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Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Phil sophy in the University of Michigan. By Volney Hunter Wells. (l) SINGLE PARAMETER SYSTEMS OF POLAR FIELDS. By Volney Hunter Wells. CONTENTS. Séction I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . II. The Pencil of Polar Fields. . . . . . . . . . . III. The Types of Pencils of Polar Fields. IV. The Range of Polar Fields. . . . . . . . . . . . V. The Types of Ranges of Polar Fields. . - VI. The First Mixed System of Polar Fields. VII. The Second Mixed System of Polar Fields. VIII. The Third Mixed System of Polar Fields. . - Page 2. 4. l6. 32. (2) I. INTRODUCTION. The early Writers on the theory of conics discuss poles and polars With respect only to a conic. In his discussion of the pole and polar relation with respect to a circle, Chasles” Says that this relation ----------------------------------------- - - - - - - - - - - - - - - - - - --------- ------------------------------------------- - - - - - - - - - - - - - - - - - - - - - - --------------------------------------------------------------------------------- - - - ------------------------------------------------------- first treats of the polar system independently of the conic. He gives the two Well-known methods for its determination. With his Work came the broader concep- tion of the polar field, making the conic a dependent part of it, not necessarily real. ------------------------ - - - - - - - ------------ - - - - - - --------------- -------- em----------------------------------------------------------------------- writers on systems of conics confined themselves to real conics. Steiner gave the first definitions of the systems of polar fields. Though he does not Wholly his work eliminate the idea of the conics, £8 brought, the reali- zation of the greater generality of the Systems of polar fields. He defines the pencil as all the polar fields having a common self-polar triangle. The systems of polar fields discussed in this paper (3) are the pencil, the range, and the three mixed sys- tems. The pencil of polar fields has been defined and studied by Jolles”. He defines first a pencil of - - - - - - - ---------------------------- * *- - - - - - - - - - - - - - - - - - - - - - * Jolles. "Einfache Synthetische Ableitung der Grund- eigenschaften eines Büschels polarer Felder' Archiv der Mathematik und Physik. Third series ll. 1906–7. OK) 72-6. ----------------------------------------- - - - - - - - - - - - - - - - - - - - - - - collinear fields, by making use of the congruence and space curve, as studied by Reye”. Then a correlation - - - - - - - --------------------------------------------------------- **Reye. Die Geometrie der Lage. Fourth edition. Vol. 3. Chapters l and 2. ---------------------------------------------------------------------------------- is set up between a field of points and each field of the pencil. The resulting system is the pencil of po- lar fields. In this paper it is proposed to define the pencil in a manner in which it is unnecessary to go outside the plane” Among the resulting theorems are the two -------------------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ***For the idea of this study, as well as for many valuable suggestions, the author is indebted to Profes- which are given by Jolles. Zacharias” continued the study of the same system, ****zabharias. "Uber die allgemeinen Eigenschaften eines Büschel polarer Felder' pp 31-9. "Uber die vers chiedenen Arten Büschiel polarer Felder' pp. 40-54. Sitzungsberichte der Berliner Mathematischen Gesellschaft. 65 Sitzung, 25. Nov., 1908. basing his work on the two theorems of Jolles. His results (4) follow this paper Without change. Börger" has also made a study of the same system. - - - - - - - ---------------------- * * * * * *---------------------------------------- ------------------ * * * * * * * * * * * * *---------------------------- He defines the pencil of polar fields as all the polar fields having in Common the involutions on tWO lines. This necessitates that the point of intersection of these two lines be a point, having the same polar in all the fields of the pencil. Since, in the method proposed the construction of such a point is of the tº third degree, its generality and advantage is estab- lished. In this paper the range of polar fields is defined, and the theorems enunciated only , since the System is the reciprocal of the pencil. The author is not aware that the mixed systems of polar fields have been pre- viously studied. The same methods are applied to these forms as to the pencil. Proofs for the theorems are not given in full. It is the intention of the author to º -- make a more extensive study of these Systems at a later date. II. THE PENCIL OF POLAR FIELDS. In a given polar field a triangle and its recipro- cal are perspective. Conversely , a polar field may be given by two perspective reciprocal triangles”; Deter- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * * * * * - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - (5) mine tWo arbitrary polar fields TT, and TT2 . Let TT, be given by the perspective reciprocal triangles ABC and A, B, C, , and Tſ, by the perspective reciprocal triangles ABC and A, B, C2. Designate a, a , by J. , b, ba by J3 , and c, ca by ſº . Theorem I. In two polar fields the points conjugate to the points of a line lie on a conic. Let D', E', F', ... represent the points conjugate in both fields to the points D, E, F, ... of the line AB . In the field TT, the polars of D, E, F, ... form the pencil C1 (D', E', F", . . ), and in the field Ta. , the pencil Cz (D', E', F", . . ). Since the range of poles is projective With each of the pencils of polars, the two pencils are projective with each other. Hence C , C2, D', E', F', . . lie on a conic. Jº and J3 lie on this conic, since they are the double conjugates of A and B respectively. Also, the common pair of conjugate points of the two involu- tions determined by the two fields upon AB, lie on this conic. Definition. The locus of the points conjugate to the points of a line is the pole conic of the line in the two fields. Determine the common pair of double conjugate points M3 and Na of the two involutions on AB; M2 and N2 of the two involutions On Ac, and M, and N of the two involu- tions on BC. In this determination there arise two cases, according as the common pair of points are real (6) or imaginary. First, they are real in case both invo- lutions are elliptic ; in case one involution is ellip- tic and the other hyperbolic ; and in case both involu- tions are hyperbolic, but the double points do not se– parate each other. Second, they are imaginary in case both involutions are hyperbolic and the double points separate each other. The pole coni CS G. , 74. , and ſº of BC, AC, and AB re- spectively are givea, since six points of each are known. At , Az , Jº , ſ , M . , and N1 determine GT : B | , Ba, JA r , M, , and N. determine gas and C, , Ca, R. , J3 , Ms., and N3 determine Gº . Theorem II. The three pole conics G: , G; , and 4 3 have three points in Common. GT and gº have the point ſ” in common, Since it is the double conjugate of C. The two conics, then, have a second point O, in common. There Will be a point on each AC and BC to which 0 | Will be doubly conjugate. If those two points be denoted by G and H, then GH is the polar of O in both fields. GH intersects AB in a point K, which is also doubly conjugate to 0, . Since, however, K lies on AB, its double conjugate must lie On 05 . Therefore, the point, 0.1 is common to the three conies. Similiarly there are two other points Oz and o, common to the three conics. Each pair of pole conics have one real point in common, and consequently one of the three COſſiſſion points must be real, while the other (7) tWO may be real or imaginary. Construct, a new triangle A 3B3 C3 as follows: Join 01 to the points f , ſ: , and I", and designate the lines by a o , bo, and co respectively. There are determined three projective pencils JR (ao , a, , a, ), ſº (b. , b, , b, ), and T (c. , e, , ca), since ſ , ſ , o, , A , and A a lie on GT f , ſ , o, , B, , and B a lie on G; ; and J. , ſ: , 0 , C, , and C a lie on 93 . Choose any ray a 3 of the pencil fl , and find the corresponding rays b% and C3 in the pencils ſº and ſ” respectively. This determines the triangle A3 B3 C3, With A 3 lying on GT , B3 on Gà , and C 3 on G3 . It is important to note, however, that it is unnecessary to determine the point 01 and the rays ae , be , and Co., for if the same ray a 3 of the pencil fl is chosen, its intersections, B3 and C3 with G. and @ respectively, can be found by the Pascal theorem. J3 C3 and ſº B3 Will intersect in A3 , and the same triangle Aa B3 C3 is determined as before. The triangle A, B3 C3 is perspective to the triangle ABC. Draw Aſ; and B fl , and let their point of inter- section be P. Also let their poles be X , and Y, in field Tr, , and X 2 and Y 2 in field iſ respectively. Then ABP and C X, Y are perspective reciprocal triangles in the field II, , and ABP and C, X, Ya are perspective recip- rocal triangles in the field iſ . (9 X , PB), (C. Y. , AP), 3.13.63 collinear, and (C, X, , PB), (C.V., AP), and (x, y, , AB) But C, X, , C. K., and PB and (x,Y, , AB) are gollinear’. (8) 2. are concurrent at J. , and C. Y. , C2. Y., and AP are con- current at ſº . Hence, RJ3 is the line of perspective for the two pairs of triangles. Since, now, Jº J3 , X, Y, , and AB are concurrent, and J'ſ , XaXa , and AB are concurrent, then X | Yi , X2: Yz , and AB are concurrent, at a point P', the double conjugate of P. But, since P' lies on AB, P lies on the conic 6. The pencil Jº (6, CaO, P ) is projective to the pencil ſ? (C, Cao, P), since R , ſ: , C , C2, 0, , and P lie on T3 . Pencil Jº (C, C.O. P) is perspective to the range a (A. A. A.; B). Pencil Jº (C, C, O, P) is perspective to the range b (B. B. B. A.). Therefore, the two ranges a and b are pro- jective, and A'B', A.B. , A, B, , AB, a, and b form a Second order pencil & . In the same manner it may be Shown that A Ti and CA intersect, in a point Q of the conic G, . The pencils R (B. B.O. Q) and ſ" (B, B2. Or Q) are projective to each other, and perspective to the ranges a (A. A. A. C.) and c (C. C. C. A.) respectively. Hence, A Cº', A.C., A.C.; , AC, a, and c form a second order pencil 8. And in the same manner also it may be shown that B ſi and C ſº intersect in a point R of the conic G1 . The pencils ſ (A, A, O, R) and ſ” (A. A.O. R.) are projective to each other, and perspective to the ranges b (B. B. B. C.) and etc. &c.B) respectively. Hence, Bºc, B, C, B. c. , BC, b, and c form a second order pencil 8, . But A; , B}, and C) are collinear on 5, , since the triangles (9) 2. ABC and A, B, C, are perspective. Similiarly A, B4, and C4 are collinear on Sz. The three second order pencils have , then, the five rays St , Sa , a, b, and c in common, and are hence the same pencil 8 . A. B., Will intersect C in the point, corresponding to the points A, and B. in the projectivity set up between the ranges a, b, and C. But this point, is C.; ; hence A: , B: , and C. are collinear. If a s , b, , and C = , the three corresponding rays of the three projective pen- cils ſº , ſ: , and ſ” , intersect a, b, and c in the points A: , B} , and C3 respectively, then A; , Bº , and C# are corresponding points of the three projective ranges. Then, as before, the three points A: , B; , and 0% are collinear. Hence, the corresponding sides of the triangles ABC and A2 B, C3 interSect in the points of a line, and the twó triangles are perspective. They determine a new polar field Which may be designated by TT 2 . AS a result of the above discussion, each of the three conics GT, , 6; , and G3 can be constructed, if de- sired, from five points which are always real, and which are always given When the two original fields 3.1°3 chosen. For instance, the points giving the conic G are A., A., ſ, , ſº , and R. The above data may be conveniently applied to the finding of a third polar field from the original two. The lines a, b, c, s , , and S 2 determine the second on- (10) - 2. der pencil § . Choose a ray S 2 of the pencil, and find its intersections with a, b, and c. Join to these three points the points R , ſ , and r" respect- ively. This forms a triangle A,B, 3, perspective and reciprocal to ABC, and hence determines, a polar field TT3. Theorem III. In the three groups of two polar fields formed from the three polar fields 0, , q, , and J.; the pole conics of any line are identical. In the fields T, and rſ; let the poléº Bc be 6'7". It is determined by the points A , As, R, ſ , and ſ”. But since these five points lie on ſi , the two pole conics of BC coincide. In the same manner it may be shown that Øſ is the pole conic of BC in the fields 7Tz and T73. Similiarly, the pole conics of AC and AB are G. and ſº respectively in the three groups of polar fields. Let the polars of a point Y on BC be y, , y2 , and ya in the fields TT, , ºria. , and T3 respectively. y , and y a intersect on G in Y', the double conjugate of Y in fields it, and Tſz. But y, and y 3 intersect on 67 in the double conjugate in the fields Tr; and TT3 . This point is y", since y, has but the points Y' and A in common with of . Let any line x intersect BC, AC, and AB in the points J, K, and L respectively. Also let the pole conic of x in the fields ºr, and tº be ºx , and in the º fields Tr; and ºr 3 be 0%'. since J lies on BC it has a single conjugate J' in the three fields. Simi- (11) 2. liarly, K and L have the same conjugates K' and L' respectively in the three fields. Hence, the pole conics Gx and G' have the points J', K', and L' in common. Since J is the intersection of BC and x, J' is a point of intersection of Q , and 0x . There is a Second point of intersection 0, which has a double conjugate on each BC and x in the fields 7T, and Tra. The polar, or the line joining them, intersects AC and AB in points which are conjugates of 0 in the two fields. But since they lie on AC and AB, 0 lies on 0. and GT3 . Hence, 0 passes through O. , 0.2, and Oa. In the same manner it may be shown that in the fields 7T, and Tra the pole conic G5 of x passes through the three points 01, Oa, and 02. Therefore the conics Gx and ox! coincide, since they have in common the six points J', K', L', O, , Oa. , and O2. Corollary. In the three polar fields TT, , T: , and T13 the three involutions formed on any line have a C Omºl Orl pair of conjugate points. In two polar fields the intersections of a line and its pole conic are the common pair of conjugate points in the two involutions determined on that line. By the theorem there is but one pole conic of a line in the three fields; hence, their intersections form the common pair of conjugate points in the three involutions on the line. (12) 2. Theorem IV. In the three polar fields 7, , T: , , and Tſa the polars of a point are concurrent. Let the polars of an arbitrary point X be repre- Sented by X, , Xz , and X 3 in the fields 77, , T2, and 773 respectively. And let x , and x 2 intersect in the point X". Then in the involutions determined on XX' by the fields 7ſ, and Ta. , the points X and X' are a common pair. By theorem III X and X’ are also a pair of C Orn- jugate points of the involution determined by 773 on the line XX'. Hence, x > also passes through the point X'. Definition. All the polar fields determined by tri- angles perspective and reciprocal to one triangle, such that the corresponding Sides of those triangles form three projective pencil S of rays, form a pencil of polar fields. The two fields are said to determine the pencil of polar fields. Any two fields of the pencil determine the same pen- cil. The sides of the determining triangles of any two fields of the pencil determine the same points ſ , J3 , and ſº ; and the lines of perspective with a, b, and c determine the same second order pencil S. - Hence, the same pole conics 67 , Ø, , and 0.5 are determined, and the same system results. Theorem V. In a pencil of polar fields the polars of a point are concurrent. This is evident from theorem IV, Where rſ3 is any field (13) Of the pencil. Theorem VI. In a pencil of polar fields the poles of a line lie on a conic. Let the polars of the points D, E, F, .. of any line x be d, , e, , f', ..., d., * fa . . . ; d. , ea , f', . . . . . . . in the fields # , , , ,,.. respectively. The polars Of the points of x are concurrent at the points X, , X2, X3, . X4, . . , Which are the poles of x in the fields TT, , 77 a , TT, , ... respectively. By theorem V d, , dz , da , . . . are con- current at D'; e, , e, , e3, ... are concurrent at E'; f, , fº , f's , . . are concurrent at F';.. . And by theorem I X, , X2, D', . E', F', ... lie on the pole conic of x. But the pencils X, , X2, X3, . . are all projective to each other, since the intérsections of corresponding rays are the points D', E', F", . . . Hence, X. , X2, X3, . . lie on a conic, the pole conic of the line. Theorem VII. The lines of perspective of the deter- mining triangles of the fields of a pencil of polar fields envelop a coni C. This follows from the discussion immediately pre- ceeding theorem III, where TT3 is any field of the pen- Cil. - Theorem VIII. The Centers of perspective of the determining triangles of the fields of a pencil of polar fields lie on a curve of the fourth order. The two second order ranges Gº (A, , Az., A , , . . ) and G-4 (B, , B, , B, , . . ) are projective, since each is per- (14) spective with the pencil ſ” . Hence, the two quadra- tic pencils A(A, , Aa, A3, . . ) and B(B, , Ba, B, , . . ) are projective. Their locus of intersection is a curve of the fourth order: Since, however, these points of in- *Emch. Introduction to Projective Geometry and its Application S. Art. 49. tersection are the centers of perspective for the deter- mining triangles of the fields of the pencil, the theorem follow S. Theorem IX. In a pencil of polar fields there are three degenerate polar fields. - If, in determining a field of the pencil, a ray of the pencil R is chosen through O. , its corresponding rays of the pencils ſº and T' will also pass through Ol. Similiarly, the sides of a second determining triangle will be concurrent at 0, , and the sides of a third at 03. These three degenerate triangles determine three degen- erate fields of the pencil. The points O. , Oa, and Oa lie on the fourth order curve discussed in theorem VIII, since they are centers of perspective for determining triangles of fields of the pencil. Theorem X. The singular points of the degenerate reas of a pencil of polar fields are the vertices of a self-polar triangle common to all fields of the pencil. The points O. , O2 . and o, have the same polars or , o, , and o, respectively in all the fields of the pencil. o, o, is the pole of O, Oz in all fields of the pencil. (15): * their conjugate must lie on the three conics ( , GT, , But 01 02 Will intersect a, b, and c in three points, each of Which is conjugate to o, o. in all fields of the pencil. Since these points lie on a, b, and c, and ſº . The point or o. is, then, O3. Hence, each ver- tex of the triangle O, O, O, is the pole of the side op- posite. In the pencil of polar fields there are four common self-conjugate points, forming a complete four-point, of Which the diagonal triangle is the triangle 01 04:03. The three pairs of double rays of the pencils O, , 0.1 . and O2 are the three degenerate conics in the three de- generate polar fields of the pencil of polar fields. Hence, the pencil of conics is determined in the pen- cil of polar fields'. ------------------------------------------- * * * * * * * * * * * * * * * * * * * * * * *--- * -- * Z 3. C h 8. r i 3. S f l O C C l t Definition. Four fields of a pencil of polar fields are harmonic when the polars of any point in those fields form four harmonic rays of a pencil. Four harmonic fields of a pencil may be determined by choosing, as sides of the determining triangles, four harmonis rays of the pencil fit . The four rays of each of the pencils ſº and ſ” will also be harmonic, because of the projectivity of the pencils of rays. Hence, four determining triangles of four harmonic fields are formed. 2. (16) Definition. Two pencils of polar fields are projective When there is a one-to-one correspondence of fields, such that the harmonic fields of the first correspond to the harmonic fields of the second. Construction 1. Construct a field of the pencil having a given pair of points conjugate. Let the given pair of points be K and L. Determine the com— non pairs M and N, and Ma and Na on KL and AC respectively for the two fields. Let KL, AC be J. Find the points J,' and J., con- jugate to J on KL. Then J.' B, and J. B. are the polars of J in the two fields, and intersect in J'. MN, KL determine the involu- tion for the new field on KL. Determine in it the point J, con- jugate to J. Let J. J', polar of J in field T3, intersect AC in J.". Then M2 N2 , J.J." determine the involution for TT3 on AC. Find in it the point A, conjugate to A. Draw A. f. , and con- struct the determining triangle of the field as usual. Construction 2. Construct, a field of the perhcil having a given point Self- conjugate. The construction follows that of l, except that the involu- tion on a line for the new field is given by a pair and a double point. III. TYPES OF PENCILS OF POLAR FIELDS.” - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ---------- * Zacharias. loc. cit. These types are given here for com- / - --- pleteness. Type I The Hyperbolic Pencil. The pencil is hyperbolic when the three conics 0 , º (17) 07., and 0 , have three real, distinct points in common. The differences in the character of the involutions determined at these points 0, , 0,..., and 0.3 give rise to five cases : l. The three involutions may be hyperbolic. 2. One involution may be hyperbolic and two elliptic. 3. One involution may be hyperbolic and two parabolic. 3 4. One involution may be elliptic and two parabolic. - 5. One involution may be undetermined and two parabolic. º Type II. The Elliptic Pencil. The pencil is elliptic When the three conics J. , 6%. , and G3 have two of the common points imaginary. Type III. The Parabolic Pencil. The pencil is parabolic when the three conics G. , Gº, , and Çı have two of the common points real and - coincident. Type IV. The pencil is of this type when the three conics G. , G; , and G. have the three common points real and coin- 2. cident. Type W. The pencil is of this type when the three conics ſº , Cº. , and GE are line pairs, and the three common points are collinear. (18) 2. and na determine Z3. IV. THE RANGE OF FIELDS. Determine two arbitrary polar fields ºr, and ºr by means of the perspective reciprocal triangles ABC, A B 1 C , and A2 B2 C2 , as Was done in the study of the pencil of polar fields. Let A, A2 be denoted by at , B, B2 by G , and C, CA by T . Theorem I. In two polar fields the lines conjugate to the lines of a point envelop a conic. Definition. The locus of the lines conjugate to the lines of a point is the polar conic of the point in the tWo fields. Determine the common pair of double conjugate lines Iſl and n 1 of the two involutions at A$ m2 and n 2 of the two involutions at B; and m3 and n 3 of the two involu- tions at C. In this determination there arise two cases, according as the common pair of lines are real or imag- inary. The polar conics z, , z, , and X: 3 of A, B, and C respect- ively are given, since Six tangents of each are known. a, , a, , (, , , , m, , and n determine Z, ; b, , b a , * , Y- , ma , and n + determine X + 3 and C1 - ca. , a , (; , m3, Theorem II. The three polar conics XI, , X a , and Žs have three tangents in common. Construct a new triangle A3 B3 C3 as follows: Denote the three common tangents of the polar conics (19) e by O. , o a , and Oa. Also denote the points of inter- Section of ol with a , (? , and r by Ao, Bo, and Co re- Spectively. There are determined three projective ranges a (Ae , A, , A a ), (*(Bo , B , B.), and r ( Co , C, , C a), Since at , a a , o, , ( , and r are tangent to #: ; b, , b , , O , o, , and Y are tangent to Ya ; and c, , ca, o, , e, , and (? are tangent to Z3 . Choose any point A 3 of the range o , and find the corresponding points B3 and C3 of the ranges (3 and Y respectively. This determines the triangle A3 B3 C3 , With a 3 tangent to Z , b 3 to Zz. , and C 3 to Y 3 . The triangle A3 B3 C3 is perspective to the triangle ABC. Hence, a new polar field T3 is determined. Theorem III. In the three groups of two polar fields formed from the three polar fields T, , Ta. , and T3 , the polar conics of any point are identical. Corollary. In the three polar fields T, , T: , and ºf 3 the three involutions formed at any point have a common pair of conjugate lines. Theorem IV. In the three polar fields Tr; , T: , and Trà the poles of a line are collinear. Definition. All the polar fields determined by tri- angles perspective and reciprocal to one triangle, such that the corresponding vertices of those triangles form three projective ranges of points, form a range of polar fields. The two fields determine the range. (2O) 2. Any two fields of the range determine the same range. Theorem v. In a range of polar fields the poles of a line are collinear. - Theorem VI. In a range of polar fields the polars of a point envelop a conic. Theorem VII. The centers of perspective of the de- termining triangles of the fields of the range of polar fields lie on a conic A . Theorem VIII. The lines of perspective of the deter- mining triangles of the fields of the range of polar fields envelop a curve of the fourth Class. Theorem IX. In a range of polar fields there are three aegenerate polar fields. Theorem X. The singular lines of the degenerate fields in the range of polar fields are the sides of a self-polar triangle common to all the fields of the range. - The pencil and range of polar fields determined by the same two polar fields have the same common Self- polar triangle. In the range of polar fields there are four common self- conjugate lines, forming a complete four-ray, of which the diagonal triangle is Oi Oi Os . The three pairs of double points of the ranges of , oz., and O 3 are the three degenerate conics in the degenerate polar fields. Hence, the range of conics is determined in the range Of polar fields. (21) Definition. Four fields of a range of polar fields are harmonic When the poles of any line in those fields form four harmonic points of a range. Definition. TWO ranges of polar fields are projective When there is a one-to-one correspondence of fields, such that the harmonic fields of the first correspond to the har- º monic fields of the second. º Definition. A pencil and a range of polar fields are pro- jective When there is a one-to-one correspondence of fields, such that the harmonic fields of the pencil correspond to the harmonic fields of the I’ārīge. Definition. A pencil and a range of polar fields are per- spective When they are projective and have their common fields self- corresponding. The pencil and range of polar fields determined by the same two polar fields are perspective. º, - Construction l. Construct, a field of the range having º a given pair of lines conjugate. Let the given pair of lines be k and l. Determine the Com- mon pairs m and n, and ma and n, at kl and B respectively for the two fields. Let kl, B be j. Find the lines j," and j, conjugate to j at kl. Then jºb and jºb, are the poles of j in the two fields, and the line joining them is j'. mn, kl determine the involution for the new field at kl. De termine in it, the line j; conjugate to j. Let j. j', pole of j in field Tº , be joined to B by j}''. Then m, n, , jj, determine the involution for 7T3 at B. Find in it the line a conjugate to a. Find the inter- section with & , and construct the determining triangle as usual. º (22) º Construction 2. Construct a field of the range hav- ing a given line self-conjugate. The construction follows that of 1, except that the involution at a point for the new field is given by a pair and a double line. W. TYPES OF RANGES OF POTAR FIELDs. Type I. The Hyperbolic Range. The range is hyperbolic When the three conics 7 t , ºf . , and Z., have three real, distinct, tangents in common. The differences in the character of the involutions de- termined on these lines give rise to five cases: 1. The three involutions may be hyperbolic. 2. One involution may be hyperbolic and two elliptic. 3. One involution may be hyperbolic and two parabolic. 4. One involution may be elliptic and two parabolic. 5. One involution may be unde termined and two parabolic. The pencil of type I-3 and the range of type I-3, its reciprocal, are identical forms. - Type II. The Elliptic Range. The range is elliptic When the three conics Z, , 2 . , and X 2 have two of the common tangents imaginary. Type III. The Parabolic Range. The range is parabolic When the three conic's Z , Y, , and X 3 have two of the common tangents real and coincident. (23) Type IV. The range is of this type when the three conics X , z., and Z3 have the three common tangents real and coin- cident. The pencil of type IV and the range of type IV, its reciprocal, are identical forms. Type V. - The range is of this type when the three conics 2 , X , , and Z3 are point pairs, and the three common tan- gents are concurrent. VI. THE FIRST MIXED SYSTEM OF POLAR FIELDS. Consider two polar fields TT, and ‘rſ: given in the usual manner by means of the perspective reciprocal triangles ABC, A, B, C , , and A.B.C2. Determine the conics F, , Zz. , and Žs as in the range of polar fields, and the conic § as in the pencil of polar fields. construct a new triangle A's B3 C2 as follo WS : Draw a tangent, S3 to 8 , and find the intersections with BC, AC, and AB. From these points draw tangents to F, , X, , and X3 respectively. This forms the new triangle A, B, C3 which is perspective to triangle ABC. Consequent- ly, it determines a new polar field 773. In the same manner other determining triangles are formed, and a Sy S- tem of polar fields is determined. The sides of the determining triangles of the fields of the system form three projective pencils of the second (24) order. The second order pencils Z, , Y, , and X 2 are perspective With the ranges a, b, and c respectively. But Since these ranges are projective with each other, the three pencils are projective with each other. + The vertices of the determining triangles of the fields of the system lie on curves of the fourth Order. The locus of inter Section of the projective Second - - - - - -------------------------------------------------- - - - - - - Definition. The system of polar fields determined by triangles perspective and reciprocal to one triangle, such that the corresponding sides of those triangles form three projective second order pencils, is the first mixed system of polar fields. With a single choice of a tangent to the conic & there are determined eight polar fields of the System. From the point of intersection of Ss With a , the two tangents ar and an are drawn to Z, ; from its intersection with b, the two tangents br and ba are drawn to Ž a ; and from its intersection with c, the two tangents c s and ca are drawn to X , . The eight triangles formed are apbfc p, apbp Cn a fºa CP . an be cf. , af ba Ca , 84 bp Ca , anbn CP and as bacA. Each is perspective to the triangle ABC, and hence each determines a polar field of the System. The points as bp , *F ba , an be , and aaba each generate a fourth (25) Order curve . Similiarly, the projective pencils Y, and Ža , and the projective pencils Z, and Z, , each generate four fourth order Curves. Hence, the ver- tices of the determining triangles of fields in the first mixed system lie on twelve fourth order curves. Theorem I. In the first mixed system of polar fields there are twenty-four degenerate polar fields. Choose AB as the targent to the conic 8 . From B there may be two tangents drawn to the conic Z, , from A there may be two tangents drawn to the conic Y.A., and from each of the four points of intersection there may be two tangents drawn to the conic ZA . There are formed, then, eight triangles, each degenerate, each perspective to ABC, each having AB as the line of perspective, and each determining a degenerate polar field. Similiarly, there will be formed eight degenerate polar fields when each BC and AC are chosen as the lines of perspective. Theorem II. In the first mixed system of polar fields there are twelve singular lines forming four groups of three on a line. From the construction of the degenerate polar fields it Will be noted that each of the twelve singular points is the singular point of two different polar fields. Hence, there are twice as many degenerate polar fields in the system as there are singular points, Specialized Systems in the First Mixed System. Theorem III. In the first mixed system of polar fields (26) 4. there are eight specialized systems of polar fields. The properties of the specialized system are as fol- low S : it, has three degenerate polar fields, the Singu- lar points of which are collinear; the sides of the de- termining triangles of the fields form three projective Second Order perm cils ; and the vertices of the determin- ing triangles of the fields lie on three fourth order curves. It is possible to determine the specialized system by choosing With the original two fields Tri and Tº each of the groups of three degenerate polar fields having their singular points collinear. The choice of a tangent to the conic 8 gave eight polar fields, one in each specialized System. When each AB, BC, and AC Were chosen, there Were formed eight degenerate polar fields, one in each specialized system. In each specialized system, then, there are three degenerate polar fields; and by theorem II the singular points are collinear. - The three projective second order pencils of the sides of the determining triangles of the fields are common to the eight, specialized System.S. The fourth order curves, on Which the vertices of the determining triangles of the fields lie, are not all different for each specialized system of the mixed system. If we suppose that the eight triangles, in the order named above, determine fields belonging the special- ized systems I, II, III, IV, V, VI, VII, and VIII re- (27) * Spectively, I and II have a fourth order Curve in com— mon. This is evident since apbp is a vertex always common to the two series of triangles determining the fields of the two specialized systems. Similiarly , I and III, I and IV, II and VII, II and V, III and V, III and VII, IV and VI, IV and VII, V and VIII, VI and VIII, and VII and VIII each have a fourth order curve in C Oţn- mon, since in each there is always a vertex of the de- termining triangles of the fields common. Theorem IV. In a specialized system or the first, mixed system of polar fields the three singular points are common points of the three fourth order curves. Theorem V. In a specialized system of the first mixed system of polar fields the polars of a point envelop a conic. - Theorem VI. In a specialized system of the first mixed system of polar fields the poles of a line lie on a fourth order Curve. Theorem VII. In a specialized system of the first mixed system of polar fields there is determined a 3-point 1-line mixed system of conics. (28) VII. THE SECOND MIXED SYSTEM OF POLAR FIELDs. The second mixed system of polar fields is recipro- cal to the first mixed system of polar fields, Consider two polar fields T1 and Tº given in the usual manner by means Of the perspective reciprocal triangles ABC, A, B, C , and A.B. c. Determine the conics G", , G; , and 0; as in the pencil of polar fields, and the conic A as in the range of polar fields. construct a new triangle A3 B3 C3 as follows: Join a point S3 on A to the points A, B, and C. Find the points of intersection With 6, , (, , , and G3 respect+. ively. This forms a new triangle A 3 B2 C3 Which is per- spective to the triangle ABC. Consequently, it deter- mines a new polar field T2 . In the same manner other determining triangles are formed, and a system of polar fields is determined. The vertices of the determining triangles of the fields of the System form three projective ranges of the second order. The Second order ranges Gº , ſ: , and Gº, are perspective With the three pencils A, B, and C respectively. But since these pencils are projective with each other, the three ranges Of the second order are projective With each other. The sides of the determining triangles of the fields of the system envelop Curves of the fourth class. The (29) lines joining the corresponding points of two pro- jective ranges of the second order envelop a curve Of the fourth Class. Definition. The System of polar fields determined by triangles perspective and reciprocal to one triangle, Such that the corresponding vertices of those triangles form three projective second order ranges, is the second mixed system of polar fields. With a single choice of a point on the conic A , there are determined eight polar fields of the system. Sº A intersects Gi in A p and An , S: B intersects G: in BP and Bn , and S3 C intersects Gº in CF and CA. The eight triangles formed are Ap BP Cp , Ap BP CA , Ap Ba Cº., AABe CA, AFB, CA AA BP Cn , A, BACP , and AA Bn Ca . Each is perspective to the triangle ABC, and hence each determines a polar field of the system. The lines Ap Bp , Ap Bº , An Be, and As BA each generate a fourth class curve. Similiarly, the projective ranges 6, and @3 , and Gº, and QT, each generate four fourth class curves. Hence, the sides of the deter- mining triangles of fields in the second mixed system are tangent to twelve fourth class curves. Theorem I. In the second mixed system of polar fields there are twenty-four degenerate polar fields. Choose C as the point on 4 . AC will intersect Gº in two points, BC Will intersect G7 in two points, and the four lines determined by these points Will each inter- sect G3 in two points. There are formed eight triangles, (30) each degenerate, each perspective to ABC, each having C as the center of perspective, and each determining a degenerate polar field. Similiarly, there will be form- ed eight degenerate polar fields When each B and A are chosen as the Center’s Of perspective. Theorem II. In the second mixed system of polar fields there are twelve singular lines forming four groups of three concurrent at a point. From the construction of the degenerate polar fields it is noted that each of the twelve singular lines is the singular line of two different polar fields. Hence, there are twice as many degenerate polar fields as sin– gular lines in the system. Specialized Systems in the Second Mixed System. Theorem III. In the second mixed system of polar fields there are eight specialized systems. The properties of the specialized system are as fol- lows: it has three degenerate polar fields, the singu- lar lines of which are concurrent; the vertices of the determining triangles of the fields form three pro- jective second order ranges; and the sides of the deter- mining triangles of the fields are tangent to three fourth class curves. It is possible to determine the specialized systems by choosing With the two original fields ºr, and T: , each of the groups of three degener- ate polar fields having their singular lines concurrent. (31) The choice of a point, on the conic A gave eight polar fields, one in each specialized system. When each A, B, and C were chosen, there were formed eight degenerate polar fields, one in each specialized sys– tem. In each specialized system, then, there are three degenerate polar fields; and by theorem II the singu- lar lines are concurrent,. The three second order ranges of the vertices of the determining triangles of the fields are common to the eight specialized systems. The fourth class curves, to Which the sides of the determining triangles of the fields are tangent, are not all different, for each specialized system of the mixed system. If we suppose that the eight triangles, in the order named above, determine fields belonging to the specialized systems I, II, III, IV, V, VI, VII, and VIII respectively, I and II have a fourth class curve - in Common. This is evident since Ap B F is a side always common to the two series of triangles determining the fields of the two specialized systems. Similiarly, I and III, I and IV, II and V, II and VII, III and V, III and VII, IV and VI, IV and VII, V and VIII, VI and VIII, and VII and VIII each have a fourth class curve in com— mon, since in each there is always a side of the deter- mining triangles of the fields in common. Theorem IV. In a specialized system of the second mixed system of polar fields the three singular lines are com- (32) mon tangents to the three fourth class curves. Theorem V. In a Specialized system of the second mixed system of polar fields the poles of a line lie on a coni C. Theorem VI. In a specialized system of the second mixed system of polar fields the polars of a point ërl- velop a fourth class curve. Theorem VII. In a specialized system of the second mixed system of polar fields there is determined a 3-line l-point mixed System of conics. VIII. THE THIRD MIXED Sys"TEM OF POLAR FITLDS. Consider two polar fields TT and TTa, given in the usual manner by means of the perspective reciprocal triangles ABC, A B, C, , and A2 B2 C2. Determine the conics Gº , G, , and G3 as in the pencil of polar fields, and the conics I, , Y, , and Z3 as in the range of polar fields. , The range CW (A, A 1.0, 0,. ) is perspective With the pencil A(A.A.o. o.). The range G1 (B, B, O, O, ) is perspective With the pencil B( B, Ba O, Oa ). But since the two pencils are projective, the two ranges are projective. And since in the two ranges the points 0 1 and 0.2 are self- corres- ponding, a perspective correspondence is set up be- tween the points of 67 and 0%. . Hence, lines joining cor- responding points form a second order pencil A- a . Sim- iliarly, the conic ſi is perspective to the conic 63 giving the second order pencil-M-2, and the conic G. º - (33) is perspective to the conic Gº, giving the second order pencil -A . In the same way the three ranges 0 , , 6-, , and Wºº are perspective each to each, with the points O, , and O2 self-corresponding. The perspectivities give rise to the second order pencils /11, M -, and / . . Also in the same Way the ranges 0 , Ø, , and ſº are perspective each to each, With the points O2 and O & self- correspond- ing. The perspectivities give rise to the three second order pencils /-, , -M-, , and -/-1. TWO tangents tº and t2 , through O1 and 0 a respective- ly, are common to the three conics -M-, , M., and M.A. Similiarly, two tangents tº 3 and tº , through O1 and 0.3 re- spectively, are common to the three conics 44, M-4-, and Alt. And the two tangents tº and t . , through Oa and Os respectively, are common to the three conics 47, -M-, , and M. . The conics -M-, , -M-4, and Mº have the tangents a and a 2 in common; the coni CS –4. y Alsº, and -/-º have the tangents b1 and b 2 in Common; and the conics –4 * , A., and A4 have the tangents c, and c 2 in common. The pencil X, (a, a. o. oa ) is perspective With the range a (A. A. O'O,'). The pencil Z.(b, b. O o a j is perspective With the range b(BE;o;04). But since the two ranges are projective, the two pencils are projective. And since in the two pencils the rays of and o 2 are self- corresponding, a perspective correspondence of tangents is set up between the conics F, and £1. Hence, the inter sections of corresponding rays form a second order (34) range 2, . Similiarly, the conic Z, is perspective to the conic Y S giving the second order range A. And the conic X. 1 is perspective to the conic Z 2 giving the second order range A . In the same way the three pen- cils X , , X, , and Z 3 are perspective each to each, With the rays of and of self- corresponding. The per- spectivities give rise to the second order ranges 2 y 2 % -, and Ae . Also the pencils Z, , z, , and X3 are per- spective each to each, With the rays ox and o 3 Self- corresponding. The perspectivities give rise to the second order ranges 47, 14, and 23. Two points T, and Tz, upon on and oz respectively, are common to the three conics A, , 7, , and A3. Similiarly two points T3 and T+ , upon o, and oa respectively, are common to the conics A., A.-, and 7 - . And two points Tº- and T., upon Oz and o, respectively, are common to the conics 77 , A & , and 7. The conics A, , 2, , and Ay have the points A, and A 1 in common ; the conics A: , A-, and 7 g have the points B1 and B2 in common; and the conics A 3 , W - , and 7% have the points C 1 and C 2 in common. Let t t intersect the coni CS A, , * * , and A 3 in the points As , Be, and Ce respectively, and let t 2 intersect the same conics in the points A, , B, , and Cº. respectively. Also let the tangents from T, to the conics -M-, , -M-, , - and As be &o y b., and c e respectively , and the tangents from Ta to the same conics be a , b, , and c; respectively. (35) The three ranges A, (A, A2A. A. T, T, ), A, (B, B2B, B. T., T.), and A2 (C, CA Ce C.T. T.) are perspective since lines joining corresponding points in the three pairs of ranges are tangent to the conics -4. , , -/-a, and A3. Hence, the six conics A , , A , As , ſ , , -/l. , and -1, are perspective to each other. The perspectivities can, however, be set up in four ways. This is evi- dent since at each of the points T, and T 2 two tan- gents may be drawn to each of the three conics. Like- Wise each of the lines t t and t 2 intersects each of the three conics in two points. As before, there may be arranged four perspectivities between the conics Aw y 2 . , 7. , 4.4 , -48 , and, Au, and also between the six conics 27, A & , 'A ,-47, 4.x, and 4-3. Construct a new triangle A3 B, C, as follows: Choose a ray as of the pencil X, , and find the cor- responding ray b% of the pencil Z. . This gives the point C3 in three positions, - on A, , Oſl At , and on 44. Determine, from the point C3 on A, , the points A 3 and B3 on A, and 2 . respectively in the four possible ways in the four perspectivities set up. De termine, from the point C3 on Aé , the points A 3 and Ba on 24 and 2 & respectively in the four possible Ways in the four perspectivities. And determine, from the point C3 on A; , the points As and B's on 2; and Wr respectively in the four possible Ways in the four perspectivities. With the single choice of a tangent to X there arise (36) twelve triangles AaB, C , . The triangles A, B3 C 3 are each perspective to the - triangle ABC. Each determines, then, a polar field. In the Same manner other series of determining tri- angles of polar fields are formed, and a system of polar fields is determined. The sides of the determining triangles of the fields of the System form second order pencils. The vertices of the determining triangles of the fields of the system form second order ranges. Definition. The system of polar fields determined by triangles perspective and reciprocal to one triangle, Such that the corresponding Sides of those triangles form perspective second order pencils, and such that the vertices of those triangles form perspective second order ranges perspective to the second order pencils, is the third mixed system of polar fields. Theorem I. In the third mixed system of polar fields there are twenty-four degenerate polar fields, twelve having singular lines and twelve having singu- lar points. In the perspectivities set up between A. , 2, , and 2: , T, and T 2 are each self-corresponding points, Hence, at each of these points the vertices of a deter- mining triangle coincide. However, at each there will be two degenerate triangles, since from each there are two tangents to each A, , M, , and -/L 5. These degener- (37) ate triangles determine four degenerate polar fields of the system. Similiarly, at each point T., , T: , T: , and Te are formed two degenerate triangles each deter- mining a degenerate polar field of the system. In the perspectivity set up between -M-, , M., and -4-2 y t and t 2 are each self- corresponding tangents. Hence, on each of these lines the sides of a determining tri- angle Will coincide. However, on each of these lines there Will be two degenerate triangles, since each tan- gent intersects each of the conics A, , 2 . , and 23 in two points. These four degenerate triangles determine four degenerate polar fields of the system. Similiarly, upon each line tº , tº , ts-, and t e are formed two de- generate triangles each determining a degenerate polar field of the system. Specialized Systems of the Third Mixed System. Theorem II. In the third mixed system of polar fields there are twelve specialized Systems. The properties of a specialized system are as follows: it has two degenerate polar fields, one having a Singular line and one a singular point ; the Sides of the determin- ing triangles of the fields form three perspective second order pensils, and the vertices of the determin- ing triangles of the fields form three perspective second order ranges. It is possible to determine the specialized systems by choosing With the two original (38) polar fields TT, and 77, the corresponding pairs of de- generate polar fields. Theorem III. In a specialized system of the third mixed system of polar fields the polars of a point envelop a conic. Theorem IV. In a specialized system of the third mixed system of polar fields the poles of a line lie on a coni C. Theorem V. In a specialized system of the third mixed system of polar fields there is determined a 2-point, 2-line mixed system of conics. Theorem VI. The third mixed system of polar fields - - is its own recipro Cal. University of Michigan. April, 1916. ſiliili RULES COVERING USE OF MANUSCRIPT THESES IN THE UNIVERSITY OF MICHIGAN LIBRARY AND THE GRADUATE SCHOOL OFFICE Unpublished theses submitted for the doctor's degrees and deposited in the University of Michigan Library and in the Office of the Graduate School are open for inspection, but are to be used only with due regard to the rights of the authors. For this reason it is necessary to require that a manuscript thesis be read within the Library or the Office of the Graduate School. If the thesis is borrowed by another Library, the same rules should be observed by it. Bibliographical refer- ences may be noted, but passages may be copied only with the permission of the authors, and proper credit must be given in subsequent written or published work. Extensive copying or publication of the thesis in whole or in part must have the consent of the author as well as of the Dean of the Graduate School. This thesis by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . has been used by the following persons, whose signatures attest their acceptance of the above restrictions. * A Library which borrows this thesis for use by its readers is expected to secure the signature of each user. 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