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 A. 7- I D 3 .1/. if/i/o 
 
 7 
 
 7673-1 
 
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. D. AMES 
 
 BALTIMORE 
 ZU fiorS <&a!timoxt (prtee 
 
 THE FRIEDENWALD COMPANY 
 1905 
 
The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924032184255 
 
Harvard College Library, 
 
 Cambridge, Mass. 
 
 To the Librarian 
 
 ENT OF SOME 
 
 SIS SITUS 
 
 I have the honor to send herewith in accordance 
 with our agreement in regard to exchange of publica- 
 tions 
 
 Dissertation. By. L, D. Ames 
 
 F HARVARD UNIVERSITY IN 
 
 THESIS FOR THE 
 
 DPHY 
 
 Respectfully, 
 
 Wm. H. Tillinghast. 
 
 Assistant Librarian. 
 
 BALTIMORE 
 C?« iSori Q^afttmore (preerer 
 
 THE FRIEDENWALD COMPANY 
 I90S 
 
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. D. AMES 
 
 BALTIMORE 
 %%t £ori <§attimott (prefer 
 
 THE FRIEDENWALD COMPANY 
 I905 
 
 J> 
 
7?7 
 
 < w u a &<u 
 
 j\, X | 3 / sr 
 
An Arithmetic Treatment of Some Problems 
 in Analysis Situs.* 
 
 By L. D. Ames. 
 
 CONTENTS. 
 
 PAGE 
 
 Introduction gig 
 
 Paet I. In Two Dimensional Space 345 
 
 Chapter I. Fundamental Conceptions 345 
 
 Chapter II. The Theorem as to the Division of the Plane 353 
 
 Chapter III. Regions, Orientation of Curves, Normals 359 
 
 Pabt 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 | 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 Cours & Analyse, 2d ed., Vol. I., §§96-103, 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 S lt S 2 and B such that a point of /Si cannot be joined to a 
 point of St 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 difference between these 
 assemblages /Si and S 2 , 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. Bliesf has 
 proved the theorem for a somewhat more general class of curves than those for 
 which Schoenflies proved it. 0. 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 Vallee Poussin, or of Veblen, but more 
 
 * Court oV Analyse infinitesimal, Vol. I., (1903), §§300-302. 
 
 t The exterior and interior of u plane curve, Bulletin of the American Mathematical Society (2), Vol. 10, 
 (1904), p. 398. 
 
 X Theory of plane curves in non-metrical analysis situs, Transactions of the American Mathematical Society, 
 Vol. 6, No. 1, Jan. 1905, p. S3. 
 
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, b- . . ■ ■ ). 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 (x, 0) is called the sc-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. A 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. 
 
 x = x 1 cos a — y' sin a + x , 
 y = x' sin a + y' cos a + y Q . 
 
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 /Si and S 2 are two complete f assemblages of points having no 
 point in common, then the distance of any point of S ± from any point of S s has a 
 positive minimum. 
 
 4. Curves. A simple curve X is an assemblage of points (a;, y) which can be 
 paired in a one to one manner with the points of the one dimensional interval 
 (t <t< t x ) in case the curve is not closed, and with the points of the circle 
 
 £ = cosJW, »7 = sinM, 
 
 in case the curve is closed; moreover, when t approaches a limiting value (7) the 
 point (x, 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 ^ are called end points. 
 
 It follows from this definition that a simple curve can be represented analy- 
 tically by equations of the form 
 
 x = $(t), y = 4{t), (t <t< *,), 
 
 where $(f) and ^(t) are single valued continuous functions, and 
 
 <K0 = $(*') and ^(t) = ^(t') 
 
 *Cf., for example, Jordan, Cours & Analyse, 2d ed., Vol. I, §30, last paragraph. 
 
 t Professor Pierpont suggests complete as the English equivalent of ahgeschlossen. 
 
 t Cf. A. Hurwitz, Verhandlungen des ersten Internationale*. Mathematiker-Kongresses, p. 102. 
 
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 1 , and are 
 not simultaneously satisfied in the case of the closed curve when t =f= if except 
 
 that 4>(<o) = $(<i) and ^( t o) = 4'( t i)- 
 
 If the curve is closed, let o = t x — t Q . It is then convenient to extend the defi- 
 nition of the functions $(t) and ^(t) to all values of t by means of the relations 
 
 <p{t + no) = Q(t) , ^(t + no) = <Wt) , 
 
 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 parameter can be so 
 chosen that the first derivatives q>'(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 
 
 » = *(«). y = *(<). (h<t<h), 
 
 where q>(t) and ^(tf) are single valued continuous functions whose first deriva- 
 tives q>'(t) and ^'(t) 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 t from above, and also when t approaches t' from below, and in each case 
 the limits approached by $'(t) and "V(t) are not both zero ; the forward limits 
 are not both equal respectively to the backward limits. It follows that one 
 sided derivatives exist at the 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 of t. 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 Kiemann surface is treated. We do not need such curves, 
 
348 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 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 {x,y), then the corresponding values of t have a limit which is a point 
 of the interval (t <t< tj), and (x, y) corresponds to i.f 
 
 The following theorem is stated without proof: 
 
 Theorem 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 =/(»). (a<x<b), 
 
 x=f(y), (<*<</££), 
 
 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 = f(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 a discussion of change of parameter, orientation of curves, etc., see Sec. III. 
 
 f Cf. Jordan, Cours d' Analyse, 2d ed., Vol. 1, §§ 64, 65. 
 
 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 = Oi t + b x , 
 y = a z t + b 2 , 
 
 «i 2 + «/>0. 
 
 (b) One of the two possible permutations, P P x or P x P , 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. P Q Pi- Taking the 
 four points P (x , y ), P^, y x ), P£(scJ, yi,), and P[{x[,y{), the two vectors 
 P Pi and P' P[ are said to be equal if 
 
 x 1 — x =x{ — x[, and y x — y =z y{ 
 
 yi>- 
 
 The length of the vector P P x is the positive number 
 
 V (x 1 — x f + (yi—yof ■ 
 
 The angle 6 from the vector P Q P x to the vector P(, P[ is defined to be any simul- 
 taneous solution of the equations 
 
 x 
 
 «o Vi — 2/o 
 <*i — aso yi — Vo 
 
 sm6 = K 
 where K is the positive number 
 
 cos = K 
 
 yi—Va —{«! — «o) 
 
 x[ — x' a 
 
 y'i—y'» 
 
 iA) 
 
 K= W{xi-xo? + {yi-y*T V(^-z67 + iy'i-ySn 
 
 If the vectors are defined by means of their equations 
 
 t x = a x t + bx , 
 P ° Pl '-\y=a 2 t + b 2 , 
 x = a[ t + b[, 
 a[t + b' z , 
 
 ( x = < 
 
 «! + «l>o, 
 (to^t^tj, 
 
 (4<t'<ti), 
 
 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 
 
 <h 
 
 a z 
 al 
 
 cos 6 = 
 
 «i 
 
 a x 
 al 
 
 where K is the positive number 
 
 « = [VaJ + o| Vaf + O" 
 
350 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 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 B A 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 27i; 
 
 (b) That if the angle from P,P 1 to P' P{isd, and that from P(>P{ to P{P{' 
 is 0', then the angle from P P t to P' 'P" is 6 + 0' + Inn, where n is a positive 
 or negative integer or zero ; 
 
 (c) That is invariant of any rigid transformation. 
 
 6. Continua. A two-dimensional continuum is an assemblage of points 
 P(x, y) such that: 
 
 (a) If P (x , y ) is a point of the assemblage, all points in the two dimen- 
 sional neighborhood : 
 
 | x — ar | <A, | y — y Q | <h 
 
 of P 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 P may be defined generally to be a continuum 
 containing P and such that the distance of any of its points from P 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 P not an interior or boundary point is 
 an exterior point, and all points near P 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 
 so that if P(x, y) is any point of the region, then 
 
 \*\ + \y\<Q. 
 
 * Cf. for instance, Godefroy, TMorie SUmentaire des Siries, Chap. 5. 
 
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 
 
 x% + y % — i* < o • 
 
 Example 2. The interior of a triangle, or more generally the assemblage S, 
 defined by relations of the form 
 
 u t {x, y) = A i x + B i y + C i >0, (*=1, 2, 3), 
 
 where these equations are satisfied by at least one point, can be proved to be a 
 continuum as follows: 
 
 (a) Let P (x , y ) be one point of S. Since u t is continuous and Ui(x , y ) 
 > , hence if P (x, y) is any point near P , ^(a:, y) > 0, and hence P belongs 
 toS. 
 
 (b) Let P (x , y ) and P r (x lt y x ) be any two points of S. Hence 
 
 u^Xq, yo) = A i *o + JBi!/o+ Ci>0, 
 
 u i {x l ,y 1 ) = A i x 1 + B iVl + Q^O, (t-1,2,3), 
 
 and therefore 
 
 where /Ij and \ are any numbers not negative and not both zero. This is a 
 sufficient condition that the point 
 
 / ^■Ep-f ^ 2 x lt 3,! y + \ y x \ 
 
 \ Aj -|- A 2 Ai + A 2 / 
 
 *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 P Pj can be expressed in this form. 
 Hence any two points of S can be joined by a straight line wholly in S. Hence 
 S is a continuum. 
 
 Example 3. Let an assemblage S (Fig. 1) of points P(x, y) be defined by 
 the relations 
 
 x a <x<x lt y=f(x)+r, 0<r<h, (1) 
 
 Fjg-1 
 
 where f(x) is single valued and continuous, and h is constant. We proceed to 
 prove that Sis a continuum. 
 
 (a) Let P(x, y, r) be any point of S. Choose 8 >• so that 
 
 2 5 <r and 2 5</j — r; 
 
 that is 23 < r <h— 2$. 
 
 Since f{x) is continuous it is possible to choose e > so that 
 
 /(»)-«< /(£) </(*) + « 
 
 when 
 
 and so that 
 
 a; — e< 
 
 X < X — £ 
 
 and 
 
 
 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 
 
 <« + £, 
 
 <*i 
 
 (2) 
 
 (3) 
 (4) 
 
 (5) 
 (6) 
 
 (7) 
 
Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 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, where < r< h. (9) 
 
 But (7) and (9) are the condition that P is in S, 
 
 (b) Let P 1 (x 1 , y x ; r t ) and P 2 (x 2 , y 2 ; r z ) be any two points in S, and let 
 r u < r 2 . They can be joined by a simple curve in S defined as follows: 
 
 x = xg, f(x) + r 1 <y< f(x) + r„ 
 
 and y = f(x) -{- r 3 , Xj<a;<a3 2 or x. i <x<x 1 . 
 
 Hence Sis a 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 = <p(t), y = ntft), 
 
 where a is the primitive period of the pair of functions ty(t) and 4{tf) . Let P (t) 
 be a variable point on the curve, and a fixed point not on the curve. Let 6(f) 
 be the angle which OP makes with the positive a -axis, or its equivalent for 
 this purpose, the vector (0, 0)(l, 0). Then 6 is an infinitely multiple valued 
 function of t for all values of t, such that any two values of 6 corresponding to 
 the same value of t differ by a multiple of 2ti. Let t Q be a particular value of t 
 and let { (£ o ) be a particular one of the values of 6(t ). Then it is possible to 
 choose from the different values of 8(t) one and only one set of values which 
 form a single valued continuous function of t taking on the chosen value 8i(t ) 
 when t = t * Call this single-valued function 6(f). The values of t and t -f- a 
 represent the same point, and the values of the multiple valued 6(t) at this point 
 differ by a multiple of 2n. Hence 
 
 B(t +a) = 6(t) + 2 n it , 
 
 * Cf. for example, Stoltz, Differential Sechnung, Vol. 3, p. 15-20. 
 
354 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 where n is a positive or negative integer or zero. Then n is defined to be the 
 order of the point 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 and not on the particular value of t 
 chosen may be seen as follows: Let * vary continuously. Then 6(t) and 
 0(*+ 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 6{t + a) 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 points near it are of order n . 
 
 Proof. Let O x be a point not on the curve, and any point near it. Let 
 0! and be the angles which 1 P and OP respectively make with the a-axis. 
 Then if Oj is sufficiently small, | X — | is less than an arbitrarily pre- 
 assigned number for all points on the curve. In particular 
 
 [0,(* + o) - 0xW ] - m + o) - 0(0] < 2 7t. 
 
 Hence the order of differs from that of O x 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 II. If 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 P (t a ) and Pi(^), (£ <C h) > °f a simple curve O 
 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 m and other points not of order m near P(t). 
 If P is not on the curve C this contradicts Theorem I. Hence, P is on the 
 curve C. 
 
 * It is sufficient 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 hy 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 hast two, and at most two continua. 
 
 First Lemma. Near any point P of a simple closed curve which satisfies Con- 
 dition A there are two 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 
 
 («) y=f{ x )> 
 
 or else by an equation of the form (b) x =f(y),. 
 
 where / is single valued and continuous. If P 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 P is not such an end point. Sup- 
 pose that the part on which P lies can be represented by the equation 
 
 y =/(*)■ 
 
 The other case is similar. Transform to new axes parallel to the original axes 
 and having P as origin (Fig. 2). All the conditions are invariant of this trans- 
 formation. 
 
 B;Co,-o 
 Fig. 2 
 
 The «/-axis has no point other than P in common with the curve near P . 
 Hence it is possible to choose B so small that if r < B, and B is the point 
 (o, r), and -Bi the point (o, — r), the segment B B x has no point on the curve 
 except P . Let P(t; x, y) be any point on the curve. Let 6 and X be the 
 
356 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 angles BP and B 1 P respectively make with the positive a-axis. Let p = BP 
 and p x = B X P, where p, p 1; and r are positive numbers. By the definition of an 
 angle, 
 
 sin 6 = - , cos v = — 
 
 P P 
 
 ■ a y -\- r a x 
 
 sin 0j = a — - — , cos p x = — . 
 
 Pi Pi 
 
 Let $ = - 0! . Then 
 
 — 2rx (x 2 + y*) — r* 
 sin d> = , cos <2> = ^ ^-^ . 
 
 PPi PPi 
 
 From these relations, if a is the primitive period and £ sufficiently small, and n, 
 n x , and n % integers, positive, negative, or zero, and x increases as t increases near 
 P , it follows that 
 
 Qfo) =(2«i+l)«, 
 
 $(t +e) >(2« 1 +l) 7 t, 
 
 <l>(t +a— s) <(2« 2 + l)7t, 
 
 <p(t + u) =(2n,+ l)?t. 
 
 When t < t < t + a, sin<£> can vanish only when x = 0, and in this 
 (x 2 -f- y 2 ) — *" 2 > and hence cos ty is positive. Hence 
 
 $(*):£ (2 n + l)»t when t <t<t + a, 
 
 and therefore n z — n 1 =- 1 . Hence 
 
 [6(t + a)) —6(Q2 - [0'(* o + o) - d'(t )] = <p(t + ) _ <j,(* ) = 2 «. 
 
 Hence the order of B exceeds that of B' by unity. 
 
 Second Lemma. Given the continuum R, and the curve AB : 
 V = /(»), or x=f(y), 
 
 where f is single valued and continuous : 
 
 case 
 
Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 357 
 
 (a) If R contains all points of the curve, except possibly its end points, which 
 may lie in the boundary of R, then the totality R~ of points of R not on AB form at 
 most two continua ; 
 
 (b) If also one or both end points lie in R, then R~ is one continuum. 
 
 Proof, (a) Suppose the curve can be represented by the equation 
 
 V =/(»), 
 
 (Fig. 3). 
 
 The other case is entirely similar. Draw a straight line GD parallel to the 
 y-axis, lying wholly in R, and bisected at a point of the curve AB, with C lying 
 above the curve. 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. 
 
 Fig. 3 
 
 Otherwise join P to D by a simple curve PD wholly in R. 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 or B. Along this arc construct 
 two continua like that of Art. 6, Example 3, one above, and one below AB', 
 lying wholly in R~, and denote them by N + and N~ respectively. Choose a 
 point F on PE 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 N + , and by similar 
 reasoning P can be joined to i by a simple curve wholly in R~. Hence 
 the points of R~ form at most two continua. 
 
358 Ames: An 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 ar-axis, to B 1 . By the 
 first case the points of B not on AB' form at most two continua. If they form 
 
 Fig. 4 
 
 one continuum the theorem is granted. If they form two 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 which 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 =/(cc), 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 op Curves, Normals, 
 and Related Topics. 
 
 10. Change of Parameter. The following theorem is stated without proof: 
 Theorem. Let a given simple curve be defined by two sets of equations 
 
 fX = $(t), _ , (x=Q(t), ,/,/, 
 
 where <£>, ^, q> and ^ are single valued, continuous functions . 
 
 (a) In the case of the open curve, if no two values of t \t'~\ yield the same point, 
 and if the values oft and t' which yield the same point of the curve are assigned to 
 each other, then t is a single valued, continuous function f(t') , 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 o/i[£'] yield the same point 
 unless 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 t' — f(t), where f(t) is single valued and continuous for all values oft, monotonic 
 and never constant. 
 
 The totality of transformations t' = 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 + of G. 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 P x (»i,2/i) be a distant point, and let P (x, y) be a variable 
 point on the curve. Let 6 be the angle P t P makes with the positive cc-axis. 
 Then by Art, 5, 
 
 cos 6 = {x — x,) j P X P, s\ne = (y — yi )l P X P. 
 
 48 
 
360 Ames: An Arithmetic Treatment of Some Problems in Analysis /Situs. 
 
 Then if /s/x\ + y\ 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 7i. Hence the order of P 1 is zero. 
 
 If the points of a continuum are all of order n the continuum is defined to be 
 of 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 neighborhood of a curve is a continuum containing all 
 points of the curve, and such that if P is a point of the continuum, and P x a 
 suitably chosen point of the curve, then PP X < 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 be 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 2 it if A is an interior point. 
 
 1 2. 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 
 P,P 2 of the end points P^t,) and P 2 (Q of a definite open arc, and P(t) is any point 
 of that arc, then we will agree to choose the parameter so that t x <t<t„ and con- 
 versely. If a change of parameter is necessary to effect this it can always be 
 accomplished by the transformation t=-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 t x t% in the positive sense if ^ is the lower limit and £ 2 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 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 it (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 have opposite 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 1 and R^ each form the interior of a simple 
 closed curve G x and G z respectively, satisfying Condition A. Let a segment o^ of C± 
 coincide with a segment <7 2 of G. 2 , then 
 
 (a) If -Rj and R % are exterior to each other, the orientation of a x is opposite to 
 that of <r 2 ; 
 
 (b) If R y is wholly interior to R z , the orientation of a 1 is the same as that of a 2 . 
 
 Proof (a). Choose a part or all of c x (or <r 2 ) which can be represented in the 
 
 form 
 
 (1) y=f(x), or else in the form (2) x=f(y), 
 
 where /is a single valued, continuous function. Let P be any point of this 
 segment not an end point. The orientation of <s x can not be the same as that 
 of <r 2 ,for suppose it were. Then by Art. 9, First Lemma, near P there are two 
 points B and B' such that the order of B with respect to either curve is greater 
 than that of B' by unity. Hence by Art. 11 B 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. Gkmrsat 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 
 
 * = *(*)» y = "W) 
 
 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 
 P (t ) at which the curve is smooth is the vector P (s ) P x (s t ) defined by the 
 equations 
 
 x = S V(to) + <p(to), ( So = o<s<s 1 ). 
 
 y = s4>i(t ) + ^(t ), K0 = = 1} 
 
 *ActaMathematica, t. IV, p. 197. Or see Harkness and Morley, Theory of Functions (1893), p. 164 
 
Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 363 
 
 A normal to the curve at a point P (t ) at which the curve is smooth is a 
 vector defined by the equations 
 
 X = — ES$'(t Q ) + $(t ), (S = <S <Sj), 
 
 y = esV(t ) + ^(t ), (f=±l). 
 
 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 + 7r/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 Pig. 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 R~ of the points 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 loop 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 : 
 
 
 Q 
 
 C 
 
 c e 
 
 L t 
 
 
 No point. 
 (3) 
 
 No point. 
 (2) 
 
 L 
 
 
 No point. 
 (1) 
 
 No point. 
 (1) 
 
 L. 
 
 
 
 
 where C { and C e denote the interior and exterior respectively of C, etc. 
 
 (1) By hypothesis, no point of L belongs to Cor to C e . 
 
 (2) No point is in C e and L t . Suppose P were in C e and L ( . Choose a 
 distant point A. This lies in G e and in L e . Hence A and P can be joined by a 
 curve wholly in C e . This curve must cut L. Hence a point of L is in C e . This 
 contradicts (l). 
 
 (3) Suppose a point P of Cwere ini 4 , then all points near P are in L ( . 
 But there are points of C e near P. This contradicts (2). 
 
 By Theorem I the points of Q consist of the curve L, and two continua 
 belonging to L t and L e respectively. From the diagram it is seen that Cis wholly 
 in L e . Hence by Theorem I the points of L e consist of the curve Cand two con- 
 tinua belonging to <7 4 and C e 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 two simple closed curves G 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 C, and 
 similarly interchanging letters; 
 
 (b) There exist points exterior to each, and these form a continuum bounded 
 wholly by G 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; 
 
 (b) A necessary and sufficient condition that they form the total boundary of a 
 finite region is that n — 1 of the curves are wholly interior to the remaining one and 
 wholly exterior to each other; the region so bounded is exterior to each of the n—\ 
 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 (x , y , z ) is a point of the assemblage, it is possible to choose 
 
 three equations 
 
 x =$(u,v), y =^(u,v), z = x (u,v), (A) 
 
 where 
 
 x = $ (« , v ) , y a = ^(u , v Q ) , z a = %(u Q , v Q ), 
 
366 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 where <£>, $, % are single valued near (u , v ), where all points given by these 
 equations near (w , v ) are points of the assemblage, and where no point (x, y, a) 
 is given by two distinct points (w, v) near (u , v Q ). 
 
 (b) The assemblage is simple, that is, if P„ (x , y , z ) is a point of the 
 assemblage, all points in the three dimensional neighborhood of P 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 P (x , y Q , z ) and P 1 (a: 1 , y lt z x ) 
 are any two points of the assemblage, then it is possible to draw a simple curve 
 
 x = h(t), y = fi(t), z = v(t), 
 
 (<0<*<*l), 
 
 having P and P x as end points and such that all points of the curve are 
 points of the assemblage. 
 
 (e) The assemblage lies in a finite region, that is, it is possible to choose a 
 constant G so that if P(x, y, z) is a point of the assemblage, then 
 
 l*l + |y| + l*l<0- 
 
 (/) The assemblage is smooth at every point, or simply smooth, that is, if 
 Pq (^o. 2A>> z o) is a point of the assemblage, it is possible to choose the equations 
 (A) so that the first partial derivatives 
 
 <K ( == ^)'^ ) >"^"' ^»' /£«< Sc- 
 
 are single valued and continuous near (u , v ,), and so that the Jacobians 
 
 J* = 
 
 
 ' Jy — 
 
 Xv <Pv 
 
 J.= 
 
 
 do not all vanish at P . 
 
 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 : 
 
 a =/(*/. z)> or y—f{z,x), or z=f(x,y), 
 
 where f and its first partial derivatives are single-valued and continuous; 
 
 (b) It is bownded 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 $. If 8 is chosen 
 sufficiently 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 (5, 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 t of a simple curve is replaced by a different parameter t', where f(t') 
 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 
 of O; 
 
 (b) All such 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 G + 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 1 = — y; the 
 totality of even and odd transformations exhaust G; moreover, the group of 
 odd transformations can be generated by the particular odd transformation 
 x' = 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 G , 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 C on which G is mapped. 
 
 19. Unilateral and Bilateral Surfaces. Orientation. Given any simple 
 surface. Let R t be any complete open region of the surface, and let G t 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 C t and G 3 are any two of these having a common segment <r , 
 
 (a) G t and Gj shall be oppositely oriented along a when R t and Rj are 
 exterior to each other, and 
 
 (b) Gi and Gj shall be similarly oriented when either R t 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; 
 
 (b) 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 = q>(u,v), y = ^(u,v), z=%{u,v), 
 
 and if a complete open region R ( of the surface, bounded by C it corresponds to 
 the region R t ' bounded by G[ of the wu-plane, then if G t and (7/ are not simi- 
 larly oriented they will be after the substitution u = — 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. Solid Angles and Order op 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 
 
 x = o sin 4> cos 6, 
 y = o sin <£ sin 6, 
 z = cos $ . 
 
 We shall speak of the line $ = and q> = it as the polar axis. Let a piece R of 
 a surface be defined by the equation 
 
 p=/(0,*), 
 
370 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 where f(6, q>) is a single valued function of 8 and $ throughout a region B' of the 
 surface of the unit sphere. Then, according to the ordinary conception, the solid 
 angle subtended by B at the origin is the area of B', which is given by the 
 
 mtegra f £ sin <pd6d<p. 
 
 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 vectores 
 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 r=/(0, <£>). 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(w, v ), 6 = ®(u,v), $ = 3>(w, v), 
 
 under suitable restrictions as to continuity, it will be observed that the Jacobian 
 
 D{0,4>) 
 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 6, <£> by 
 the new variables u, v we obtain the integral 
 
 // 
 
 sin d> r, ) ' tv d u d v * 
 T 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. 
 Given any oriented bilateral surface B 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) z = Z(u,v), 
 where the parameters are so chosen as to satisfy the requirements of Art. 19. 
 
 *See, for example, Goursat, Cours & Analyse, Vol. I, §138. 
 
Ames : An Arithmetic Treatment of Some Problems in Analysis Situs. 371 
 
 Let O(x , y , z ) be any fixed point not on the surface. Change to a new 
 system of rectangular axes with as origin by a transformation having a 
 positive determinant. Then change to a system of spherical coordinates having 
 O as origin and defined by the equations given above. R can now be 
 represented by one or more sets of equations of the form 
 
 o = P(u,v), <£ = <£(«, tf), 6 = @(u, v). 
 
 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 R at to be the integral 
 
 / / sin d> _. ) ' ™( du dv , (A) 
 
 J Jr' r D(u,v) v ' 
 
 where R' is the totality of the regions in the w-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 P^a^, y y , z-y, p^ on the polar axis, the integrand 
 is not defined at that point as the Jacobian may become infinite. Let a point P 
 approach P x . Then it can be shown that 
 
 hmSm * D(u,v) pi 2? (u,v) ' 
 
 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 is accomplished by 
 introducing 6' 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 q> 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: 
 
 Theorem. Given any smooth surface, and any point Onot on the surface; then 
 it is possible to draw through a straight line not tangent to the surface, making an 
 arbitrarily small angle with a given line. 
 
 Proof The condition that a line through O is tangent to the surface, repre- 
 sented by spherical coordinates, is that D(6, <p)/D(u, v) = 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: (1)p = Jl(M)) Qr ( 2 )0 = ^ f p), or (3)* = „( Pl 0), 
 
 where A, p, 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 D(6, Q)/D(u,v) =fc in the part. Hence 
 no line through 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 
 sufficiently small, D(d, <f)/D(u,v) < 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 O, 
 
 a 
 
 * ./Xl ■*♦!$$ *•* 
 
 - J *>s-\D(u, v) 
 
 = e ffs- dudv ^'fJ s dudv = E K, 
 
Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 373 
 
 where K is independent of the method of division. Hence by taking the parts 
 sufficiently 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 as center. 
 Then it is possible to draw a straight line through 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 = /(p, 6), (3) = A*, p)- 
 We shall show first that 
 
 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(0, p)/D(u, v) =f= 0, and hence 
 has a constant sign in the part. Hence 
 
 yy S in p D D A' ») du dv =ff sin p dd dp , ( G) 
 
 where dd dp has a constant sign, the same as D{6, p)jD{u, v). We may think 
 of the transformation d — 6(u, v), q> = <p(u, v), as a transformation from the uv 
 plane to the dp plane. Suppose D(B, <p)/D(u, «)< 0, and hence dd dp < 0. 
 Apply Green's theorem. We obtain 
 
 f /sin p dd dp = / cos p 
 
 dd 
 
 extended along the boundary of the region in the dp plane in the positive 
 Bense, or 
 
 — / cos p 
 
374 Ames: An Arithmetic Treatment of Some Problems in Analysis /Situs. 
 
 extended in the negative sense. Since D(6, $)/D(u, v)< 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(6, <p)/D(u, v) > similar reasoning leads to the same result. 
 Hence the solid angle is given by the integral 
 
 — / cos $ dd 
 
 taken in the positive sense, for every region of the first class. 
 
 Consider those parts which can be represented in the form <£» = /(p, #)• 
 Then 
 
 By reasoning similar to that just used the same result is obtained. 
 
 We shall adopt a different 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, <p) and (p, 6' , $>') having the same origin 
 and such that no point of R or G is a point of either polar axis. If R is suffi- 
 ciently small it is possible to construct a conical surface having G 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 G. 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 =/(0,4>), or <p = f(p,6), 
 
 and also by an equation of at least one of the following forms : 
 P=/i(0'> <*>')> or *' =/!(?, 00- 
 
 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 = / cos $' d&. 
 Hence in computing the value of / cos $ dd around the boundaries of all the 
 
Ames: 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,*) or $>=/(p,0). 
 
 Hence for any sufficiently small region not containing a point of the polar axis 
 
 the solid angle equals — / cos q> 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 q t 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 0~ 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, 
 
 / _cos # dd -f 2 _/}< cos $ dQ = 0. 
 
 But by taking the parts sufficiently small 
 
 — / _ cos q> dd 
 
 and hence its equal 
 
 ^ / cos <£ d6 
 
 can be made arbitrarily near to the solid angle. But at the same time the latter 
 sum can be made arbitrarily near to 
 
 2 * hi 
 
 dd 
 
 where m = =fc 1. Hence the solid angle equals 
 
 ]£*./>■ 
 
 50 
 
376 Ames:, An Arithmetic Treatment of Some Problems in Analysis Situs. 
 On the other hand 
 
 L de +I,fu dd = o > 
 
 since the integral is extended along every segment once in one sense and once 
 in the opposite. But 
 
 I 
 
 This may be shown by taking the parts sufficiently small, and showing that the 
 total variation of 6 in any one part is less than 2n, and hence equal to zero. 
 Hence 
 
 But 
 
 / dd= 2 Hi 71, 
 
 where n { is an integer, positive, negative or zero, and the solid angle V ^ / dd 
 
 d6 by twice the contribution of those integrals for which ^ is 
 
 negative. Hence the solid angle equals 4 nn, where n is an integer, positive, nega- 
 tive or zero. Then define the order of the point 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. If the order of a point not on the surface is n, then all points 
 in the neighborhood of 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 is 
 continuous when is not on the surface. But the order can vary only by a 
 multiple of unity. Hence it is constant, 
 
Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 377 
 Corollary. The 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 op 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 Q 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 P , there is a point of the surface near P at which it is smooth, and which 
 may be used instead of P . Hence we may assume that the surface is smooth at 
 P . Transform to a new set of rectangular coordinates with origin at P , and so 
 chosen that the surface near P 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 + (o, o, S) and 
 0~(o, o, — 8) where & is so chosen that no point of the surface except P lies on 
 the segment 0~ + of the z-axis. Now refer to two systems of spherical coordi- 
 nates having the origin at + 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 $ dd, 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 P to the angle at + differs 
 from the contribution of the same part to the angle at 0~ by a number arbitrarily 
 near to 4 n. That of the remaining parts in the two cases differ by an arbitrarily 
 
378 Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 
 
 small number. But the orders of 0+ and 0~ can differ only by integers, hence 
 they differ by unit}-. 
 
 Second Lemma. Given any three dimensional continuum R, and a surface S: 
 z —f(x,y), or y=f(z,x), or x—f{y,z), 
 
 where f is single valued and continuous : 
 
 (a) If R contains all points of S except possibly its boundary points which 
 may lie in the boundary of E, then the totality Br of points of R not on S form at 
 most two continua ; 
 
 (b) If also S has a simple regular boundary one point of which is in R, 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 R) . Draw a straight line CD 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 Dby 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 in R, and containing E and the point 
 
Ames: An Arithmetic Treatment of Some Problems in Analysis Situs. 379 
 
 common to CD and the surface S, Define two assemblages N + and N~ analog- 
 ous to that of Art. 6, Example 3, as follows : 
 
 z =/(»• V) + r, (x, y) in S', 
 
 < r < h for N+, 
 
 —7t </•<() foriV~. 
 
 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 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 B~, which is contrary to hypothesis. Hence F must lie in N + , 
 and by similar reasoning P can be joined to G by a simple curve wholly in B~. 
 Hence the points of B~ 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 P interior to B. 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 B. Hence we may assume that it is not a vertex. 
 Let the surface now be extended slightly past P . By reasoning similar to that 
 of the first case it can be proved that the points of B 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 Condition B form two continua, 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 : 
 
 x=f(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 
 two 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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