l>e any two points of S. Hence
Mi(a;o, 2/o) = ^.a-o + 5i2/o+ C'i>0,
u,{x„y,) = Ax, + B,y,+ Q>0, (*-i,^^j.

and therefore

where ;ii and \ are any numbers not negative and not both zero. This is a
suflBcient condition that the point

/\ Xq + a^ zi , ^iyQ±2^yy\

\ Aj -|- Ag Ai "T A2 /

*Cf. Jordan, ibid., §23.

47

352 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

belongs to S. But any point of the segment Pq Pi can be expressed in this form.
Hence any two points of S can be joined by a straight line wholly in S. Hence
S IB & continuum.

Example 3. Let an assemblage S (Fig. l) of points P{x,y) be defined by
the relations

iCo<a;<JBi, 2/=/(a5)+r, 0<r<A, (1)

Fi£. 1

where f{x) is single valued and continuous, and h is constant. We proceed to
prove that >S'is a continuum.

(a) Let P{x, y, r) be any point of S. Choose 5 >■ 0 so that

25<r and 25<A — f;

25< r <A — 25.

that is

Since /(x) is continuous it is possible to choose e >■ 0 so that

Ax)-S< Ax) <A^) + 8

when

and so that

x — e<^ x -eC^x + e,

We now proceed to show that any point P{x, y) in the neighborhood

X — e<

of P lies in S. From (4) and (5)

X

y

<3-l-

(2)

(3)
(4)

(5)
(6)

(7)

Ames: An Arithmetic Treatment of Some Problems in Analysis iSitus. 353

Adding (6), (2), and (3) and simplifying by means of (1) we obtain

/(^)< y <f{x) + h, (8)

that is, y=.f{x)-\-r, whereO<r<A. (9)

But (7) and (9) are the condition that P is in S,

{b) Let Pi(xi,yi; r^) and ^3(0:2, y^; r^ be any two points in S, and let
ri, <^ rj. They can be joined by a simple curve in S defined as follows:

x = X2, /(x) + ri<f/</(a;) +rj,

and 2/=/(a;) + »•], x-i<x<X2 ov Xi<x<xi.

Hence S\s & continuum.

II. The Theorem Relating to the Division of the Plane by a Simple Closed

Curve.

7. Order of a Point. Given any closed curve whose equations are

x=:^{t), y = ^t),

where a is the primitive period of the pair of functions ^{t) and 4'{t) • Let P (t)
be a variable point on the curve, and 0 a fixed point not on the curve. Let 6{t)
be the angle which OP makes with the positive cc-axis, or its equivalent for
this purpose, the vector (0, 0)(1, 0). Then 6 is an infinitely multiple valued
function of ^ for all values of ^, such that any two values of d corresponding to
the same value of t differ by a multiple of 2 71. Let t^ be a particular value of t
and let 6i{t^) be a particular one of the values of 6{to). Then it is possible to
choose from the different values of 6{t) one and only one set of values which
form a single valued continuous function of t taking on the chosen value 6,(<o)
when t = t^.* Call this single-valued function 6{t). The values of t and t + u
represent the same point, and the values of the multiple valued d{t) at this point
differ by a multiple of 27t. Hence

e{t + o) = e{t) + 2 n 7t ,

•Cf. for example, Stoltz, Differential Seehmmg, Vol. 2, p. 15-20.

354 Ames: An Arithmetic Treatment of Some Problems in Analysis Sitiis.

where n is a positive or negative integer or zero. Then n is defined to be the
order of the point 0 with respect to the particular parametric representation of
the curve.* The order is not defined for any point on the curve.

That the order depends only on 0 and not on the particular value of t
chosen may be seen as follows: Let t vary continuously. Then 6{t) and
${t-\-a), and hence n vary continuously. But n can vary only by integers.
Hence n is constant. That n is independent of the particular one of the possible
single valued functions chosen is seen in a similar manner. That n is invariant
of a rigid transformation follows from the fact that 6{t) and d{t + Ui) are invariant.

8. We proceed to prove some theorems about the order of a point.

Theorem I. If a point is of order n with respect to a given closed curve, then
all Joints near it are of order n.

Proof. Let Oj be a point not on the curve, and 0 any point near it. Let
0, and B be the angles which 0^ P and OP respectively make with the a;-axis.
Then if Oi 0 is suflBciently small, \ d^ — B \ is less than an arbitrarily pre-
assigned number for all points on the curve. In particular

\_Q,{t + CO) - e,{t) ] - \e{t + (o) - 0(0] < 2 n.

Hence the order of 0 diflfers from that of Oi by less than unity. But both are
integers. Hence they are equal.

Corollary. The points of the order of a given point form one or more continua.

Theorem IL // two points are of different orders with respect to a given closed
simple curve G , any simple curve joining them has a point in common with G .

Proof Let the end points Po(^o) and Pi{ti), {t^-C 0> o^ ^ simple curve G'
be of orders m and n respectively with regard to the closed curve G. Consider
the upper limit t of the values of t corresponding to points of order m.
Then there are points of order to and other points not of order m near P{i).
If P is not on the curve C this contradicts Tlieorem I. Hence, P is on the
curve C.

*It is sulBclent for the present chapter to consider only one particular parametric representation. See
Art. 10 for a discussion of the invariance of n with respect to a change of parameter.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 355

9. The theorem relating to the division of the plane by a closed simple
curve will be proved by the aid of two lemmas corresponding to the statements
that the curve divides the plane into at least two, and at most two continua.

First Lemma. Near any point Pq of a simple closed curve which satisfies Con-
dition A there are ttco points of orders differing by unity.

Proof. The curve consists of a finite number of parts, each of which can be
represented by an equation of the form

(«) 2/ =/(»).
or else by an equation of the form (b) x =/(y),

where / is single valued and continuous. If Pq is an end point of one of
these parts, then there is a point near P^ which is not such an end point. Hence
we may assume without loss of generality that Pq is not such an end point. Sup-
pose that the part on which Pj lies can be represented by the equation

The other case is similar. Transform to new axes parallel to the original axes
and having Pq as origin (Fig. 2). All the conditions are invariant of this trans-
formation.

The y-ax'iB has no point other than Pq io common with the curve near P^.
Hence it is possible to choose R so small that if r < B, and B is the point
(o, r), and Bi the point (o, — r), the segment BBy has no point on the curve
except Pj. Let P{t; x, y) be any point on the curve. Let $ and dy be the

356 Ames: An Arithmetic Treatment of Some Problems in Analysis Sittis.

angles BP and Bj P respectively make with the positive a-axis. Let p = BP
and pi = BiP, where p, pu and r are positive numbers. By the definition of an
angle,

Sm d = ^ , COS 6 =■ —

P P

sm Bi = ^— ! — , COS 01 = — .

Pi Pi

Let <^ = 0 - 01 . Then

sm d) = , cos A = ^^ —^-^ .

PPi PPi

From these relations, if o is the primitive period and e sufficiently small, and n,
ni, and n, integers, positive, negative, or zero, and x increases as t increases near
P,, it follows that

^{to+e) >(2ni+l)7t,
^{to+ci—e) <(2n2+l)7«,
^(<o + 6)) =(2nj+l)7t.

When <o<C ^ <C ^0 + "> sin^ can vanish only when a; = 0, and in this case
(x* + 2/^) — r', and hence cos ^ is positive. Hence

^{t)^{2n+ 1)71 when <o<^<^o+".

and therefore «2 — nj = 1 . Hence

[0(<„ + 0)) — 0(0] - [0'(«o + ") — ^'(^o)] =^{to + ") - <p{to) = 2 7t.

Hence the order of B exceeds that of B' by unity.

Second Lemua. Griven the continuum B, and the curve AB :
y = /{x), or x=^f{y),

where /is single valued and continuous :

Ames: An Arithmetic Treatment of Some Problems in Analysis /Situs. 357
P

(a) If R contains all points of the curve, except possibly its end points, which
may lie in the boundary of B, then the totality Br of points of B not on AB form at
most two continua ;

(b) If also one or both end points lie in B, then B" is one continuum.

Proof, (a) Suppose the curve can be represented by the equation

y=^f{^)^

(Fig. 3).

The other case is entirely similar. Draw a straight line CD parallel to the
y-axis, lying wholly in B, and bisected at a point of the curve AB, with C lying
above the curve. Let P be any point of B" which cannot be joined to Z> by a
simple curve wholly in B~. If there is no such point the theorem is granted.

Fig. 3

Otherwise join P to D by a simple curve PD wholly in B. This curve will have
a point in common with the curve AB. Let PE be an arc of PD having one
extremity E on the curve AB, but containing no other point of this curve,
Choose an arc A'B' of AB which contains E and also the point common to AB
and CD in its interior, but does not contain A ov B. Along this arc construct
two continua like that of Art. 6, Example 3, one above, and one below A!B',
lying wholly in B~, and denote them by Ar+ and N~ respectively. Choose a
point F on PE so near to E that it lies either in iV"+ or N~. Suppose it lay in
N~. Choose a point G on CD in N~. Then F and G can be joined by a
simple curve wholly in N~. Hence the simple curve PFGD lies wholly in Br,
which is contrary to hypothesis. Hence F must lie in N"^, and by similar
reasoning P can be joined to J^ by a simple curve wholly in B~. Hence
the points of B~ form at most two continua.

358 AuES: Ati Arithmetic Treatment of Some Problems in Analysis Situs.

(b) Suppose the curve is represented as in the first case but let B (Fig. 4)
lie in B. Extend the curve AB slightly parallel to the y-axis, to B' . By the
first case the points of E not on AB' form at most two coutinua. If they form

Fig. 4

one continuum the theorem is granted. If they form tvro continua, by the
adjunction of the points of BB' exclusive of B these can be annexed to each
other, thus forming one continuum.

Main Theorem. The points of the plane not on a given simple closed plane
curve satisfying Condition A form two continua of each of ichich the entire curve is
the total boundary.

Proof. In the neighborhood of any point of the curve there are two points
of different orders with respect to the curve (First Lemma). Hence the points of
the plane not on the curve form at least two continua (Art. 8, Th. I and Cor.,
Th. II, also Art. 6). Divide the curve into a finite number of parts each of which
can be represented by an equation of the form y =f[x), or else by an equation
of the form x=f{y). Construct these in the order in which they appear in the
curve. By the second part of the Second Lemma each of these except the last
does not divide the region consisting of the plane less the points already cut out.
The last divides the plane into at most two continua. Hence the points of the
plane not on the curve form just two continua.

Any point of the curve is a boundary point of each continuum (First Lemma,
and Arts. 8 and 6). Any point not on the curve belongs to one of the continua,
and hence cannot be a boundary point.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 359

III. Regions, Orientation of Curves, Normals,
AND Related Topics.

10. CJiange of Parameter. The following theorem is stated without proof:
Theorem. Let a given sim2J'le curve he defined by two sets of equations

\;Zlt '.<><.. and |-|W; ,<e<,.

where ^, t//, (^ and 4> are single valued, continuous functions.

(a) In the ca,se of the open curve, if no two values oft\t!'\ yield the same point,
and if the values oft and t which yield the same point of the curve are assigned to
each otlier, then t is a single valued, continuous function fit'), monotonic and never
constant throughout the interval of definition ; and the same two points are given as the
end points in each case ;

(b) In the case of the closed curve, if no two values of t \t''\ yield the same point
imless they differ by a period of the pair of functions, the values of t and t' which
yield the same point of the curve can be assigned to each other in such a way
that i' = f{t), where f if) is single valued and continuous for all values oft, monotonic
and never constant.

The totality of transformations H =f{t) thus defined form a group G. Such
a transformation is said to be even if an increase in t yields an increase in t'. The
even transformations of G form a subgroup G+ o{ 6. Any transformation of G
is an even transformation or is equivalent to an even transformation followed by
the transformation t! ^ — t. The order n of a point with respect to the curve
is invariant of any even transformation. If t is replaced by — t' the sign of the
order of a point is reversed. Then n* is invariant of any transformation of G.

1 1 . Interior and Exterior.

Theorem. All sufficiently distant points are of order zero with respect to a
given closed curve.

Proof Let Pi {xi,yi) be a distant point, and let P {x, y) be a variable
point on the curve. Let Q be the angle PjP makes with the positive cc-axis.
Then by Art. 5,

cos Q = {-x — x,)l P,P, Bin 6 = {y — y{) I P^P.

48

360 Ames: An Anthmetic Treatment of Some Problems in Analysis Situs.

Then if Va^ + 2^] is taken sufficiently large either cos 6 or sin 6 never
changes its sign as P varies. In either case the maximum variation of 6 is less
than 71. Hence the order of Pj is zero.

If the points of a continuum are all of order n the continuum is defined to be
oi order n. The exterior of a simple closed curve is defined to be that one of the
two continua into which the curve divides the plane which contains all
sufficiently distant points. The other continuum is defined to be the interior. It
follows that the exterior is of order zero, and the interior of order ± 1 . If the
interior is of order — 1 , the parameter can be so chosen that the order of the
interior will be + 1. The neigJiborhood of a curve is a continuum containing all
points of the curve, and such that if P is a point of the continuum, and Pi a
suitably chosen point of the curve, then PPj <[ h, where A is a positive constant
previously chosen as small as either party to a discussion wishes.

We have proved incidentally the following theorem, which for greater
clearness we state somewhat freely in geometric language.

Theorem. Let P he a variable point on a simple closed regular curve, and A
any fixed point not on the curve. Then when P traces the curve and returns to its
initial position, the angle which AP makes with the positive x-axis, varying continu-
ously returns to its initial value if A is an exterior point of the curve, and is changed
by in if A is an interior point.

12. Orientation of Curves. The conception of an oriented curve is a gener-
alization of that of a vector. It is often desirable to distinguish the positive from
the negative sense along a curve. The process or the result of making this dis-
tinction we will call orientation. More explicitly, we define an oriented curve
and then define the positive sense along such a curve. An oriented simple curve is
defined to be an object determined by the two following phenomena:

(a) A simple curve;

(b) One of the two possible permutations AB or BA of the end points of any
one open arc of the curve.

If the orientation of a given simple curve is defined by the permutation
PjPg of the end points Pi{t^ and Pz{t^ of a definite open arc, and P{t) is any point
of that arc, then we will agree to choose the parameter so that <, < << <8. and con-
versely. If a change of parameter is necessary to effect this it can always be
accomplished by the transformation < = — t'. Thus a permutation of the end

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 361

points of any open arc is uniquely determined. Hence a simple curve can be
oriented in two and only two distinct ways, and one of these is fully determined
by a permutation of the end points of an arbitrary arc. With this agreement as
to the choice of parameter, a point is said to trace a simple curve in the positive
sense if its parameter increases continuously. A line integral is said to be
extended along the arc ti t^ in the positive sense if t^ is the lower limit and t^ the
upper limit of integration. The sign of the order of a point with respect to a
curve is reversed by reversing the orientation of the curve.

In general the orientation of an open curve is entirely arbitrary. If a
simple curve is considered as part or all of the boundary of a definite region, we
will agree that the curve shall be so oriented that the region shall be of order
one greater than the region from which the curve separates it (see Fig. 5). If a

closed curve is not explicitly considered as a boundary of a region exterior to it,
it shall be oriented so that its interior is of order one greater than its exterior,
in other words, so that if 0 is an interior point and P traces the curve in the
positive sense returning to its initial position, the angle which OP makes with a
fixed line varying continuously shall be increased by 2 7t (see Art. 11).

If a one-to-one relation is established between the points of two simple
curves and the parameters have been chosen as above, then they are said
to have the same orientation, or to be similarly oriented if the parameter of one is
an increasing function of that of the other. They are said to hQ.\eopposite orien-
tations, or to be oppositely oriented if the parameter of one is a decreasing func-
tion of that of the other. A special case of this is that in which two curves
coincide along a given arc. In this case coincident points in the two curves are
assigned to each other, unless otherwise specified.

362 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

Theorem. Let two plane regions R^ and R^ each form the interior of a simple
closed curve Cj and G^ respectively, satisfying Condition A. Let a segment Cj of Cj
coincide with a segment a^ of C^, then

(a) If Ry and i2, are exterior to each other, the orientation ofoiis opposite to
that of (T, ;

(b) If Ri is wholly interior to R^, the orientation ofOi is the same as that ofcz-

Proof (a). Choose a part or all of (Tj (orc^) which can be represented in the

form

(1) yz=.f{x), or else in the form (2) x=f{y),

where /is a single valued, continuous function. Let Pq be any point of this
segment not an end point. The orientation of (Tj can not be the same as that
of (Tj, for suppose it were. Then by Art. 9, First Lemma, near Pq there are two
points B and B' such that the order of B with respect to either curve is greater
than that of .6' by unity. Hence by Art. 11 jB is interior to each curve, which is
contrary to hypothesis. The second case is proved similarly.

The property of the plane stated in this theorem is later taken as the defini-
tion of a bilateral surface. It is not true on a unilateral surface. Thus it will
follow that the plane is bilateral. Goursat tacitly assumes this theorem or an
equivalent one in his proof of Cauchy's Integral Theorem.* It is assumed
whenever an integral taken around any region R is assumed to be equal to the
sum of the integrals taken around the mutually exclusive regions of which R
consists, and in analogous cases involving variation, or analytic continuation
along closed paths in the study of multiple valued functions.

13. Tangents and Normals. If a simple closed regular curve is represented

by the equations

x = ^{t), y = ^{t)

and if its orientation has been defined, and the parameter has been chosen
according to the specifications of Art. 12, then the positive tangent at a point
Pq (<o) at which the curve is smooth is the vector Pq (*o) Pi (*i) defined by the
equations

x = 8^'{t,) + q>{t,), (s, = Q<s<s,).

•Acta Matbematica, I. IV, p. 197. Or see Harkness and Morley, Theory of FuncHont (1893), p. 164

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 363

A normal to the curve at a point Pq (^o) ^t which the curve is smooth is a
vector defined by the equations

X = — es^'(0 + ^(<o) . («o = 0 S « ^ si) ,

y= es^{t,) + i^{t,), (e==bl).

If this enters the interior of the curve in the neighborhood of the curve it is
called the inner normal. It follows from the proof of the first lemma, Art. 9,
that in the case of the inner normal e = + 1, and the inner normal makes an angle
of + 7t/2 with the positive tangent. Even when the region is exterior to one of
its boundaries, these conclusions are equally true of that normal which enters the
interior of the region provided the orientation of the boundary is chosen accord-
ing to the specifications of Art. 12 (see Fig. 5).

14. Regions and Boundaries. As an illustration of a class of theorems which
are often assumed without even mention, but which are by no means trivial, the
following theorems are stated, mostly without proof. A loop-cut is defined to be
a simple closed curve lying wholly in a continuum under consideration.

Theorem I. The totality B~ of the paints of a plane continuum R not on a
given loop-cut L satisfying Condition A form two continua, one of which is wholly
interior and one wholly exterior to the loop-cut, and every point of the hop cut is a
boundary point of each.

The proof is similar to that of the main theorem of Art. 9, which is a special
case of this.

Theorem II. If a loop-cut L is drawn in the interior of a closed simple curve
C, each satisfying Condition A :

(a) The interior of L lies wholly interior to C, and is wholly bounded by L;

(b) The exterior and perimeter of C lies wholly exterior to L, and the exterior of
C is bounded wholly by C ;

(c) There exist points exterior to L and interior to C, and they form a continuum
of which C and L form the total boundary.

Proof. The main points of the proof may be exhibited in outline as follows:

364 Ames*. An Arithmetic Treatment of Some Problems in Analysis Situs.
Each point of the plane belongs to one of nine mutually exclusive classes:

a

c

c.

L,

No point.
(3)

No point.
(2)

L

No point.
(1)

No point.
(1)

A

where Q and C, denote the interior and exterior respectively of C, etc.

(1) By hypothesis, no point of Z belongs to Cor to Cg.

(2) No point is in C, and Zj. Suppose P were in C, and Xj. Choose a
distant point A. This lies in 0^ and in Z,. Hence A and P can be joined by a
curve wholly in C^. This curve must cut L. Hence a point of Z is in C,. This
contradicts (l).

(3) Suppose a point P of Cwere in Z,, then all points near P are in Z^.
But there are points of C<, near P. This contradicts (2).

By Theorem I the points of Q consist of the curve Z, and two continua
belonging to Z( and Z« respectively. From the diagram it is seen that Cis wholly
in Z,. Hence by Theorem I the points of Z, consist of the curve Cand two con-
tinua belonging to (7< and C^ respectively. Hence there are points of each class
left blank in the diagram. The first clause of (a), (b), and (c) can now be read
off from the diagram. The remainder of the proof is left to the reader.

Theorem III. If tico simple closed curves C and L each satisfying Condition
A are wholly exterior to each other:

(a) The interior of C is wholly exterior to L, and is wholly bounded by G, and
similarly interchanging letters;

{b) There exist points exterior to each, and these form a continuum bounded
wholly by C and L .

The proof is similar to that of Theorem II.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 365

Theorem IV. If n simple closed curves satisfying Condition A have no point
in common :

(a) A necessary and sufficient condition, that they form the total boundary of an
infinite region is that they lie wholly exterior to each other; the region so bounded is
exterior to each;

(6) A necessary and sufficient condition that they form the total boundary of a
tinite region is that n — 1 of the curves are wholly interior to the remaining one and
wholly extei-ior to each other; the region so hounded is exterior to each of the n — 1
curves and interior to the remaining one.

This can be proved by mathematical induction.

Part II.
IN THREE DIMENSIONAL SPACE.

I. — Fundamental Conceptions.

15. Some of the fundamental conceptions made use of in the following
chapters have been discussed in the introduction for space of two dimensions.
Those definitions and principles will now be extended to space of three dimen-
sions by the addition of a third variable, without further comment, whenever no
difficulty presents itself in so doing. We shall prove the theorem that a simple
closed surface divides space into two continua, first for a very restricted class of
surfaces, and later indicate how the proof can be extended to more general cases.
The proof follows a method similar to that used in two dimensions. A prelimi-
nary discussion of certain fundamental conceptions is necessary.

16. Surfaces. A smooth simple closed surface is an assemblage of points
P{x, y, z) defined as follows:

{a) If P^{xq, 2^0, Zq) is a point of the assemblage, it is possible to choose
three equations

X =^[u,v), y ='4,{u,v), z=x{u,v), (A)

where

Xo = ^(mo, Vo), yo = ■4'(Wo. Vo), Zq = ;t(?«o. ''o).

366 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

where 4», 4'> X ^^^ single valued near (mo- Vo)> where all points given by these
equations near (t^, Vo) are points of the assemblage, and where no point {x, y, z)
is given by two distinct points (u, v) near {uq, Vq).

(b) The assemblage is simple, that is, if Po {xq, y^, Zq) is a point of the
assemblage, all points in the three dimensional neighborhood of Pq can be given
by one set of parametric equations (A) as just defined.

(c) The assemblage is complete, that is, if P(x, y, z) is a limiting point of
the assemblage, then it shall belong to the assemblage.

(d) The assemblage is connected, that is, if Po{xq, y^, Zp) arid Pi{xi, y^, Zj)
are any two points of the assemblage, then it is possible to draw a simple curve

x = 7.{t), y = (i{t), z = v{t), {fo<t<ti),

having Pq and Pj as end points and such that all points of the curve are
points of the assemblage.

(e) The assemblage lies in a Jlnite region, that is, it is possible to choose a
constant G so that if P(a;, y, z) is a point of the assemblage, then

\x \ + \y \ + \z\<^G.

(/) The assemblage is smooth at every point, or simply smooth, that is, if
-fo (^oi Z/o' 2o) is a point of the assemblage, it is possible to choose the equations
(.4) so that the first partial derivatives

1>u (—9^)'^"' '^■" '^'" Z"' JC"

are single valued and continuous near {uq, Vq,), and so that the Jacobians

J.=

'i'uX''

1 Jy

4'« ^„ I 7- _ i «?'u '4'u

do not all vanish at Pq.

By virtue of (a), (c) and (e) the assemblage is said to be closed.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 367

17. Dissection of Surfaces.

Theorem I. A smooth simple closed surface can be divided into a finite number
of parts, each of which has the following properties :

(a) It can be represented by an equation of one of the following forms :

«=/(y.z). or y=f{z,x), or z=f(x,y),

where f and its first partial derivatives are single-valusd and continuous;

(b) It is bounded by one simple regular curve;

(c) It can be included in a sphere of arbitrarily preassigned radius.

To prove this we divide space into cubes of edge h. If 5 is chosen
BuflBciently small, either the part of the surface in any cube consists of a finite
number of prices each of which satisfies the requirements of the theorem, or, in
case the surface is tangent to a face, the part of the surface in the two cubes
having this face in common satisfies the requirements of the theorem. The
details of the arithmetization are omitted.

Most of the discussions which follow apply to any simple surface which can
be dissected as specified in Theorem I, We shall describe such a surface by say-
ing that it satisfies the following condition :

Condition B: A surface is said to satisfy Condition B if it consists of a finite
number of parts each of which answers to the description in Theorem I, and if,
moreover, the surface satisfies conditions (6, c, d, e) of Art. 15. If it also satisfies
(a) it is said to be closed.

18. Parametric Representation of Surfaces. The following theorems relate to
the possibility of representing a given surface by two different systems of para-
metric equations, and to the relations of the two parametric planes. The trans-
formations involved are analogous to the transformation t =f{t') by which the
parameter < of a simple curve is replaced by a different parameter t', where /(<')
is monotonic and never constant (Art. 10). The theorems are given without
proof.

Theorem II. If two planes R and R are mapped in a one to one and continu-
ous manner on each other, and a simple closed curve C in R is mapped on a simple
closed curve C in R', each of which is oriented, then
49

368 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

(a) Any interior {exterior) point of G is mapped on an interior {exterior) point
ofO;

{h) All sitch transformations form a group G; of these there are transforma-
tions, called EVEN transformations, for which the curves of every such pair have the
same orientation, and these transformations form, a subgroup (r"*" of G ; there are
transformations called odd transformations for which the curves of every such pair
have opposite orientations, in particular the transformation x! ^=- x, y= — y; the
totality of even and odd transformations exhaust G; moreover, the group of
odd transformations can be generated by the particular odd transformation
a/ = X, y' '=■ — y in succession with each of the even transformations ;

(c) If the Jacobian of such a transformation is defined and continuous at every
point of the regions involved, and does not vanish in them, then the necessary and
sufficient condition that the transformation is even is that the Jacobian is positive at
every point.

Theorem III. If a finite simple surface region R including its boundary,
which is a simple closed curve C, is mapped in a one to one and continuous manner
on a portion R' of a plane, then R' forms the interior and boundary of the closed
curve G' on which G is mapped.

19. Unilateral and Bilateral Surfaces. Orientation. Given any simple
surface. Let i2, be any complete open region of the surface, and let (7,- be a
simple closed curve forming part or all of its boundary. If, having oriented one
such boundary, it is then possible in one and only one way to orient every such
boundary so that if (7^ and Gj are any two of these having a common segment a,

(a) Gi and Gj shall be oppositely oriented along <t when i2, and Rj are
exterior to each other, and

[b) Gi and (7, shall be similarly oriented when either Ri or Rj is interior
to the other,

then the surface is said to be bilateral. If this is not possible the surface is said to be
unilateral. We will agree that if one such curve on a bilateral surface is oriented,
then the orientation of every such curve on the surface shall be consistent with
the above specifications.

Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 369

Let us now extend to surfaces the idea of orientation. "We define an
oriented simple surface to be an object determined by the three following
phenomena :

(a) A simple bilateral surface;
(6) A definite complete open region of the surface;

(c) An oriented simple curve forming part or all of the boundary of that
region.

It follows that a simple bilateral surface can be oriented in two and only
two distinct ways. If a simple bilateral surface is oriented and is represented by
three equations of the form

x = ^{u,v), y = ^{u,v), z=;i^(m, v),

and if a complete open region 5, of the surface, bounded by C^, corresponds to
the region R! bounded by (7/ of the mu -plane, then if Oi and C/ are not simi-
larly oriented they will be after the substitution m = — u', v = v'. We shall
assume that the parameter has been so chosen. Then by the theorem of Art. 12
and Theorem III of Art. 18 the same is true for every such curve. Thus the
different parts of the surface may be given by different analytic representations
and the validity and definiteness of our definitions be not affected.

II. SouD Angles and Order of a Point.

20. Solid Angles. Let us start from the ordinary conception of a solid
angle. Define a system of spherical coordinates by the relations

a; = p sin ^ cos 6,
^ = p sin <^ sin 6,
s = p cos (^ .

We shall speak of the line 4» = 0 and ^ = 7t as the polar axis. Let a piece R of
a surface be defined by the equation

370 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

where /(0, <p) is a single valued function of 6 and <^ throughout a region R' of the
surface of the unit sphere. Then, according to the ordinary conception, the solid
angle subtended by R at the origin is the area of R', which is given by the

^™ y r, sin 4><£0d<^.

We wish to extend this definition in such a way that the solid angle shall be
susceptible of an algebraic sign. To illustrate our purpose, consider the convex
surface of a circular cylinder so situated that the origin is exterior to it, and so
that it is not pierced by the polar axis, and so that some of the radii vectoies
cut it in two points. It can be divided into two parts so that each can be repre-
sented by one equation of the form p =/(0, 4)). We wish to define the solid
angle subtended by the cylindrical surface so that the contribution of one of
these parts shall be positive, and the other negative, and so that the total solid
angle shall be the algebraic sum of the solid angles subtended by these two parts.
If this surface is represented parametrically by the equations

p = P{u,v), d = @{u, v), ^ = ^{v, v),
under suitable restrictions as to continuity, it will be observed that the Jacobian

D {u, v)

will be positive throughout one of these parts of the cylinder, and negative
throughout the other. If now in the double integral above we replace Q, <p by
the new variables u,v we obtain the integral

//

sin d> r.) ' -< du dv,*
^ D {u, v)

extended over the total cylindrical surface. In this form the Jacobian takes
care of the sign, and thus yields in one integral the result desired. Guided by
this illustration we shall proceed to formulate a general definition of a solid angle.
Giveu any oriented bilateral surface R referred to a system of rectangular
coordinates, and represented by one or more sets of equations of the form

X = X{u, v), y ■= Y{u, v) 2 = Z(u, v),
where the parameters are so chosen as to satisfy the requirements of Art. 19.

*S«e, for example, Qonrsat, Court cP Analyu, Vol. I, |138.

AiOCS: An Arithmetic Treatment of Some Problems in Analysis Situs. 371

Let 0{xq, i/q, Zq) be any fixed point not on the surface. Change to a new
system of rectangular axes with 0 as origin by a transformation having a
positive determinant. Then change to a system of spherical coordinates having
0 as origin and defined by the equations given above. R can now be
represented by one or more sets of equations of the form

We shall at first require that the surface shall not be pierced by the polar axis.
Later we shall remove this restriction. We define the solid angle subtended
by ^ at 0 to be the integral

where R' is the totality of the regions in the ww-planes corresponding to R, and
where dudv is essentially positive. It follows that the solid angle is invariant
of any change of parameter made by a transformation having a positive Jacobian
at every point. In particular it is immaterial whether the surface is represented
by one or many sets of parametric equations.

If the surface has a point Pi(a;i, y^, z^, p^) on the polar axis, the integrand
is not defined at that point as the Jacobian may become infinite. Let a point P
approach Pj . Then it can be shown that

which is finite and defined. The integrand at such a point shall now be defined
to be that limit. Then the integral is a fully defined proper integral. We now
define the solid angle in all cases to be the integral (A) .

Any rotation of the system of spherical axes about 0 is accomplished by
introducing ff and ^' in place of 6 and ^ by a transformation having a positive
Jacobian. By simply making this substitution it can be shown that the integrand,
and hence the solid angle, is invariant of this transformation. It follows that
the solid angle is independent of the particular choice of polar axis, and is
invariant of any change in the original system of rectangular axes made by a
transformation with positive Jacobian.

21. Solid Angle in Terms of a Line Integral. We shall now express the solid
angle by means of the line integral / cos ^ dd taken around the boundary of the

372 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

surface. The possibility of doing this is suggested by analogy with Green's
theorem in the plane. In fact, in the simpler cases this can be accomplished by
a direct application of Green's theorem. From Art, 19 it follows that, if the
surface is divided into parts, this integral extended along the entire boundary of
the surface is equal to the sum of the integrals of the same function extended
along the boundaries of the parts, taken in the positive sense of the curve in each
case. This follows since along the common boundary of any two of the parts the
integral is extended once in one sense and once in the opposite, and these two
integrals cancel each other. We need the following theorem:

Theobem. Given any smooth surface, and any point 0 not on the surface ; then
it is possible to draw through 0 a straight line not tangent to the surface, making an
arbitrarily small angle with a given line.

Proof. The condition that a line through 0 is tangent to the surface, repre-
sented by spherical coordinates, is that D{d, ^)ID{u, ?;) = 0 at the point of contact,
excluding from consideration points of the surface in the neighborhood of the polar
axis. This will be no limitation on the generality of the proof, as the polar axis
maybe changed if desired. Now divide the surface into a finite number of parts
each of which can be represented by an equation of at least one of the following

forms: (j) ^ ^;^(0^ ^)^ ^^ (2) 0 = ^(<?,, p), or (3) <?. = r(p, 0),

where ^, fi, v, and their first derivatives are single valued and continuous. That
this is possible follows directly from Art. 17. The condition that a part
can be represented in the first form is that Z)(0, ^)/D(u,v) :^ 0 in the part. Hence
no line through 0 can be tangent at a point of a part of the first class. Discard all
parts which can be represented in the first form. Then if the parts are taken
suflBciently small, D{d, ^)jD{u, f ) < e throughout the remainder of the surface,
which we will denote by S~, where e is an arbitrarily preassigned positive number.
Then if a is the solid angle subtended by /S" at 0,

a = / / sin d) r,/ X du dv
J Js- ^ I){u, v)

< f f \Bmq>^^\dudv

= J Ja-\ ^ D{u, v)

- J Js-\J){u, v) I

^vX-'^"^^ <^f£dudv = eK,

4^ES: An Arithmetic Treatment of Some Problems in Analysis Situs. 373

where K is independent of the method of division. Hence by taking the parts
suflBciently small the solid angle subtended by S~ can be made less than that
subtended by a preassigned region of the surface of a sphere with 0 as center.
Then it is possible to draw a straight line through 0 piercing this region and not
touching S~. This proves the theorem.

Such a line meets the surface in at most a finite number of points. Let the
polar axis be so chosen. Now divide the surface into a finite number of arbitra-
rily small parts by Art. 17. At most a definite number independent of the size
of the parts, contain points on the polar axis. If the parts are taken sufficiently
small the contribution of these parts to the solid angle can be made arbitrarily
small. Each of the remaining parts can be represented in at least one of the
three following forms:

(1) p =/(0,^), (2) ^ =/(p, e), (3) e = /(4.. p).

"We shall show first that

//«^° ^ ?^j ^" ^" = -f''' ^ ^^ (^)

in each part which can be represented in the first form. The line integral is to
be extended around the boundary of each part in the positive sense. The con-
dition that a part can be so represented is that D(d, ^)/D{tc, v) =1= 0, and hence
has a constant sign in the part. Hence

ry sin 4, g (^' ») du dv =ff sin ^dQd<i», ( (7)

where dO d^ has a constant sign, the same as D{Q, ^)/D{u, v). We may think
of the transformation 6 = d{n, v), ^ = ^{u, v), as a transformation from the uv
plane to the 6ip plane. Suppose D{d, ^)/D{u, v) < 0, and hence dO d^ < 0.
Apply Green's theorem. We obtain

I I s'm pdd d^=^ I cos <^

dd

extended along the boundary of the region in the 6^ plane in the positive
sense, or

^de

■ j cos ^ ^

374 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

extended in the negative sense. Since D{d, ^)/D{u, ■»)< 0, by Art. 18, Th. II c,
this corresponds to the positive sense of the curve in the uv plane, and by Art. 19,
this corresponds to the positive sense of the boundary of the region on the
surface. If D(d, ^)jD{u, v) >■ 0 similar reasoning leads to the same result.
Hence the solid angle is given by the integral

— / cos 4) dB

taken in the positive sense, for every region of the first class.

Consider those parts which can be represented in the form ^ =/(p, B).
Then

By reasoning similar to that just used the same result is obtained.

We shall adopt a diflFerent method for the third case. Let R be any one of
the regions already discussed, and Cits boundary. Let it be referred to any two
systems of spherical coordinates (p, 6, ^) and (p, 6' , ^') having the same origin
and such that no point of i? or C is a point of either polar axis. If R is suflS-
ciently small it is possible to construct a conical surface having 0 as directrix
and a point V as vertex so chosen that no element intersects either polar axis.
Consider that part of this surface which lies between V and C. Let this be
divided into arbitrarily small regions. Then each of these regions can be repre-
sented by an equation of at least one of the following forms :

p=/{d,^), or ^ = f{p,6),

and also by an equation of at least one of the following forms :
p =/,{$', ^'), or ^'=/,{p,e').

Hence by each system of coordinates the solid angle is equal to the line integral
along G. But the solid angle is invariant of the system of axes. Hence

/ cos ^ dd =J cos ^' d&.
Hence in computing the value o{ jcoB^dO around the boundaries of all the

AuES: An Arithmetic Treatment of Some Problems in Analysis Situs. 375

regions of the given surface we may choose the axes arbitrarily for each region.
But the axes can always be chosen so a given small region can be represented
in one of the forms

p=/(0,4.) or ^=Ap,d).

Hence for any sufficiently small region not containing a point of the polar axis

the solid angle equals ■^— / cos ^ dd extended along its boundary in the positive
sense.

22. Order of a Point. Now suppose that the surface is closed. We shall
show that the solid angle is a multiple, positive, negative or zero, of4 7t. Let
the curves qt be the boundaries of those parts, finite in number, and each
arbitrarily small, which have a point in common with the polar axis. These
parts may be so chosen that the point in common with the polar axis is an
interior point of the part. Let C~ be the total boundaries of the remaining
parts. Then since the integral is extended along every boundary once in one
sense and once in the opposite,

y^_cos 4> tZ0 + 2 / . cos ^ (Z0 = 0.

But by taking the parts sufficiently small

— J _C08 ^ dd

and hence its equal

^ / cos <^ do

can be made arbitrarily near to the solid angle. But at the same time the latter
sum can be made arbitrarily near to

where i;, = ± 1. Hence the solid angle equals
50

376 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.
On the other hand

X-'^« + 2X'^^=o-

since the integral is extended along every segment once in one sense and once
in the opposite. But

" d6 = 0.

t

This may be shown by taking the parts suflBciently small, and showing that the
total variation of 0 in any one part is less than 2n, and hence equal to zero.
Hence

But

where «( is an integer, positive, negative or zero, and the solid angle ^ >7> J,( ^^
differs from V / dd by twice the contribution of those integrals for which rii is
negative. Hence the solid angle equals Ann, where n is an integer, positive, nega-
tive or zero. Then define the order of the point 0 with respect to the surface to
be the number n. The order of a point on the surface is not defined.

The solid angle is invariant of the particular choice of the polar axes, and of
any change of the parameters u, v in any part of the surface, provided the trans-
formation has a positive Jacobian. If the parameters are changed at all points
of the surface by transformations having negative Jacobians, the sign of the
solid angle is changed but its absolute value is invariant. Hence the same
statements are true of the order of a point. For a like reason the order of a
point is invariant of any change in the rectangular axes to which the surface is
referred, effected by a transformation with positive Jacobian.

Theorem I. 1/ the order of a point 0 not on the surface is n, then all points
in the neighborhood of 0 are of order n .

Proof. The integral defining the solid angle is the integral over a finite
region of a uniformly continuous function of all the variables involved, including
the coordinates of 0. Hence the solid angle, and therefore the order of 0 is
continuous when 0 is not on the surface. But the order can vary only by a
multiple of unity. Hence it is constant.

Altes: An Arithmetic Treatment of Some Problems in Analysis Situs. 377
Corollary. Tlie points of the order of a given point form one or more continua.

Theorem II. If two points are of different orders with respect to a given
surface, then any simple curve joining them has a point in common with the surface.

The proof is the same as in two dimensions (Art. 8).

III. The Division of Space by a Closed Surface.

23. The theorem that a closed bilateral surface which satisfies Condition B
(Art. 17) divides space into two continua is proved by the aid of two lemmas
entirely analogous to those used in two dimensions.

First Lemma. If P^is a point of a closed bilateral surface which satisfies
Condition B, then near P^ there are two points whose orders with respect to the given
surface differ by unity.

Proof Let the surface be arbitrarily oriented. If the surface is not smooth
at Pq, there is a point of the surface near Pq at which it is smooth, and which
may be used instead of Po- Hence we may assume that the surface is smooth at
Pg. Transform to a new set of rectangular coordinates with origin at Pq, and so
chosen that the surface near Pq can be represented by one equation z=f{x, y),
where / is single valued and continuous. The axes can at the same time be so
chosen that the z-axis has only a finite number of points in common with the
surface (Art. 21, Theorem). It is possible to choose two points 0"'"(o, o, 6) and
0~{o, 0,-6) where 5 is so chosen that no point of the surface except Pq lies on
the segment 0~ 0^ of the z-axis. Now refer to two systems of spherical coordi-
nates having the origin at 0"*'and 0 ~ respectively, and the positive z-axis as
positive polar axis. Cut out from the surface small regions, each containing one
of the points common to the surface and the polar axis. In each case the sum

of the integrals / cos ^ dQ, extended around these regions in the positive sense, is

arbitrarily near to the solid angle subtended by the surface at the origin
(Art. 21). The contribution of the part containing Pq to the angle at 0+ differs
from the contribution of the same part to the angle at 0~ by a number arbitrarily
near to 4 7t. That of the remaining parts in the two cases differ by an arbitrarily

378 AuES: Aji Arithmetic Treatment of Some Problems in Analysis Sitm.

small number. But the orders of 0"*" and 0~ can differ only by integers, hence
they differ by unity.

Second Lemma. Given any three dimensional continuum R, and a surface 8 :
2 =f{^, y), or y =/{z, x), or x —f{y, z),

where /is single valued and continuous:

(a) If R contains all j^oints of S except possibly its boundary points which
may lie in the boundary of R, then the totality R~ of points of R not on S form at
most two coniinua ;

{b) If also Shas a simple regular boundary one point of which is inR, then R~
forms one continuum.

Proof, (a) Suppose S can be represented by the equation z = f{x, y).
The other cases are similar. (See Fig. 6, in which the surface S is represented

but not the boundary of /jf). Draw a straight line CZ) parallel to the z-axis,
lying wholly in R, and bisected at a point of the surface S, and such that Cis
above the surface S. Let P be any point of R~ which cannot be joined to D by
a simple curve wholly in R~. If there is no such point the theorem is granted.
Otherwise join P to Z) by a simple curve PD wholly in R, This curve will have
a point in common with the surface S. Let PE be an arc of PD having one end
E on the surface S, but containing no other point of S. Choose a region S' of S
whose interior and boundary lies wholly \n R, and containing E and the point

Ahes: A71 Arithmetic Treatment of Some Problems in Analysis Situs, 379

common to GD and the surface *S'. Define two assemblages iV+ and N~ analog-
ous to that of Art. 6, Example 3, as follows :

z =/(a;. y) + r, {x, y) in S',

0 < r < A for iV+,

— 7i < r < 0 for N'.

These can be proved to be continua in a manner analogous to that just referred
to. Choose a point F on the arc PE, and so near to E that it lies either in N'^
or N~. Suppose it lay in N^. Choose a point G on GD in N~. Then F and G
can be joined by a simple curve wholly in N~. Hence the simple curve PFGD
lies wholly in R~, which is contrary to hypothesis. Hence F must lie in iV^+,
and by similar reasoning P can be joined to C by a simple curve wholly in R~.
Hence the points oi R~ form at most two continua.

(b) Suppose the surface is represented as in the first case, but let it be
bounded by a simple regular curve having a point Pq interior to R. If this
point is a vertex there is a point of the boundary of S near it which is not a
vertex, and which lies in R. Hence we may assume that it is not a vertex.
Let the surface now be extended slightly past Pq. By reasoning similar to that
of the first case it can be proved that the points oi R not on this enlarged surface
form at most two continua. If they form one continuum the theorem is granted.
If they form two continua, they can be annexed to each other by the adjunction
of the points added to the given surface, thus forming one continuum.

Main Theorem. The points of space not on a given simple closed bilateral
surface which satisfies Gondition B form two contintia, of each of which the entire
surface is the total boundary.

Proof. In the neighborhood of any point of the surface there are two points
of different orders with respect to the surface (First Lemma). Hence the points
of space not on the surface form at least two continua (Art. 22, Th. II). Divide
the surface into parts each of which can be represented in at least one of the
following forms :

a:=/(y.z). or y=f{z,x), or z—f{x,y).

Discard these, one at a time in such an order that each part after the first when
discarded shall have a portion at least of its boundary in common with a part

380 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs.

already discarded. Then replace them in reverse order. By the second part of
the Second Lemma each of these except the last replaced does not divide the
region consisting of all space less the points already cut out. By the first part of
the same lemma the last part replaced divides the resulting region into at most
ttoo continua. Hence the points of space not on the surface form just two
continua.

Any point of the surface is a boundary point of each continuum (First Lemma,
and Art. 6). Any point not on the surface belongs to one of the continua and
hence is not a boundary point. This proves the theorem.

A discussion of interior, exterior and normals might be made analogous to
that in two dimensions. More general surfaces, having edges or vertices may be
defined in a manner analogous to the definition of a smooth surface given in
Art. 16. If such a surface satisfies Condition B (Art. 17), then the foregoing
discussion applies to it.

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