TABLE OF CONTENTS INTRODUCTORY CONSIDERATIONS. CHAPTER I INTRODUCTORY THEOREMS UPON ORTHOGONAL SPHERES AND CIRCLES PAGE.. 1 Dimensionality... Definitions of Orthogonality. General Theorems.... Circle Cut out of Sphere by Plane..... Intersections of Orthogonal Spheres are Orthogonal Circles Dimensionality of Assemblage. Circle Fixed by Two Points... Corresponding Points... Summary.... ~.. 2. 2. 2... 3 3.4.... 4. 6 6 7 9 CHAPTER II COORDINATES OF THE CIRCLE A. Pentaspherical Co6rdinates. B. Point-pairs......... 4 C. Dual Interpretation of ai= 0..... D. Coordinates of a Circle as Envelope of Spheres.. E. Co6rdinates of a Circle as Locus of Point-pairs.. F. Condition that Two Circles Intersect......... G. Conditions that a Given Circle may Lie on a Given Sphere or Pass through a Given Point-pair..... 9 11 13 14 15 16 17 CHAPTER III THE LINEAR COMPLEX OF CIRCLES A. Linear Complex of Circles Regarded as Envelopes of Spheres. B. Linear Complex of Circles Regarded as Loci of Point-pairs C. Polar Spheres and Pole Point-pairs.. D. Second Proof of Fundamental Correlation. E. Comparison between Circle Geometry and Associate Theories iii. 18. 19. 20. 21. 23 iv TABLE OF CONTENTS CHAPTER IV CONJUGATE CIRCLES, POLE POINT-PAIRS AND POLAR SPHERES A. General Theorems...........24 B. Special Complexes of Circles.......... 25 C. Co6rdinates of Conjugate Circles......... 26 D. Construction of the Complex..,...28 CHAPTER V LINEAR CONGRUENCES OF CIRCLES AND PROBLEMS UPON THE ASSEMBLAGE A. The Linear Congruence........... 30 B. The Surface of Intersection of Two Congruences of a Complex...32 C. Radius of a Circle, Circles in Involution, Degeneration of S....33 CHAPTER VI FAMILIES OF SPHERES.. 36 CHAPTER VII TRANSFORMATIONS OF THE ASSEMBLAGE A. Generalization of Co6rdinates yi...... 40 B. Transformation of Fundamental Spheres....... 41 C. Invariance of (A)............43 D. Projective Transformations by Means of Complexes.....44 CHAPTER VIII THE COMPLEX OF SECOND DEGREE' A. Tangent and Polar Linear Complexes........46 B. Intersection of a Pencil of Circles with Cs..... 48 C. Generalization of Pliicker........... 49 BIBLIOGRAPHY......49 INDEX iii THE GEOMETRY OF CIRCLES ORTHOGONAL TO A GIVEN SPHERE BY CHARLES SAVAGE FORBES INTRODUCTORY CONSIDERATIONS It is well known that the progress of geometry in the past century has been due in large measure to recognition of the fact that the choice of element is arbitrary. To one trained in the Pliicker line geometry, the conception of space as an assemblage of lines becomes both convenient and natural. We have chosen the circle as an element because in the domain to which we invite the reader's attention the circle is the natural element. We treat of the assemblage of circles orthogonal to a given sphere. At the outset we regard this system as a part of ordinary space, but once the foundations are laid we shall think and move in a space of circles. Koenigs has roughly outlined the six dimensional theory of circles in space. Our space is related to his much as the plane is related to ordinary point space. We have restricted our domain, but have thereby rendered possible a more searching inquiry. The geometry of circles orthogonal to a given sphere is closely related to the Plucker line geometry. Both are four-dimensonal, and like the geometry of the planes of a point in space of four dimensions each possesses a self-reciprocal element. The circle is dually regarded as an envelope of spheres, or as a locus of pointpairs. The sphere and the point-pair are reciprocal notions, and the properties of configurations of circles are stated dually in terms of these fundamental concepts. Two sets of six circle coordinates arise, and a linear relation between either set defines a complex of circles. Point-pairs and spheres are correlated by each complex as pole and polar, and either contains all the circles of the complex belonging to the other. Hence the circles of a complex lying upon a sphere constitute a pencil of circles. Each circle has a conjugate circle with respect to every complex. Two complexes intersect in a congruence and three complexes in a circle surface, closely related to the hyperboloid of one sheet. The geometry of families of spheres is found similar to that of systems of parallel planes. The linear transformation of the spheres of reference, the transformation by inversion, and the projective transformations are utilized; and there is set up a geometry of polar and tangent linear complexes with respect to a complex of second degree. 1 2 C. S. FORBES: THE GEOMETRY OF CIRCLES I am indebted to Professor C. J. Keyser of Columbia University for the suggestion of the topic and much kindly criticism. CHAPTER I INTRODUCTORY THEOREMS UPON ORTHOGONAL SPHERES AND CIRCLES In this chapter we establish certain elementary theorems upon which the subsequent theory is based. THEOREM I. There are oo planes in three dimensional space. The equation of a plane is Ax + By+ Cz+1 =0, which involves three parameters, and gives to the plane three degrees of freedom. THEOREM II. There are oc6 circles in three dimensional space. The equation of a circle in a plane involves three parameters. No two planes have a circle of finite radius in common. Hence (Theorem I), there are 003 X 03= 006 circles in space. THEOREM III. There are 004 spheres in three dimensional space. The equation of a sphere, x2 + y2 + z2 + 2ax + 2by + 2cz + d= 0 involves four parameters. DEFINITION I. Two spheres are orthogonal, when the square of the distance of their centers is equal to the sum of the squares of their radii. THEOREM IV. There are oo3 spheres orthogonal to a given sphere. Definition I imposes one condition upon the four parameters of the arbitrary sphere, leaving to it three degrees of freedom. DEFINITION II. Two intersecting circles are orthogonal when their tangents at the point of intersection are perpendicular. DEFINITION III. A circle is orthogonal to a sphere when it is orthogonal to every great circle of the sphere passing through one of the points of intersection. THEOREM V. A circle orthogonal to two great circles of a sphere at their point of intersection is orthogonal to the sphere. For the tangents to the great circles through the point of intersection all lie in a plane tangent to the sphere at the point. The tangent of the circle being perpendicular to two of these tangent lines is perpendicular to the plane and therefore to them all. Furthermore this tangent line being perpendicular to a ORTHOGONAL TO A GIVEN SPHERE 3 tangent plane at the point of tangency goes through the center of the sphere. A circle and its tangent line lie in a same plane. From this follows: THEOREM VI. Every circle orthogonal to a sphere, lies in a plane passing through the center of the sphere. THEOREM VII. Every sphere passing through a circle orthogonal to a sphere is orthogonal to that sphere. Let C be a circle orthogonal to a sphere S. Pass a plane II through C and point 0 the centre of S (Theorem VI). II cuts out of S a circle C', orthogonal to circle C (Definition III). Let A be the centre of circle C. Let P be one of the points of intersection of circles C and C'. Draw the tangents OP and AP to C and C'. These tangents are perpendicular. Pass any sphere S' through circle C. The center O' of S' is in a perpendicular A O' to the plane I. O'P is therefore perpendicular to OP. Hence o o'= op2 + op-2. But 00' is the distance of the centres of spheres S and S'; and OP and O'P are their respective radii. Therefore S and S' are orthogonal (Definition I). We next find the analytical condition (rectangular coordinates) that the spheres (s ) x2 + y2 + z2 = R_, (S) + +^ \ (S') x2 + y2 + z2 + 2ax + 2by + 2cz + a2 + b2 + c2=r2, shall be orthogonal. The condition is (Definition I) a2 + b2 + c2 R2 + r2. Whence r2= a2 + 2 + 2 _ 2 and S' becomes x2 + y2 + z2 + 2ax + 2by + 2cz = - R2. Hence the result: THEOREM VIII. A sphere S' is orthogonal to a sphere S, whose center is the origin, when the absolute term of S' is equal to the square of the radius of S. THEOREM IX. Every plane passing through the center of a sphere S, cuts out of a second sphere S', orthogonal to S, a circle C', orthogonal to the circle C cut out of S by the plane. Let the equations of the spheres be (S) x2 + y2 + 2 = R2 S') x2 + y2+z2 + 2ax + 2by + 2cz = - R2 4 C. S. FORBES: THE GEOMETRY OF CIRCLES Since the property is geometrical, it is independent of the choice of axes. The cutting plane may therefore be taken as z=0. This gives the circles (C) 2 + y2 = R2, (C') X2 + y2 + 2ax + 2by = - R2. These circles are orthogonal, for they lie in a plane, and a2 + b2 R2 a2 + b2 _ R2. That is, the square of the distance of their centers is equal to the sum of the squares of their radii. Hence their tangents at the points of intersection are perpendicular, and they are orthogonal (Definition II). THEOREM X. Two spheres orthogonal to a same sphere intersect in a circle orthogonal to that sphere. Let the spheres (St) x2 + y2 + z2 + 2ax + 2by + 2cz = - R2 (") x2 + y2 + z2 + 2a'x + 2b'y + 2c'z= - 2, be orthogonal to (S) x2 + y2 + 2 = R2 The intersection of S' and S" is a circle C lying in the plane (II) (a-a')x +(b- b')y+ (c —c') =O. This plane II passes through the center, 0, of sphere S. Let P be one of the points where circle C intersects sphere S. Pass a plane II' through P and 0, perpendicular to plane 11. II' cuts out of sphere S a circle C' orthogonal to C (Definition II). We have the Lemma: Every circle lying in a plane which passes through the center of a sphere is orthogonal to a great circle of that sphere at each of its points of intersection with the sphere. C is also orthogonal to circle C" cut out of S by 1 (Theorem IX). C being orthogonal to two great circles C' and C" of sphere S, is orthogonal to S (Theorem V). Theorem X may be analytically established as follows, whether the circle of intersection be real or imaginary. The equation of the fundamental sphere is (8) x2 + y2 + 2 = R2. ORTHOGONAL TO A GIVEN SPHERE 5 Any two spheres orthogonal to S are given by (S') 2 + y2 + z2 + 2a'x + 2b'y + 2c'z + R2 = 0, (S") 2 + y2+z2 +2(a+ a')x + 2(b + b')y + 2(c + c')z+ R2= 0. The axial plane of S' and S" is (n) ax + by+ cz = 0, and of S and S' is (H') a' + b'y + c' + R2 = O. The intersections of II, II' and S are the points of intersection of the three spheres, i. e., the points where the circle (S', S") pierces S. Solving simultaneously for x', y' and z' the coordinates of these points we obtain, [c(c'a-ca') + b(b'a - ba')]R2 i R(c'b-cb' )V/F F' [a(a'b-ab') + c(c' b-cb')]R2 - R(a'c - ac ')VF y F'' a- F' where F = (ac' - a'c)2 + (b' - b'c)2 + (ab' - a')2 - R2(a2 + b2 + c2), and T = (ac' - a'c)2 + (bc' - b'c)2 + (a'b - ab')2. It is observed that F. and Ft are symmetric in a, b, c, a', ', c'; that x', y', z' are finite (for finite a, b, etc.) unless a:b: c =a': b': c'; and that x', y', z' are real or imaginary according as F_ > 0, or F8 < 0. The tangent plane to S at (x' y' z') is a ( - ) + a,,a(Y-Y) + (Z-Z)=. ax/(X- )+ayf(v- )+~,(Z~- )=:09 or since,2 +y,2 + 2 R2 (II") xx' + yy' + zz' = R2. This plane contains the tangent lines of all great circles of S, passing through (x' y' To prove a circle orthogonal to S, itlis only necessary, therefore, to prove the tangent to the circle at the point (x' y' z' ) perpendicular to this tangent plane at (xy'z' ). The plane is real onlyCwhen F = 0. The tangent to the circle of intersection of S' and S" at (x'y'z') is the intersection of plane 6 C. S. FORBES: THE GEOMETRY OF CIRCLES II and the tangent plane of one of the spheres, say S', at the point. This tangent plane is a' (x-x)+ y a' (y-y')+ a' (z- ')=0. or (X? + a')(x - x') + (y' + b')(y - y') + (z' + c')(z - z')= O. We have the relations,2 y,2,2. R2 a'x'+ b'y'+ c''= - R2. Hence the above reduces to (I+ a')x + (y'+ b')y + (z'+ c')z = 0. Since both this plane and plane II pass through the origin and (x'y'z'), their line of intersection does the same, and is perpendicular to the plane II", tangent to S. But this line is the tangent of the circle (S', S") at the point of intersection with S. Hence circle (S', S") is orthogonal to every great circle of S passing through (x'y'z') and is orthogonal to S. At no point in the argument have we assumed that (x'y'z') was a real point. Hence Theorem X holds for both the real and imaginary cases. THEOREM XI. Every circle orthogonal to a sphere at one point of intersection is orthogonal at the other also. This follows at once from the fact that the circle lies in a plane passing through the center of the sphere (Theorem VI), and from the symmetry of the construction. THEOREM XII. There are o4 circles orthogonal to a given sphere. Theorem V imposes two conditions upon the six parameters of the circle, leaving four degrees of freedom. Again there are Co2 points on the sphere. Through each point P and the center of the sphere, oo planes may be passed. In each of these planes oo circles may be drawn through P, orthogonal to the intersection of the plane and the sphere, and hence orthogonal to the sphere (Definition III; also see proof of Theorem X). Therefore through any point on the sphere, oo2 circles may be passed orthogonal to the sphere. Hence, since each circle cuts the sphere in a finite number (2) of points, there are o02 X 002 = o04 circles orthogonal to a given sphere. THEOREM XIII. Two points on the sphere fix one of the orthogonal circles. Pass a plane through the points P and P', and the centre of the sphere. In ORTHOGONAL TO A GIVEN SPHERE 7 this plane one and but one circle may be passed through P and P', and orthogonal to the circle formed by the intersection of the plane and the sphere. THEOREM XIV. All those circles orthogonal to the same sphere, which have one point in common, have a second point in common related to the first point by inversion with respect to the sphere. Let the circles C' and C" be orthogonal to a sphere S, and intersect in a point P. P may be taken without the sphere. Let S' be any sphere passing through C' but not through C". Spheres S and S' are orthogonal (Theorem VII). Circle C" lies in a plane II drawn through P and 0, the centre of S (Theorem VI). This plane cuts out of sphere S' a circle C"' orthogonal to sphere S (Theorem XI; Lemma, Theorem X). Draw OP. Plane II cuts out of S a circle C. C" and C"' therefore lie in the same plane II with circle C and are orthogonal to it. C"' cuts OP a second time at B. Therefore by the theory of orthogonal pencils of circles * C" also passes through B. That is P and B are reciprocal poles with respect to the circle C. Circle C' lies in the same plane with OP and therefore cuts OP a second time. But C' lies on sphere S' and therefore must cut OP in the point in which OP cuts S'. This point is B since C"' lies on S' and passes through B. Hence C' and C" intersect a second time in B. We next establish that the analytical relation between the points P and B is that of inversion with respect to the sphere S. Assume 0, the center of 8, as the origin. The equation of S is x2 + y2 + z2 = R2 Since the configuration is independent of a rotation of axes, the plane II may be assumed as _= 0. The circle C cut from S by II is x2 + y2 = R2. Let OP be the axis of X and P be the point (K 0 0). Since the circle C"' cut from S' by II, is orthogonal to C, its equation is of the form x2 -+ y - 2ax - 2by - - 2. Since C"' passes through the point (K 0 0), K2 2aK= -_ 2, i. e., K2 4- I-2 * KLEIN, Einlitung in die hiohere Geometrie, vol. I, p. 87. 8 C. S. FORBES: THE GEOMETRY OF CIRCLES The equation of C"' is therefore K2 + R2x y2 2y= R K X + - 2by =-_. This circle cuts the axis of X in the points (P and B) (KO 0) and (R2/K 0 0). Since the coordinates of B do not involve the coefficient b of the equation defining C"', it is seen that P and B are uniquely related, i. e., since any two circles orthogonal to S and intersecting in P, also intersect in B, all such circles passing through P will also pass through B. The coordinates of P and B satisfy the formulae of inversion X 'R2 y'R X - 2,2 -,2,2 x +y x +y and are therefore reciprocal poles with respect to the circle C. Since the configuration is independent of a rotation of axes, the relation between P and B is unaltered by a rotation of plane II about OP as axis. P and B are therefore reciprocal poles with respect to the sphere, and their coordinates satisfy the formulae of inversion If X'R2 y'R2 'R2 X+y+z x -,2 +, +,2 y -,2 +, 2 - 2 + y,2 +y 2 For it has been shown that if OP = K, then OB = R2/K. That is OP x OB = R2. Now let P be the point (x'y'z'). The point (x"y"z") lies upon OP. For the equation of the line OP is xy z yx' - zx'y'. Substituting the values of the coordinates of (x"y"z"), we have the identity xy'zIR2 x'y'z'R2 x'y'z'R2,2 2 2, 2 +2,2, t 2,2,2,2' x +y +z +y + +z x +y +z Furthermore the distance OP = Vzx2+ y,2 + 2'2; and if (x"y"z") is denoted by B, (x'2 + Y2 +Z2)2 OB =,2,2 2 X + +Zy..OP X OB=R2. ORTHOGONAL TO A GIVEN SPHERE 9 Hence to every point without a sphere S there corresponds a point within. These two points are related by the formulae of inversion. If two circles orthogonal to the sphere S pass through a same point without the sphere, they pass through the corresponding point within. Two circles orthogonal to a same sphere, therefore intersect in two points or not at all. No two such circles can intersect in two points without or within the sphere, else they intersect in two other points corresponding to these two, and therefore coincide. It is readily seen that the relationship between corresponding points is reciprocal. In other words throughout the argument the terms point without and point within may be interchanged. Particular attention is called to Theorems X, XII and XIV, which are of especial consequence in the subsequent chapters. We repeat these theorems for the convenience of the reader. THEOREM X. The intersection of two spheres orthogonal to a given sphere is a circle orthogonal to that sphere. THEOREM XII. There are QQ4 circles orthogonal to a given sphere. THEOREM XIV. All those circles orthogonal to a same sphere, which intersect in one point, intersect in a second point the reciprocal pole of the first point with respect to the sphere. It is noted in passing that Theorem X is but a generalization of the theorem: The intersection of two planes perendicular to a third plane is a line perpendicular to the third plane. We pass from these preliminary investigations to the main body of our paper, and shall henceforth make exclusive use of systems of circle-co6rdinates derived from the Darboux pentaspherical coordinates. CHAPTER II COORDINATES OF THE CIRCLE A. Pentaspherical Coordinates 1. Choose five mutually orthogonal spheres 81, 92, S3, 94, 8S. Let their radii be R1, R2, R3, /R, R5. Any point is determined by five Darboux * pentaspherical coordinates, X1, x2, X3, X4, X5; where P. * '=U Pr (i=1,, 2u,.5); * DARBOUX, Theorie des Surfaces, vol. 1, book II, chap. 6. 10 C. S. FORBES: THE GEOMETRY OF CIRCLES Pi denoting the power of the point with respect to the sphere S, and p being an arbitrary factor. The five coordinates are homogeneous, and are connected by the identity 2 2 3 2 2 (1) + 2 + + + _ 0. Any equation of the form 5 (2) a1 1+ a22 + a33 + a4x + ax, C aix = 0, where the quantities a. are constants, represents a sphere. * The equations of the five fundamental spheres of reference, S!, are (3) x =0, x2 0, x3 -, x,=0, 5= 0. These spheres intersect in ten circles (4) Xi. = = 0 (i,j=1, -, 5; ij); and in ten pairs of points (5) x.= = Xk=- 0 (i,j, k=l,.., 5; i+j+k). 2. The condition that two spheres 5 5 (6) ax = o, Eb.x. = 1 1 shall be orthogonal is (7) alb ab + ab2+ b b + ab, + aab5 = 0. 3. We are considering the geometry of circles orthogonal to a given sphere. The choice of the five fundamental spheres, Si, allows that the given sphere be taken as any one of the five. We choose to take it as S5, and to give it a real positive t radius. Although this latter restriction is of no advantage algebraically, it serves to clarify the geometrical conceptions. 4. The equation of a sphere orthogonal to S5, is of the form (8) a, a x+ 2 + a33 + ax aixclx 0. [See (3) and (7).] There are oo3 such spheres, since (8) contains three parameters (Cf. Theorem IV). Any two such spheres intersect in a circle, real or imaginary, orthogonal to S^. This has been proved in Theorem X for both the case where the circle of intersection is real, and where it is imaginary. There are oo4 circles orthogonal to S, (Theorem XII). *Cf. KOENIGS, Contributions 4 la theorie du cercle dans l'espace, Annal es * Toulouse, 1888, p. 1; DARBOUX, op. cit. t One of the five spheres must have a negative radius. See KLEIN, Einleitung, vol. 1, p. 102. ORTHOGONAL TO A GIVEN SPHERE 11 Although there are six parameters involved in the equations of the two spheres intersecting to form the circle, the latter can be thus generated in 002 ways. All the circles orthogonal to S5 may be generated by the intersections of the spheres orthogonal to S5. 5. Since we treat exclusively of the spheres and circles orthogonal to 5., we shall use the terms sphere and circle to mean respectively, sphere orthogonal to S5, and circle orthogonal to S5. The configuration of reference consists therefore of the four mutually orthogonal spheres 1S, S2, S3, 4S, i.e., x= 0 (i=1,...4). These spheres intersect in six circles xi = x.= 0 (i,j- 1,.4; i+j), and in four pairs of points, Xi= Xj = =0 (ij, k==l,...4; i=j+=k). Attention is called to the similarity between this tetrasphere of reference and the tetrahedron of reference in line geometry. The following notions correspond. TETRAHEDRON TETRASPHERE 4 faces planes spheres 6 edges lines circles 4 vertices points pairs of points It is noted that the four spheres of reference belong to the assemblage of spheres orthogonal to S5, but that S5 itself does not. The coordinates of a point referred to this configuration are tetraspherical. B. Point-pairs 6. Three spheres of the assemblage intersect in two points, one within and one without S5 (Theorem XIV). All those circles (or spheres) which pass through one of the points pass through the other also. Conversely, if two circles of the assemblage intersect in two points, these points are related by inversion. We denote two points so related as a point-pair, or simply as a point. The pentaspherical coordinates of the point-pair may arbitrarily be taken as those of the outer element, but we now show that inasmuch as a point is fixed by the ratios x,: x2: x3: x4, it makes no difference whether the inner or outer element is selected. Let (x'y'z') and (x"y"z") be the elements of a point-pair. We have by the formulae of inversion with respect to the fundamental sphere 12 C. S. FORBES: THE GEOMETRY OF CIRCLES (A) the relations,, tX'R2 - 2,2,2 x +y + x2 + y2 + R2,! y'R2 Y,- 2,,2 +,2, x+y +z z' R2,2,2 +,2 -x +y + z Let Si, one of the fundamental spheres, not S5, be 2 + y2 + Z2 + 2ax + 2by + 2cz + R2= 0. Si is orthogonal to 85. With reference to (x'y'z') p x2 + Zy 2 + 2ax' + 2by' + 2cz' + R2 xi=P R P R ( i-,.-., 4; p arbitrary). With reference to (x"y"z") (x,2, y2 + Z,2) 2ax' + 2by' + 2cz' ('2 4- + 2 ) +- Y + Rz i =iP -7 R2 -P x2 +,y2 + -,2 x 2 +y 2 2 + 2ax' + 2by' +2cz'+ BR2 Ri2 Hence we have the relations (9) R2, X. + i,-, 2,2,2,;i Z + y +z /2,,2 +,2 72 P,2?+y - p x p5 - p -y-2,2- ^ t x5= PR -=p R,2,2,2 X1 + y2 + R- ( + +^ +y __ + n A(X f + y12 + Z2) -.5 i. e., (10) F R R2 — P,2,2 +,2 X +y +%,2, 2,2 2 x + y + 'z R I R2 5 -,2,2,2 5? 4/y +,2 x;y + +7 The theorem follows: The pentaspherical coordinates of the elements of a point-pair are respectively proportional, save that the coordinates x5 differ in sign, i. e., x:x:x:x:x x: x *:x 4* 5 *X1 2 *S X3 4 X ' XC ' — X * X5 ' 2 3 X4 *- X5' The coordinate x% of the inner point is always negative.* For if the inner point is (abc) P5 = a2 b2 +c2- 2 < 0, * For p positive..'. 5 =P pR < 0 ORTHOGONAL TO A GIVEN SPHERE 13 Therefore to every point-pair there is a definite set of ratios x,: x2: x3: x4 and conversely, to every set of these ratios, there is a definite point-pair. The 5 coordinate x5 may be determined from the identity xs = 0, the sign being i according as the f oter according as the { iner } element of the point-pair is taken. CORRESPONDENCE I. To every point without a sphere S5 corresponds a unique point within, and reciprocally. To the center of the sphere, however, corresponds the plane at infinity and reciprocally. If the radius of the sphere is zero, the sphere is a point which corresponds to every point of space. 4 C. Dual Interpretation of E atxi = 0 1 4 7. The equation L aix = 0, may be interpreted in a dual sense. If we regard 1 the quantities at as constant, and the quantities x. as variable, equation (8) represents a sphere as the locus of its points. By virtue of the identity (1), a point is known when four of its coordinates are given. (Cf. Section 6.) Three points x1, * * *.,4, x1, * * '.., 4, *.. 4 X determine the sphere, since they determine the ratios of the quantities ai uniquely, provided the points are not on a same circle, i. e., x X2 X3 X4 (11) X1 X2 X3 X4 40. X1 2 X3 X 4 The same sphere is equally well determined by any three points, whose coordinates are of the form (12) X = XXI + X2XI+ X3x' (i=1, ~.*, 4;;1,;2, X3 arbitrary). Conversely every point whose coordinates are of the form (12) is on the sphere.* Reciprocally if the quantities x; are regarded as fixed, equation (8) represents a point-pair xc as the envelope of the spheres which pass through it. Three sets of sphere coordinates a,,... a49 a,,, a4... al, '"i ad al i a al, * ( A, a4 fix the point, provided the three spheres do not pass through a same circle, i. e., al a a3 a4 1 2a4 (13) a, a2 a,3 a 4 0. _______ Ia1 2 a3 a4 a If af" alff a' * Cf. KEYSER, Plane Geometry of the Point in Space of Four Dimensions, pp. 311-313; KRONIGS, La Geometrie Reglee, Annales-Toulouse, 1889, p. 11, et al. 14 C. S. FORBES: THE GEOMETRY OF CIRCLES The point is equally well determined by any three spheres whose coordinates are of the form (14) a = X 1a + X 2a + X3ca (i = 1, ~ 4; x,, 2,, a3 arbitrary). Conversely every sphere whose coordinates are of the form (14), passes through the point. D. Coordinates of Circle as Envelope of Spheres 8. All the spheres passing through a given circle constitute a pencil of spheres. If the circle is given by the intersection of the spheres aixi =O, E bx= 0 the coordinates of the spheres of the pencil are of the form a = X ai + X2 b (a1, a2 arbitrary). The circle is equally well defined by any two spheres of the pencil E(ai+ Xbi)Xi = (a = =2 ). 9. In particular it is defined by any two of the four spheres of the pencil, which are orthogonal, each to one of the four fundamental spheres x,= 0, x2=, x= 0, x4 O. The equations of these four spheres of the pencil are + P2x2 +P 3 X3 + P14 4= 0, (15) P21X1+ * +P23X3+P24X4-= P31X1 +P32X2 + * + 34 X4 0, P2X1+P42x2+P43x3+ * = 0, where = abk - ab (, k=l,..., 4). Obviously Pii =; Pik= -Pki Expanding the determinant al a2 a3 a4 b, b3 b4 = 0, a, a2 a3 a4 b_ b2 b3 b4 *Cf. KOENIGS, La Geometrie Reglee, Annales -* Toulouse, 1889, p. 6. ORTHOGONAL TO A GIVEN SPHERE 15 in terms of its second order minors, we obtain (16) w (p) P)12P34 + P13P42 +P P14~23 0 Although two of equations (15) are sufficient to determine the circle, all four are retained for the sake of symmetry, and the six * coordinates Pk are taken as the homogeneous coordinates of a circle orthogonal to the fundamental sphere S5. The identity (16) reduces the degree of freedom of the system to four as is required by Theorem XII. E. Coordinates of Circle as Locus of Point-Pairs 10. Reciprocally two point-pairs axi = O, Eaiyi = 0 determine a circle, and the same circle is determined by any two points of the range ( xi + Xyi ) ai = 0 (]A arbitrary). (By range of points is meant all the point-pairs on a circle which contains the generators of the range.) In particular, the circle is determined by any two of the four points of the range lying on the fundamental spheres. The equations of these points are * + q12 2 + q,3 a3 + q4 a4 = O, a?21, + * + 23a3 + q24a4 = - q31l + q3a2 + * + q34 a4 =O, q41 al + q42 a2 4+ q 3 + *= 0, where 9iR = ZiY,- XyYi (i, k=,..., 4). qii = 0, qik = - qk' The identity exists (18) co() ql12?34 + q13242 + q14?23 = 0~ 11. As shown in the precisely similar algebra of line geometry,t if Pik and qjk are the coordinates of a same circle, (19~) \P12 13__ P14 _ 34 P42 P23 q34 q42 q23 q12 913 q14 * Cf. PLUCKER, Neue Geometrie des Raumes, p. 2; CAYLEY, Six Coirdinates of a Line, Collected Works, vol. 7, p. 66. t PLiCKER, Neue Geometrie, p. 5. KOENIGS, La Geometrie Reglee, Annales.. Toulouse; 1889, p. 8. 16 C. S. FORBES: THE GEOMETRY OF CIRCLES Hence six quantities 7,i such that 712 = PP12 - 'q34 713 = PP13 = uq42, etc. (p, a arbitrary ), may be assumed as the homogeneous coordinates of a circle, without regard to the method of generation, or if the coordinates are taken in a certain order they give the circle as an envelope of spheres, or if the same coordinates are taken in a different order the circle is regarded as the locus of its points (i. e., point-pairs). Conversely, given six quantities Y7k, which satisfy the identity o(7)=O, they may be taken as the coordinates of a circle, and be interpreted in a dual sense according to the order in which they are taken. F. Conditions that Two Circles may Intersect 12. Two circles intersect * when their generating spheres (points) have a point (sphere) in common. Let the given circles be 721 X1+ * +7233 + 724x4= 0 These four spheres (points) have a point (sphere) in common, provided 0 712 713 714 (22) 721 0 723 724 -. ~. 12 713 714 721 ~ 723 724 By the aid of 7ik = - 7ki and w(7)= 0 this reduces to (23) w)(, 7' ) 712 734 + 713 742 + 1 + 71 + 2734 + 7r1342 + 71423 = This is therefore the condition that the circles 7r, and 7lk shall intersect in a point-pair, or dually shall lie upon a same sphere. *Cf. KOENIGS, La Geometrie Reglee, op. cit., p. 9. ORTHOGONAL TO A GIVEN SPHERE 17 G. Conditions that a Given Circle may lie on a Given Sphere or pass through a Given Point-pair 13. We seek the condition that a given circle (envelope of spheres) may lie upon a given sphere. Let the sphere be given by (24) cx1 + C2X2 + C3X3 + C4x4 = 0; and the circle by (25) a'x, + a2 2 + a3 3 + a4 = 0, (25) bx b2 + b b3x + b4xa= 0. The condition is that the sphere is one of the spheres of the pencil of spheres, which is defined by (25), i. e., ci =1 ai + 2 bi ( 21, 2 arbitrary), This involves the two conditions a1 b1 c1 (26) a2b2c2 =0. a3 3 c3 a4 64 c4 14. Reciprocally, the condition that a circle (locus of points) given by two points shall pass through a third point, is that the point shall belong to the range defined by the two points which give the circle. Let the point be (27) alZl + a2Z2 + a33 + a4z = 0, and the circle be r a,x + ax2 + ax3 + a4,x, 0, ~(28) r alYl + a2Y2 + aa3 + a44 = 0. The condition is zi X xi + \2yi (1, 2 arbitrary). This involves the two conditions X, y1 1 (29) =2 Y2 2 =0. X3 Y3 Z3 c4 Y4 z4 15. The condition that a given circle locus of points shall lie on a given sphere, is that both points which define the circle shall lie on the sphere. If the sphere be ax, + a22x + a3x. + ax = 0, 18 C. S. FORBES: THE GEOMETRY OF CIRCLES and the points be y, and zi, the conditions are (30) i aly, + a2Y2 + a3y3+ a4 Y4 0 al1 z + a2z2 + a 3 + a44 O. 16. Reciprocally, the condition that a given circle, envelope of spheres shall pass through a given point, is that both spheres which define the circle shall pass through the point. If the point is ai,1 + a 22 + a;x + a4, = 0, and the circle is defined by the spheres bi and c, the conditions are ~(31) I 1~~f blZ, + b,x, + bx,3 + b4x 4 0 c1x + c2x2 + C3 3 + c44 = 0. 17. The last four sections exhibit the similarity between the reciprocal algebras of the circle regarded as an envelope of spheres and as a locus of points. The necessary and sufficient conditions for the union of the circle and sphere, and the circle and point in the one theory, are algebraically identical with the necessary and sufficient conditions for the union of the circle and point, and the circle and sphere in the reciprocal theory. CHAPTER III THE LINEAR COMPLEX OF CIRCLES 18. Equations of relation between the coordinates of the circle serve to select special systems of circles from the oo4 circles of the space. The system defined by a single equation is called a complex. If the equation be linear, the complex defined by it is called linear. A. Linear Complex of Circles Regarded as Envelopes of Spheres 19. The equation of a linear complex of circles regarded as envelopes of spheres may be written * (32) A12P2 + A3p13 + A14p14 + A2323 + A42p42 + A3434 = 0. Expanding, we obtain A,2 (a,b2- a2bl) + A3(ab3 a3b,) (33) '+ A14(a,b4 - a4b,) + A3(a2b3 - a3b2) + A42(a4b,2- a2b,) + A34(a364 - a4 b) = O. * Cf. PLUCKER, Neue Geometrie, p. 27. ORTHOGONAL TO A GIVEN SPHERE 19 Rearranging according to terms in a, we obtain (+ A12 b2 + A13b3+ A14 4)al + (- A12b1 + A23 b - A42b4) a (34) +- ( - A13bl- A23 b2 + A3 b4) 3 + (- A14b, + A42b2 -A3 b,)a4 =. If the quantities bi are fixed, they define a sphere (35) ob2 xi=0. If we regard the quantities a2 as variable, equation (34) is the equation of a point, the envelope of all the spheres of the circles of the complex, lying upon the sphere (35); in other words all circles of the complex lying upon the sphere (35) pass through the point (34). Hence, to each sphere of space corresponds with respect to each complex one point. The point lies upon the sphere since (36) ai- b satisfies equation (34). Therefore All the circles of the complex lying on a sphere pass through a point of the sphere and form a pencil of circles. B. Complex of Circles Regarded as Loci of Point-Pairs 20. Reciprocally, the equation of the complex may be written (37) A12q34 + A13q42 + A14q23, + A23 4 + + A42?q3 + A,34q12= 0. [See (19).] Expanding, we obtain A12(3Y4 - 4y3) + A,(x4y,2 - x2Y4) (38) + A14(xy, - xy2) + A23(1Y4 - X4YI) + A42( xy, - xy) + A,4(x1y, - xy) = 0. Rearranging according to terms in x., we obtain ( + A23Y4 + A42 Y3 + A34Y2) xl + ( - A134 + A143 - A341) X2 (39) + ( + A12y4 - Ay 1 2- A4 y, ) x3 + (- A12y3 + A13y2 A A23 Y) A xC = 0. 20 C. S. FORBES: THE GEOMETRY OF CIRCLES If the quantities yi are fixed they determine a point (40) Eby, =0. Then, if we let the quantities xi vary, equation (39) represents a sphere, the locus of the points of the circles of the complex, which go through the point (40), in other words, the locus of the circles themselves. The point (40) is on the sphere (39), since (41) xi = yi satisfies (39). Therefore To each point of space corresponds, with respect to each complex, a sphere passing through it, and all the circles of the complex passing through the point lie on the sphere. C. Polar Spheres and Pole Point-Pairs 21. We denote a point and a sphere related as in the last two sections as pole and polar with respect to the complex. CORRESPONDENCE II. Every complex correlates the point-pairs and spheres of the space, so that, with respect to the complex, to every point-pair there corresponds a sphere polar to the point-pair, which contains all the circles of the complex which pass through the point-pair, and to every sphere corresponds a point-pair, pole of the sphere, through which pass all the circles of the complex lying upon the sphere. CORRESPONDENCE III. A triple correspondence is set up by each complex between the point-pairs and the spheres of the space and the circle pencils of the complex, such that a point-pair, a sphere and a circle pencil correspond uniquely and are united in position. The pole of the sphere (42) b x.= 0 is, from (34), the point whose coordinates are Y1= A12b2 + A13b3 + A14b4, ~(43) Y2 = -- 12 bl + A23 3 - A42 4, - A13bl - A23b2 + A34 b4 Y4 = - A14 b + A42 b - A34 b3. The polar of the point yj is the sphere whose coordinates are [b = A23Y4 + A42y3 + A34Y2, (44) -A 4 + A43-A341[See (39). b3- A12Y4 - A,42 - A421, b4 = - A12y3 + A13y2 - A23y1. ORTHOGONAL TO A GIVEN SPHERE 21 Let (45) o(A) =A A2A34 + A,13A4, A 14 A23. If the last three of equations (43) be multiplied respectively by A34, A42 and A23, and the results be added, we find - w(A) b = A34y2 + A42y3 + A23y4. Similarly we obtain - (A) b = -A34y + A14y - A3y4, - w(A)b3 =- - A - y2 + A12Y4, - ( A )b4= - A23y1 + A13y, - A12Y3. Since only the ratios of the quantities bi are of concern, these equations are equivalent to (44). Similarly (43) may be gotten from (44). D. Second Proof of Fundamental Correlation 22. The notion of pole and polar with respect to a complex, is so important that we give another proof of this fundamental correlation. The equation of a linear complex is * (46) Aikpik 0 (i, k- 1,..., 4, i k); ik7 those of two spheres are (47) axi= 0, ixi = 0. These two spheres, S and S', intersect in a circle, whose coordinates are ik = ai bk -a7-bi' The condition that this circle belong to the complex is that its coordinates satisfy equation (46), i. e., (48) Aik( ai b - a b,) = 0, ik or (49) Z(bEAka,,)=O, i k provided the summation is made under the restrictions (50) A — Ak; A.. =O. Consider another sphere 8", (51) lii = 0, where * Cf. KOENIGS, Contributions, etc., Annal es *Toulouse, 1888, p. F. 12. 22 C. S. FORBES: THE GEOMETRY OF CIRCLES (52) l = ik al k The equation [see (49)] (53) E(bi Aik ak) - Ebl -= 0, i k expresses that the spheres S' and S" are orthogonal. Furthermore, from (48) and (49), (54) E(aiEka) - Ela =O, i k i. e., the spheres S and S" are orthogonal. The circle of intersection of S' and S, is therefore orthogonal to S"; but the equation of S" E (E Aik ak)Xi = 0, i k is independent of bi, and in general different from 58. Hence the theorem: All the circles of a linear complex which lie upon a sphere S are orthogonal to a second sphere S", which is called the conjugate of S with respect to the complex. Hence the circles of the complex lying upon S are orthogonal to two spheres S" and S5. These circles therefore lie in planes, which pass through the centers of both spheres (Theorem VI). These planes form an axial pencil, whose axis (i. e., the line joining the centers of S" and S,), cuts S in two points. The circles of the complex lying upon S all cut the axis twice and therefore must pass through these two points. The theorem follows: All the circles of the complex which lie on a given sphere pass through a point-pair, the pole of the sphere. The pole of a sphere (55) aiZx, = 0 may be found as follows: Construct the sphere S" (56) E ( E Aiak) xi 0 i k and join its center to the center of S5. The intersections of this line with (55) are the point-pair, pole of (55). 23. It is seen that, in general, but one of the circles of a pencil of circles lying upon a sphere will pass through the pole of the sphere, and belong to the complex. We now prove * that if the circles of a complex, whose equation is (57) F(pk), *Cf. PLUCKER, Neue Geometrie, p. 18. ORTHOGONAL TO A GIVEN SPHERE 23 are so distributed that but one in general of the circles of a pencil belongs to the complex, then F is a linear function. Let XlPik + X2Pik (, 2 arbitrary ), be the coordinates of the pencil. The condition that but one of the circles of the pencil shall belong to F is that the equation F(X1pic + \X2k) = ( for some X: X2) shall be of first degree in X: \X. F is therefore of the form A.ikpi = 0. E. Comparison between Circle Geometry and Associate Theories 24. Analytically the two theories are seen to be identical. Transition from one theory to another is effected by the following exchange of notions: ASSEMBLAGE OF CIRCLES ASSEMBLAGE OF LINES Circle Line Sphere Plane Point-pair Point Each system is a four-dimensional assemblage. A circle is determined by two spheres or two point-pairs, just as a line is determined by two planes or two points. The analytical condition that two circles or two lines intersect is the same. The polar theory of sphere and point-pair is precisely that of plane and point. The oo circles of a complex lying on a sphere constitute a pencil, as do the oc lines of a complex lying in a plane. The relationship can be shown more clearly however, and an actual correspondence be set up between the lines of space and the circles orthogonal to a sphere by means of the double-point geometry of COSSERAT.* He takes as element a pair of points upon the surface of a sphere, and as configuration of reference four circles upon the sphere. This double-point has six coordinates pi,, and if the four circles of reference be taken as the intersection of our four spheres of reference, S., with S5, the coordinates pik of the double-point of COSSERAT are precisely the coordinates of the orthogonal circle of S5 fixed by the double-point (see Theorem XIII). Furthermore, COSSERAT shows that if the four centers of the spheres S be taken as the vertices of a tetrahedron of reference the coordinates Pk of the line through the double-point are, save for a factor of proportionality, the coordinates pik of the double-point. Hence the coordinates of a line and of an orthogonal circle through the same double-point differ only by a factor of proportionality. * COSSERAT, Sur le cercle comme un element generateur de l'espace, A n n a 1 e s - T o u os e, 1889, E 40. 24 C. S. FORBES: THE GEOMETRY OF CIRCLES CORRESPONDENCE IV. To every sphere orthogonal to S5 corresponds uniquely the circle of intersection in double-point geometry, and the plane of that circle in line geometry. To every circle orthogonal to S5 corresponds uniquely a double-point upon S,, and the line through the double-point. Furthermore with respect to any complex the pole of any plane is the point where it is cut by the line joining the elements of the point-pair, pole of the corresponding sphere with respect to the circle complex whose equation is the same; and reciprocally. Care must be taken not to confuse point-pair and double-point. The former is two points, one within and one without the sphere 85, and uniquely related by the formula of inversion with respect to S5; the double-point is a pair of points upon the surface of S5, either of which may be again paired with any of the 0o2 points of the surface. There are oo4 double-points and but oo3 pointpairs. 25. It is of interest to note that the plane * geometry of the point in space of four dimensions belongs to the same family of reciprocal, four-dimensional geometries, as does both line geometry and the present theory of orthogonal circles. Transition may be effected by the following exchange of notions: ASSEMBLAGE OF CIRCLES ASSEMBLAGE OF PLANES Circle Plane Sphere Line Point-pair Lineoid We return to the development of circle geometry, bidding the reader keep in mind the associate theories to which we have called attention. CHAPTER IV CONJUGATE CIRCLES, POLE POINT-PAIRS AND POLAR SPHERES A. General Theorems 26. If a point and a sphere are united in position, their respective polar and pole are united in position.t For let x' and a' be respectively a point and a sphere, such that (58) ~a"x =O. *KEYSER, Plane Geometry, etc., op. cit. tCf. KOENIGS, La Geometrie Reglee, chap. 2. MOBIUS, Uber Figuren im Raume, C rel e' s Journal, vol. 10, p. 321. ORTHOGONAL TO A GIVEN SPHERE 25 Let a' and x" be the polar and pole respectively, of x' and a" with respect to a complex C. To prove a a'x= 0. The circle of intersection of the spheres a' and a" being contained in a' and i i i containing x' (58), the pole of a', belongs to the complex. Therefore since it lies upon a?, it contains x'. Therefore a' x"= 0. 27. Let pi, be any circle. Every generating sphere S' of pi is united in position with every generating point P of pi. Hence every pole P' of the spheres S', is united in position with every polar S of the points P. That is, the points, P', and the spheres, 8, generate one and the same circle p',. Two circles, p2k and P'k, so related that each of them is the locus (or envelope) of the poles (or polars) of the generating spheres (or points) of the other, are called conjugate circles. 28. The following theorems * result at once by analogy with the line geometry of ordinary space, and the plane geometry of the point in four-space. Every circle of a complex is self-conjugate with respect to that complex. Every self-conjugate circle belongs to the complex. Two distinct conjugate circles cannot pass through a same point, else they would belong to the complex and be self-conjugate. Two distinct conjugate circles cannot lie on a same sphere. If two conjugate circles have each a sphere (point) in common with a third circle, this third circle belongs to the complex. lies on Ihein-pairo two IfT a circle of the complex {pasesough} the same {ponepar} with one of two conjugate circles, it {palies og) a same {(sper} with the other also. All the circles intersecting two conjugate circles belong to the complex. 29. If two circles intersect, their conjugates also intersect. For let p' and p" be two circles intersecting in the point P. The conjugates of p' and p" lie on a sphere S, polar of P, and therefore intersect. CORRESPONDENCE V. To the c02 circles of a point-pair correspond the oo2 circles of a sphere polar to that point-pair with respect to a complex, and reciprocally. B. Special Complexes of Circles A complex is called special when all its circles intersect a same circle, the directrix of the complex. The condition that two circles Pik and p'k may intersect is (59) P12P34 + P13P42 + * * + P34Pl2 = [See (22).] *Cf. KOENIGS, La Geometrie Reglee, chap. 1. KEYSER, op. cit., p. 417. MOBIUS, op. cit., etc. 26 C. S. FORBES: THE GEOMETRY OF CIRCLES If the quantities P'k are fixed this is the equation of a complex, all of whose circles intersect Pi,, i. e., a special complex whose directrix is pk. The conditions that the complex AikPi = 0 may be special is therefore (60) O(Ak)=0. 32. Every pair of complexes,* C and C', of which one C' is special, determines another special complex C" such that the assemblage of circles common to C and C", is identical with that common to C and C'. The director circles of C' and C" are conjugates with respect to C. C is one of the pencil of complexes determined by X1 C' + X2 C" (2, 2 arbitrary). From this may be derived the coordinates of a circle p" conjugate to Ptk. ( ao,(A) w(A) (61) IXp,, 2P'. =A,, i;pPi Also given a complex C and a circle Pik, the theorem follows: pk; is self-conjugate or not, according as it belongs, or does not belong to C; if pk and p,, be any two circles, their conjugates are distinct, or not, according as C is non-special, or special. In case C is special the directrix is conjugate to all circles, itself included. 33. The following theoremst result at once from the definitions of pole and polar. The polars of the points of a sphere pass through a common point, the pole of the sphere. The poles of the spheres of a point lie upon a same sphere, the polar of the point. The conjugates of the circles lying upon a sphere all pass through the pole of the sphere, and the conjugates of the circles passing through a point, all lie upon a sphere, the polar of the point. A circle passing through two points has for its conjugate the circle formed by the intersection of the spheres, polar to these points. fhe circle formed by the intersection of two spheres, has for its conjugate the circle passing through the poles of the spheres. C. Coordinates of Conjugate Circles 34. We derive the coordinates of a circle conjugate to a given circle, by a method slightly different from that used in section 32, and based directly upon the principles of section 33. Given a complex * Cf. KEYSEB, Plane Geometry, etc., pp. 318-320. KOENIGS, La Geometrie Reglee, op. cit., p. 24. t Cf. MOBIUS, Uber Figuren im Raume, pp. 329-336. ORTHOGONAL TO A GIVEN SPHERE 27 E Aik pik = O and a circle p, defined by the spheres, S and S', (62) f p12x2 + P13X3 + p14X4 = 0, LP21 l1 + P23X3 + P24X4 = 0, we seek to find the coordinates of P'k conjugate of pik. The conjugate of Pik is the locus of the poles of the generating spheres of P,k and is fixed by any two of these poles, particularly by the poles of (62) themselves. These poles are [see (43)], [X = A12 12 + A13p13 + A14 p, (63) 1c2 A23p13 -A42P4 x3 -A23P12 + A,34P14 X= A4212 -A34P13; and [ =_ A13p23 + A4 p24, g(64) 12=- A12 P21 + A23P23 - A42P245 I = - A13 p21+ A34 P24 4=- A14P A 21 A34P23. A circle passing through these two points is fixed by any two spheres passing through the points, in particular by the spheres passing through these points and one of the points (65) 2 = X3 = 4 = 0 = x = = 0, i. e., one of the vertices of the configuration of reference [see (5)]. The equations of these two spheres are Xl CX2 X;3 X;4 t! 2 3! 4 (66) I = 0 X, X2 X3 X4 1 0 0 0 and X X2;3 X4;el X2 X, X4 (67) 2 3 4 =0. 1 2 x3;4 0 1 0 0 28 C. S. FORBES: THE GEOMETRY OF CIRCLES Expanding we obtain (68) 34x2 + La42X3 + -C23X4 = 0, I a431 + a14 + a3 = 0 where a. _xx -xx 5ik - X - k i' Since the spheres (68) are orthogonal to the fundamental spheres, whose equations are x1-~0 x-Q x1 =2 0, = O, the quantities ac may be taken as the coordinates of the circle (see section 9). That is P12 - 34 - 3 X4 - X4 X3 P13 -= 42 = X4 X2 - X32 X 4 (69) P14 a 23 -' - - 3i 2 P23 - 14 — 14 - X4 U1 c r42 a 13 - 1 i3 - 3 XFrom the form of these equations and the condition @0(p)=0, it follows that P34 -2- 5= 1 2 X2 2 X1 But these are the coordinates qi of the circle regarded as a locus defined by the two points x' and xI. In fact, the last theorem of section 33 states: That the conjugate of a circle formed by the intersection of two spheres is the circle fixed by the poles of these spheres. The theorem follows: If a circle is given by coordinates pi7 and its conjugate by coordinates qi, the spheres which define the coordinates Pi are the polars of the points which define the coordinates i,,, and reciprocally. It is found on trial that the results of (61) and (69) agree save for a factor of proportionality. D. Construction of the Complex 35. Let the symbols (P, P') and (S, S') denote the circles drawn through the points P and P', and lying on the spheres S and S' respectively. Let the symbol (P', P", P"') denote the sphere drawn through the three points P', P", P"' and (S', S", S"') denote the point of intersection of the spheres S', S" and S"' ORTHOGONAL TO A GIVEN SPHERE 29 36. Five non-intersecting circles of a complex being known, the complex may be determined. Analytically, they give five equations, from which the five constants of the complex may be determined. Geometrically, four of them are cut by but two circles, conjugate with respect to the complex. A different four give another pair of conjugates. Two pairs of conjugates are sufficient to fix the pole of any sphere, or the polar of any point, for every circle cutting a pair of conjugates belongs to the complex, and of the circles cutting a pair of conjugates, one in general passes through any point, or lies on any sphere. The polar or pole is fixed by the two circles of the complex passing through the point or lying on the sphere. Three spheres and their poles being known, the complex is determined provided not more than one of the circles of intersection belong to the complex i. e., not more than two of the poles lie upon the circles of intersection. For the conjugates of the circles of intersection are the circles determined by the poles of the corresponding pair of spheres. In the exceptional case two of the circles coincide with their conjugates, and the configuration determines but one pair of conjugates. The general case may be treated as follows: Given three spheres S', 8", S"' with poles P', P", P"', we seek the polar of a point P. Draw the three spheres, (P, P', P"), (P, P', P"'), (P, P", P"'). They cut the three circles of intersection, ( S'"), (S', 8"'), ( ", 8"'), in the points P1, P2, P3 respectively. These points are the poles of the spheres respectively, for (P', P") and ( ', 8") are conjugate circles, etc. P1, P2, P3 being the poles of three spheres passing through a same point P, all lie on a same sphere (P1, P,, P3), polar of P, and therefore containing P. The reciprocal problem, given S to find its pole, may be similarly treated. The point ( S', S", S"') is on the sphere (P', P", P"'), since the poles of all the spheres passing through a point, lie on a sphere which contains that point as pole. The following table shows these relations: SPHERE POLE SPHERES CONTAINING POLE S' P' (PP'P") (PP'P"') (P'P"P"') S" P' (PP'P") (P"") (PP'P"P"') '"' P" ' (PP'P"') (PP"P"') (P'P"P"') (PP2P,) P (PP'PP) (PP'P) (PP"P"') (PP'P".) P1S' S" (PlP2P3) (PP'P ") P2 S' S" (PP2P3) (PP"P"') P " S "' (P, PP,) (P'P"P"') (S' S"S'") ' S" S'" 30 C. S. FORBES: THE GEOMETRY OF CIRCLES The configuration * therefore consists of two groups of four spheres, the poles of each of which are the intersections of three of the spheres of the other group. If such a group be called a tetrasphere, the configuration consists of two tetraspheres, each of which is both inscribed in and circumscribed about the other. For the poles of the spheres of each are the vertices of the other. CHAPTER V LINEAR CONGRUENCES OF CIRCLES AND PROBLEMS UPON THE ASSEMBLAGE A. The Linear Congruence 37. A linear congruence consists of the circles whose coordinates satisfy two linear equations of the first degree, i. e., the circles common to two complexes. Let a pair of complexes be A -2-AikPik, 0 (70) B 1 BI=0. B -- E;Bik pik 0 Upon a sphere there lie in general one pencil of circles satisfying A and another pencil satisfying B. The circle belonging to both of these pencils is in general the only circle on the sphere belonging to the congruence. Reciprocally, through a given point there passes in general one circle of the congruence. 38. The given complexes A and B may be replaced by any two of the pencil of complexes X1 A + X2 B = 0 ( 1, 2 arbitrary). Of the complexes of a pencil, two are special, for there are in general two solutions to the quadratic condition (71) wo(XA +X2B)= 0. All the circles of a special complex intersect the directrix (see section 31). Hence the circles of the congruence, being common to the two special complexes, consist precisely of the oo2 circles intersecting the two directrices of the special complexes of the pencil. The directrices are called directrices of the congruence. The circles passing through any point of one directrix, and intersecting the other, lie on a sphere of which the point is the pole with respect to each of the complexes of the pencil, since the directrices are conjugates with respect to each of the complexes of the pencil. Hence the directrices of a con*Cf. MOBIUS, Uber Figuren im Raume, Crelle's Journal, Vol. 10, p. 324. ORTHOGONAL TO A GIVEN SPHERE 31 gruence may be defined * as the {envel} of the ) spheres} whose { les} are the locus J of tepoints [polars) are the same with respect to every complex of the pencil, of which the congruence is the common part. Any two circles conjugate with respect to a complex may be taken as directrices of a congruence belonging to the complex. Again, the directrices may be defined as, the locus of points (envelope of spheres) through which oo circles of the congruence pass. The equation of condition (71) co(XiA + X2B)= 0 may have two real roots, giving a congruence with real directrices, two imaginary roots, giving imaginary directrices, or two coincident roots, in which case the directrices coincide, the double directrix belongs to the congruence, and if a bilinear relation be set up between the generating points and the generating spheres of the directrix, then the assemblage of circles obtained by taking all and only the pencils that are contained on the spheres corresponding to the points, constitute the congruence.t In case (71) gives an indeterminate solution for X1: X2, the congruence has an infinity of directrices, which constitute a pencil of circles. The congruence consists of the circles of the point common to the circles of the pencil, and of the circles of the sphere upon which the pencil lies. The theory of the congruence of circles is seen to be identical in form with those of the congruence of lines, and of the congruence of planes in point geometry of four space. $ 38. We now turn to certain special problems relating to the complex, and first find the condition that a complex shall contain a given congruence. Let the complex be given by AikPik = 0 and the congruence by EBikPk =O, E CiPikk. The condition is, that the complex shall be one of the pencil of complexes defined by (X\ Bi + X2 Ci ) = 0, i. e., for some value of X1: X2 the six equations X1 B+k + X2 Ck = Ai shall be consistent. This is equivalent to the four independent conditions given by * Cf. PLUCKER, Neue Geometric, p. 78. t Cf. KOENIGS, La Geometrie Reglee, chap. 3, Annales.. Tonlouse, 1892; PLUCKER, Neue Geometrie, p. 62. t See references to PLUCKER, KOENIGS, KEYSER, etc. 32 C. S. FORBES: THE GEOMETRY OF CIRCLES A12 B12 C12 A13 B13 C13 A 14 B14 C14 (72) A 4 =4 0. 23 B23 23 I A42 B42 C42 A34 B34 C34 A complex may be subjected to five conditions. There are therefore oo complexes containing a given congruence i. e., the complexes of the pencil whose directrices define the congruence. The condition that a complex may contain a given circle p', is evidently (73) Akp =. Similarly a complex will contain a given pencil XlPik + X2Pik if it contains the generating circles of the pencil, i. e., EAikpk P =, Aikpilt= 0 -The systems defined by the equations Bp 0=O, *.. Bp= 0 (, 4), belong to the complex, if A12 B' ~.. 12 1 2 B12 * 2 (74)..........0, A34 B34. B34 giving (6 - n) independent conditions. In the case n = 4, the four equations above, and the quadratic identity define two circles, which accounts for the 6 - 4 = 2 conditions. B. The Surface with Circles as Generators Formed by the Intersection of Two Congruences 39. Two congruences of a complex of lines intersect in a ruled surface, the hyperboloid of one sheet, having two systems of straight-line generators. In section 24 (p. 23) we set up a one-to-one correspondence between the lines of space and the assemblage of circles orthogonal to a given sphere. The analytical condition that two circles intersect is precisely the condition that the two corresponding lines intersect. Hence by analogy we derive the properties of the surface formed ORTHOGONAL TO A GIVEN SPHERE 33 by the intersection of two congruences of circles of a complex. Consider two congruences of a complex, as defined by two pairs of circles p,, p[ and p2, P2, conjugate with respect to the complex. Any circle intersecting two conjugate circles, belongs to the complex, and to the congruence of which the two conjugate circles are directrices. Of these oo2 circles, ool circles intersect any other circle, and two of these 002 circles in general intersect any pair of circles. If this latter pair are conjugate, with respect to the complex, all the circles of the complex which intersect one intersect the other also. The theorem follows: 001 circles of a complex intersect two pairs of circles conjugate with respect to the complex, i. e., two congruences of a complex intersect in a configuration composed of ool circles of the complex. These circles generate a surface such that through every point on the surface pass two generating circles. The circles are arranged in two systems constituting a net of circles. No two circles of the same system intersect, but every circle of the one system intersects every circle of the other system. A sphere which contains one circle of the surface contains a second circle of the surface, and may be considered as tangent to the surface at the point-pair in which the two generators intersect. Since no two circles of the same system intersect no sphere can contain two generators of the same system. In general two congruences do not belong to a same complex. The condition is that one of the generating complexes of one congruence, shall be a generating complex of the other congruence. Three complexes, however, intersect in a surface of the above character. Its equations are therefore given as three equations of the form E Akikpk = 0. C. Radius of a Circle, Circles in Involution, Degeneration of S5 We treat in this division several special problems upon circles and the assemblage. 40. The radius p of a sphere a axi = 0 is (7a5) p t a c isgivti i Every sphere t passing through a circle pi, is given by the equation (76) E(X)Pai + Ikpg i )xi = 0 X where a, /3 are two of the indices 1, 2, 3,4, and X and /. are arbitrary. The radius of this sphere is * DARBOUX, Theorie des Surfaces, yol. I, p. 227. t KoENIGS, Contributions, etc., Annales - Toulouse, 1888, p. F 6. 34 C. S. FORBES: THE GEOMETRY OF CIRCLES E (Xpai + wup)2 (77) P2 i + ]2 When this is a minimum the radius of the sphere will be equal to that of the circle. Its value will be 2 (78) P2= \ When p= 0 (79) p2=j (p)=O. This is a complex of the second order, and consists of the points of 85, regarded as circles orthogonal to AS.. Each of the points is taken infinity times and in fact is a degenerate pencil of circles, having as axis the radius of S5 through the point. When p = oo, (80) 0()2 This is therefore the equation of the straight lines passing through the center of S5, each taken oo times, i. e., regarded as a degenerate pencil of circles orthogonal to S5 and of infinite radius. If R2 _ _2 - -R2 equation (80) degenerates into rEPii= 0. (,j=,...4). i 41. The polar of the form S (p), (81) E(p, p') = a Pk EPikp 2- i apk5ik has been shown by Koenigs * to play an important role in the theory of circles in space. Let there be given two circles pi and p'. If thorough one of them a sphere can be drawn orthogonal to the other, then through the second a sphere can be drawn orthogonal to the first, and the circles are said to be in involution. The analytical condition is found to be (82) (P C')= 0 * KOENIGS, Contributions, etc., Annales *... Toulouse, 1888, p. F 9. ORTHOGONAL TO A GIVEN SPHERE 35 The transition from the theory of circles in space, to the theory of circles orthogonal to a given sphere, consists solely in a proper choice of fundamental spheres, and the limiting of the range of i, k to the values 1, 2, 3, 4, instead of 1, 2, 3, 4, 5. If one of the circles, Pik be fixed, the circles in involution with it form a special complex, since they are given by the equation (83) P, =0 (83) EPikPik Pik [see (82)], and the coordinates of the complex are the coordinates of a circle [see (60)]. The quantities pk are not the coordinates of the directrix of the complex, regarded as an envelope of spheres; but writing in full equation (83), P12P12 + P13P13 + P14P14 + P23P23 + P42P42 + P3434 = 0, and the condition that the circles p'7 and pj intersect [see (22)], P34P12 + P4.2P13 + P23P14 + P14P23 + P13P42 + 32P34 = 0, it is readily seen that the coordinates P'k are proportional to the coordinates qi, which give the directrix as the locus of points [see (19)]. Hence the theorem: The circles in involution with a circle P', form a special complex whose directrix is given by coordinates q'k proportional to the coordinates pk', The same reasoning applies to the coordinates Aik of any special complex [see (60)] E AikPik = 0. The theorem follows: The coordinates Aik of a special complex are proportional to the coordinates qik of the directrix. 42. The angle between two spheres E ai =0, and E bxi = 0 is given by (84) ~ a.b. (84) v = cos-1 taib - * Be atI/ Y b2. If the spheres are tangent externally or internally Eab,= i e/ Ea2/ Z b2 '- bi 4- ai i' Squaring and expanding we obtain (85) E (abk - a bi) = Ep2i = 0. Conversely, if Ep2' = 0, the spheres are tangent. Two spheres, however, can only be tangent at a point on the sphere S5. Hence (85) is the equation of the points of S5, regarded as * DARBOUX, Theorie des Surfaces, vol. I, p. 228. 36 C. S. FORBES: THE GEOMETRY OF CIRCLES circles orthogonal to S5. Each point is taken ox times, since a pair of spheres may be tangent at any point on S5 in oo positions. If the quantities pik are real, they must all equal zero, and their ratios be indeterminate. These results are identical with that of (79). 43. The system as a whole may degenerate in two ways, - the radius of S5 may become zero or infinity. In the former case S8 becomes a point. Each of the o3 spheres and 004 circles passing through the point may be regarded as orthogonal to it. The inside elements of all the point-pairs coincide and this point taken 003 times is regarded as related by inversion to every point of space. The formule of inversion of course become indeterminate. Secondly, if the radius of S5 becomes infinite, S5 becomes a plane. The X03 spheres and 004 circles orthogonal to the plane, all have their centers in the plane. Points on opposite sides of the plane, and symmetrically placed, form a point-pair, their axis being perpendicular to the plane, just as the axis of a pointpair in the general case is a radius of the sphere and therefore perpendicular to the surface of the sphere. CHAPTER VI FAMILIES OF SPHERES 44. The spheres for which the set of ratios a,: a2: a3 is constant are said to form a family of spheres. We shall show that families of spheres exhibit many of the characteristics of families of parallel planes in line geometry. The pole of the sphere an axi = 0 is, [see (43)], X1 = Al12a2 + A13 a3+ A 14a4, (= -2 A12 a, + A23a 3- A42 a4 (86) 3 - A a A2 al - A + 34 a, X4 - A14 a + A42 a2 A34 a3 A. The sphere (87) al,1 + a2x2 + a3 - = 0 belongs to the same family and is orthogonal to the fundamental sphere S4. The pole of (87) is ' xf= Aa +A, IX = - A12a + A13a3, (88)2 23 3 X / - - A^ d - AL^ a, X = - A13 al A 23 2 i -A -14a A a3 + A A4 A. ORTHOGONAL TO A GIVEN SPHERE 37 From (86) and (88) we obtain x1 -.A14a,, (89) x- X= -A a, x 3-3 = A3 a4 Dividing the several terms of these equations by X = x A, we obtain x1 1 A14a4 X4 x4 A " 424 (90) 2 A42 a4 x3 x3 A34 a4 X4 X4 whence xxzc4 -- xx — — xx — 3 ---j --- X1 4 - 4 1 2 4 - 4 a2 3 4 - X3 (91) ( 1) A A 42 A34 If we regard xi as variable, it is seen that these are the equations of a circle, the locus of points x, the poles of the spheres of the family al: a2: a3. For since the equations are independent of a4, they are satisfied by the coordinates xi of the pole of any sphere of the form (92) a1x + a2x2 + a33 + X4 = ( arbitrary). Similarly the poles of the spheres of the family (93) blx, + b2x2 + bx + Xx4 = 0 lie on a circle whose equations are (94)n 1Y4 - X4Yt _2Y4 -:4Y2 c XaY4 - X4Y3 (94) 14 42 34 where the point y' is the pole of the sphere blxl + b2x2 + b3 3 = 0. We shall now show that the two circles (91) and (94) lie on a same sphere. Expanding we have as the equations of the spheres, 8 and 8', defining (94), (95) X- Ayx1 - A 14yx2 + (A42y + A14Y2)4 = 0, A34y4x A14y43 +- (A14Y3 - Ayl)4 x= 0. 38 C. S. FORBES: THE GEOMETRY OF CIRCLES Similarly from (91) we obtain, (96) F- A424 1 - A144 2 + (A42X; + A14X)X4- 0, A34XX1 - A144x3,+ (A14x3 - A34)= 0. In order that the circles may lie on a same sphere, it is necessary and sufficient that one of the pencil of spheres S+ S' ( arbitrary), defined by (95), should be identical with one of the pencil S" + -uS ( i arbitrary), defined by (96), for some values of X and a. That is, equating ratios of coefficients r - A42y4 + XA 34y - A42x4 + -A34x1 -X= -F', y4A14 4A14 (97) (A42Y + A14y) + X(A14y3-A34y ) l 14(A42x + + A14x) + pA(A,4x - AA3x) L XC~ ~ ~ ~ ~ ~~ g4 A14 The second of these equations of condition requires that X- = /. Substituting this value in the first equation, it is seen to be satisfied identically whatever the value of /. There remains but one equation and JL may be taken so as to satisfy it. The theorem follows: The loci of the poles of two families of spheres are two circles lying upon a same sphere. We call such circles diametral circles. 45. From the last of equations (97) we obtain 4(X A42 + A14) - x (y A,2 +2 A214) Xf (yf A,-!'A - y — (x A - x A -4 (3 14 - 1l y 34) - 4 -(3 A14- 1 A34) which reduces to Al4w(A)P13 - A42CO(A)P23 A =- A14o(A)p12 - A34 (A)P23' A14P3 - A42 P23 A14P12 A34P23 When the complex is special, p is indeterminate. Otherwise p is a function of A14, A42 and A34 and of al, a2, a3 and b, b2, b3, and changes as these latter quantities change, i. e., is different for different families. It follows that the ORTHOGONAL TO A GIVEN SPHERE 39 circles associated with different families do not all lie on a same sphere, but have a relation similar to that of parallel straight lines in space. Any two of these lines lie in a plane just as any two of the circles lie in a sphere, but in general three lines or three circles do not lie in a same plane or on a same sphere. PLUCKER* has shown that the locus of the poles of a system of parallel planes is a straight line (" Durchmesser"), and all these diameters are parallel. A family of spheres in the circle theory is seen to correspond to a system of parallel planes in the line theory. 46. There are o2 families, since a,: a2: a3 involves two parameters. There are xo2 diametral circles, since in general each family defines its diametral circle uniquely. Upon each of the generating spheres of a diametral circle, lie oc diametral circles. There are o02 circles, not lying on the common sphere of two given diametral circles, but co-spherical with each of them separately. These circles are co-spherical with every diametral circle. For consider a circle given by a sphere of the pencil (95) (98) + XS', and by a sphere of the pencil defined by (96) (99) S " + tS"'. Let the equations of any other diametral circle be r -A x A x2+(A +A x=0 (100) F A42~x4 1-A14z4x2 + (A42 +1 A14z)24 - 0, A3^ x, - A1 Z x,3 + (A,3 - A3 ) x, =Z 0. The condition that the circle defined by spheres (98) and (99) shall intersect the diametral circle (100) is the vanishing of the determinant of the coefficient of these four spheres i. e., the condition that they have a point in common, A34 - pA424x - pA14x - A144 pA42x1 etc. 1A3y4 - pA42y — pA14y4 -A14y PAY42Y etc. - A42z - A1z4 0 A42zI etc. A34 z 0 - A4z4 A14z3 etc. Taking the factor - A14 from the second and third columns and replacing it by - A42 and A34, we obtain * PLUCKER, Neue Geometrie, etc., pp. 35 et al. Also MSBIUS, Crel le's Journal, vol. 10, pp. 329 et al. 40 C. S. FORBES: THE GEOMETRY OF CIRCLES A3 x' etc. - pA42 x -A2 A34y etc. - pA42x 14 A42A34 A - A24 4 4 A34 4 0 Adding third column to second, we obtain A34 X 34y4 A34z4!i4 pA42x etc. A x - pA x' A x - pA 'x A34 4-pA 42 4 34 4 42 4 A14 A34 - pA42y4 A34x4- pA42zX A42 A344Y2 z/ -A4 - z A4 A4 A34z4 A34Xz A344. A34y4 * = 0. 0 34 4 The theorem follows: A circle co-spherical with two diametral circles, but not lying on their common sphere, is co-spherical with every diametral circle. In line geometry every line co-planar with two diameters but not in their common plane, i. e., parallel to them, is a diameter. Hence by analogy each of these oo2 circles is a diametral circle. CHAPTER VII TRANSFORMATIONS OF THE ASSEMBLAGE A. Generalization of Co6rdinates yi7 47. "The coordinates ryk" of section 11, "admit of generalization.* We know from the theory of forms, that the new variables v4 in the transformation, P7ik = Cik, 1v + C:k, 2 2 +- Ck, 6 ^6, where the modulus is not zero, are connected by a homogeneous quadratic identity (v) = 0, where l (v) is the transformed of co (y). Moreover the polar form o (y, r') of o(y() is converted by the same transformation into the polar form,.(v, v'), of Q2(v). It is well known that o(7y), regarded as a quadratic form, lhas a non-vanishing discriminant, and that it is possible to find a linear tlansformation which will convert this form into any quadratic form l (v), whose discrirm*See KEYSER, Plane Geometry, etc., p. 310. KOEXIGS, La Geomct.rie ReBl.'e. ORTHOGONAL TO A GIVEN SPHERE 41 inant does not vanish. Accordingly we may employ for homogeneous" circle "coordinates any six variables vi connected by the quadratic identity Q (v), where &2 (v) has a non-vanishing discriminant. Hence the condition that the" circles " i and v' shall intersect in a" point pair "or lie on a same" sphere "is that S(vy, v')) o." B. Transformation of Fundamental Spheres 48. We now consider the general transformation of the fundamental spheres Si. The equation of a sphere in space in pentaspherical coordinates is (102) S aix, + a2X2 + c4-a3 + a4x4 + a5x = 0, where Pi (103) Px R' Pi being the power of the point xi with respect to the sphere Si, and Ri being the radius of Si. Consider the linear transformation a == a1a[ + ana + a13a + a;a + a 5a, a2= an a1 + a22a + a23a + a2a ~ a25a, (104) a =a3la a+a32a2 3a + a33a3 + a3a4 + a35act ( a55= 1 ), a =-a, a1 + a422 + 1 + a3 t4 a + a5 as, a5 _= a51 a + a52 a - a53 a + a4 + a55 a, (105) S' i ak ak -- a' E a - a = X 0 is 1k 1k!i 1k where 5, a x.. 5-k _ E c -- i -5 We now seek the conditions that a point P (xyz) represented by xi, shall be represented by xa. That is P (106) px, z- (k=l, -, 5), where P' is the power of the point P(xyz) with respect to a new fundamental sphere Sk, and R' is the radius of that sphere., Expanding x. we obtain 2 + y2 + z2 + 2Ax +22~y + A2 222 X2 +2A+2Bi2Cz+A + Ci — R (107) x a R, 42 C. S. FORBES: THE GEOMETRY OF CIRCLES whence cX ^ = C ikX E (x2ik + 2 + 2) (108) + 2 E- A+ E Rik (A2 + B. + _C - R2) i i i R i Write aik/R, /,,ik, and divide numerator and denominator by Ei/ik, 2+ y2+ +2 2EA, /, +...+ZLk ( A + T + i )-I E3k R (109) =ik 1 Adding and subtracting (110) ( Ai in the numerator, it becomes of the form Hx'(111) = (x + a)2 + (y + b)2 + (z + C)2 r2 where if x' is of the form (106) pn2 = r2 (p arbitrary), i. e., P _ ( AI 3A,)2 + ) 2 + 3,, R- z;, i,,(A + C + C ) (112) ( f, = ( Z/}) (E aink )2 (" ik )2 This imposes one condition upon the quantities i/3, i. e., on at,. Hence in general that the five quantities x' may represent the same point P(xyz) that xi represents, it is necessary and sufficient that the quantities aik be subjected to four conditions of the form (112), and the transformation is equivalent to a change of fundamental spheres. We next impose the condition that the sphere S5 may remain unchanged, i. e., that the system of orthogonal circles which compose the space may remain unchanged. It is necessary and sufficient in order that X5 = X-5 that (113) a15 =a= 5 = a3 45 =0 and 55 = 1. Impose further the ten conditions that the five spheres, S', shall be mutually orthogonal. Since the four fundamental spheres s1' 2 ' 3' 4 ORTHOGONAL TO A GIVEN SPHERE 43 are orthogonal to S5 S5, it follows that every sphere of the form 4 (114) Eakx= 0 1 is orthogonal to S5. The theorem follows: The system of circles orthogonal to a given sphere may be transformed into themselves referred to form new fundamental spheres of the system, by a general linear transformation of the quantities ai, the twenty-four parameters of the transformation being subject to eighteen conditions. 49. As a special case let the sphere 85 be unchanged, and the quantities a, (i = 1, 2, 3, 4) be transformed by the relations a~ _ all al + * - - a4 a1l + 111 t14 4 =15 a3 a +.. a= ),a = a_ at + * - - a atv /4 ~41 + ' 44 a4 This involves fifteen parameters. S becomes 4 4 4 4 (116) S' Z x,' E aa - E aiE aix,i-E akxk 0= li 1k 1k li k If xk and xi represent the same point P(xyz), the quantities, ai,, are subjected to three conditions. If the new fundamental spheres S', (i = 1, * 4), are orthogonal, six more conditions are imposed. As before there are left six degrees of freedom. C. Invariance of C(A) 50. We now show the invariance of the fundamental form, o(A), of the complex. Consider the complex (117) EAikPk =0. A linear substitution (118) a. = a a + ai a + a a3 + ai a (i=,. 4) has been shown to be equivalent to a change of fundamental spheres. Let (119) a=i ik aa, bi aik b. k k By this substitution Pik = aibk - ak bi becomes (120) ~~~~2(120) Pik E likjrPjr Jr where (121) P-r aJb: - a b li'r = lr akr i jr j r r j' 2ikjr — )%ijcr - %kr 1ij~ 44 C. S. FORBES: THE GEOMETRY OF CIRCLES Substituting these values in the equation of the complex, we obtain (122) A.;. = 0, where (123) A'r=ljr A ik From these form the invariant wo(A') -wr(1ir) A, +.jr(lik, lir)AikAik + 1(jr 12jr 134jr)A12 A3 + Oj ( 3 142r) A13A42 + (jr (14jr, 23jr)A14 A23 a11 a12 a13 a14 2121 22 a23 a24 (124) w (A) A- Aw (A). a31 a32 a33 a34 a41 a42 a43 a44 For terms of the first two types vanish identically, and the coefficients of the last three terms are all equal to A. Since w (A) is in general not zero, only four of the coefficients of a non-special complex can become zero through any transformation, and the two remaining must not have a common subscript. The complex may take any of the three forms A2P122 + A34p34 = (125) A13p3 + A4 p42 0, A14p14 +- A 23p2 0. There are still two degrees of freedom among the coordinates of the transformation. It will be remembered, however, that the reduction of the complex of lines to its simplest equation, is of such a character that the complex may undergo symmetrical translations and rotations with respect to its axis, the equation remaining in its simplest form. Hence in both the cases of line and circle complexes there remain two degrees of freedom. 51. Since the elements of a point-pair are simply interchanged by the formulae of inversion, it follows that every point-pair, sphere or circle of the assemblage is invariant under this transformation. The theorem follows: Every configuration of the assemblage is invariant under the transformation by inversion with respect to the fundamental sphere S,. D. Projective Transformation by Means of Complexes 52. We have * seen that any linear complex C sets up a one-to-one correspondence between the spheres and the point-pairs of the assemblage. Let p and * Compare KEYSER, Plane Geometry, etc. ORTHOGONAL TO A GIVEN SPHERE 45 ' be two circles conjugate with respect to C. Each is the {nelope of the {pols} of the { eit-'irs} of the other. Consequently if 81, S2, 83, S be four spheres of p, their poles P1, P2, P3, P4 lie upon p' and (8 S2 3 84) = (Pl P2 P3P4), i. e., every complex sets up a projective transformation between the spheres and point-pairs of each pair of circles conjugate with respect to it, and between the spheres and point-pairs of each self-conjugate circle, i. e., each circle belonging to the complex. If p be a circle common to two complexes, and S,, **, S4 be any four of its spheres whose poles with respect to one of the complexes are P1... P4, and with respect to the other P, ***, P4, then (P1 P, P,3P4)= [(, S S S,)] = (P P2' P3' P). Similarly ( S1 S2 S3 S4),-,, = [(P_,P,3P4)] ( S; 8S S St). The theorem follows: Any two linear complexes set up a projective transformation between the {phepars } of each of the circles of their common congruence. Accordingly any circle of the congruence may be regarded as two superimposed {penglss of {pOahirs } associated one with each of the complexes which define the congruence. Any circle of the congruence is common to all the complexes of the pencil of complexes E(x\ C + 2 C,)P = 0 which define the congruence. Let p be any circle of the congruence. Its pointpairs and spheres are transformed by every complex '1: 2-= m of the pencil. Since the director circles of the congruence are conjugate with respect to every complex of the congruence, it follows that the (pfls} of the {pnt-si comon to p and to one of director circles, are the same with respect to every complex of the pencil. Hence these {point-pairs } will be the foci of every projective transformation of the {Preairs} of p by means of any pair, ml and m, of the complexes. Let ml, m2, ms, m4 be any four complexes of the pencil, and P1, P, P, PI be the corresponding poles of any sphere 8 of p, then (mI,2mzam4) == (PI P2PP4). In particular if 81 and S2 be the special complexes of the pencil, ( ml m, 2 82) = (P1Pdl P2 Pd2 ) =, since P1 and P2 are any pair of corresponding elements and P,, and Pd2 are the foci. Hence the {pPhrpar} transformations effected upon all circles com 46 C. S. FORBES: THE GEOMETRY OF CIRCLES mon to a pair of complexes by the complexes, are formally identical. Reciprocally a transformation is effected among the complexes of a congruence by the {polS} Of any {pnt-airP } of a circle of the congruence. The foci are the special complexes. 53. The equation of the transformation of the point-pairs of a circle with respect to a pair of complexes assumes its simplest form when referred to the foci. We have the relation (PPlP2,Pd) =k; i. e.,* P2 - P1d P/d2- P1 P2 -- Pd Pd2 PdLet P2 = oo, and Pdl = 0. We have -P P2 _1 P2 or P = (1- k)P,, which is the required transformation. Similarly (1 Sdl 82 Sd2) = k and S1= (1-k)S2. Hence the point-pair and sphere transformations effected upon the circles common to two complexes by the complexes, are formally identical. CHAPTER VIII THE COMPLEX OF SECOND DEGREE A. Tangent and Polar Linear Complexes 54. The circles whose coordinates p, satisfy a homogeneous equation of the second degree form a complex of the second degree. The equation may be written C2 a A1212P + A1313P3 + A1414p14 + A223p233 + A4242p42 + A434p34 + 2A1213Pl2P13 + 2A214Pl12P14 + 2Al223P12P23 + 2A1242+P2P42 (126) + 2A1234P12P34 + 2A1314 P13 14 + 2A1323P13P23 + 2A1342P13P42 + 2A1334P13P34 + 2A41423P14P23 + 2A1442P14P42 + 2A1434P14P34 + 2A2342p23P42 + 2A2334P23P34 + 2A4234 P42P34 = 0~ *Notation of DOEHLEMANN, Projektive Geometrie. t The P's are of course to be replaced by their coordinates if a metrical interpretation is desired. ORTHOGONAL TO A GIVEN SPHERE 47 Let the polar form of C2 be (127) (, P ') -- E_2P ik - ik jO ik Two circles, pik and p'k, which satisfy the equation (127) C2(p, p') 0, are said to be associate with respect to C2. Holding P'k fixed and letting p, vary, (127) is the equation of a complex. The theorem follows: All the circles associate with a given circle with respect to a complex of second degree, form a linear complex. The circle is called the pole of the complex, and the complex is the polar of the circle. It follows at once from (127): If p'k is in the polar of pik, then p, is in the polar of P'. If pik satisfies C2 it is among the circles of its polar complex, for the polar equation reduces identically to C2(p') when Pik -Pi k' The polar complex is then said to be tangent to the complex of second degree. A linear complex does not in general possess a pole with respect to a given complex of second degree. For while the six equations ao,(p') (128) pA I -a ', define in general the ratios of the six quantities pik uniquely, these quantities p, do not in general satisfy the identity (co()=0, and hence are not the coordinates of a circle. There are in fact o5 complexes, and but oo4 circles. Two, however, of the complexes of a congruence possess a pole. For let the congruence be defined by the pencil (129) ( X atk + X2 A"ik)Pik = 0O The seven equations (p)= (p0 involve seven unknown quantities, pi and X,:,, and are in general capable of two solutions. If a circle is associate with respect to two circles pk and pk, it will be associciate with every circle of the pencil defined by these two circles. For (131) C2(p, Xp' + X2p") — 1 C2 (p,p') + X2 C2(p,p") = 0 48 C. S. FORBES: THE GEOMETRY OF CIRCLES B. Intersection of a Pencil of Circles with C2 55. A pencil has two circles in common with a complex of second degree. For the equations of a pencil are (132) P12 — P122 P13- X13 _ _P34 - X34 (132) if =... = f 2 P12 P13 - P34 The circles common to the pencil and C2 are found by eliminating Pik between these equations. We find (133) C,(p")X2 + 2X1 pk (p ) + C2(p') O. This is a quadratic equation in X2: X\, and is satisfied by two values of this ratio, i. e., the pencil and complex have two circles in common. If these two circles coincide, the pencil is said to be tangent to the complex. The condition that (133) has equal roots is (134) C(i )-4)-C2(p )C(p )=0. If C2 (p') = 0, the condition becomes (135) P apk -- 0. The theorem follows: If pi be a circle of a complex of second degree, the circles P k which generate with p' pencils tangent to the complex of second degree form a linear complex. A similar theorem evidently holds for P'k. This complex is seen from its equation to be the tangent complex. Reciprocally: If p'" be a circle not belonging to a complex of second degree, the circles pk of that complex, which with P"k generate pencils tangent to the complex, belong to a linear complex polar to p^, with respect to the complex of second degree. 56. Attention is called to the similarity between the theory here developed, and the theory of polars with respect to a conicoid. The following elements enjoy corresponding properties in this regard: CIRCLE GEOMETRY. POINT GEOMETRY. Circle Point Pencil Line Linear complex Plane Complex of second degree Conicoid ORTHOGONAL TO A GIVEN SPHERE 49 C. Generalization of Pliicker 57. The deductions of section 54 are not strictly general. Following PLtCKER * we may proceed as follows. Every circle whose coordinates satisfy a complex of second degree C2 also satisfies the equation C2 + x ( p) = 0 (2 arbitrary). The polar form of this is (136) C2(pp') +\X(p, p'). Hence to every circle there are an oo of polar complexes, and to every complex there is a pole. Likewise if p', is one of the circles of C2, there are co complexes tangent at pk,. Since co(p, p' ) = 0 is special, pk' is one directrix of the congruence defined by (136). The other directrix is the conjugate of p'k, and is the same with respect to all the complexes of the congruence. The second special complex is given by a C2 aC2 a2 aC2 C2 a'c (137) X 1 app12 aP'+4 OP]3 CP 42 apI4 p23 (137) X= -- C2(p ) If C. (p') = 0, both directrices coincide, and X-= o or - Circles of C2 for which X = - are called singular circles of the complex. A treatment of this subject would exceed the limits of this paper. The theory of circles orthogonal to a given sphere might be pushed much further, but in showing the similarity of the theory to that of line geometry, and the development of its fundamental problems, we have accomplished our object. BIBLIOGRAPHY The following articles treat of topics more or less intimately associated with the subject of the present paper. Those which have been of particular value to us are marked with a (*). CAYLEY. On the Six Coordinates of a Line. Collected Mathematical Papers, vol. VII, p. 66. Sixth Memoir on Quantics. Op. cit., vol. II, p. 561. COSSERAT. Sur* le cercle consider6 comme element generateur de 'espace. Annales de la Faculte des Sciences de Toulouse, vol. III, 1889, p. E 1. * PLUCKER, Neue Geometrie, pp. 287-296. 50 C. S. FORBES: THE GEOMETRY OF CIRCLES DARBOUX. Sur une classe remarkable de courbes et de surfaces algebriques. Theorie * des Surfaces, vol. I, book II. chap. 6. Sur les relations entre les groups de points, de cercles, et de spheres dans la plan et dans l' espace. Journal de Liouville, 2nd series, vol. I, 1872. DEMARTRES. Sur les surfaces z generatrice circulaire. Annales de 1' ]Icole Normale, 3rd series, vol. II. p. 123. ENNEPER. Die cyklischen Flichen. Zeitschrift fiir Mathematik und Physik, p. 393, 1869. KEYSER. The* Plane Geometry of the Point in Space of Four Dimensions. American Journal of Mathematics, vol. 25, No. 4. KLEIN. Uber der sogenannte Nicht-Euclidische Geometrie. Mathem atische Annalen, vol. 4, p. 573, vol. 7, p. 531, vol. 37, p. 544. Einleitung * in die Hbhere Geometrie, vol. I. Uber Liniengeometrie und metrische Geometrie. M a t hematische Annalen, vol. 5, p. 257. KOENIGS. Contributions * a la theorie du cercle dans l'espace. Annales de la Faculte des Sciences de Toulouse, vol. II, 1888, p. F. 1. La* Geometrie Reglee. Op. cit., vol. III, 1889: vol. VI, 1892; vol. VII, 1893. Sur les proprietes infinitesimales de l'espace regle. Annales de l'Iocole Normal, 1882. LIE. Uber * Complexe, inbesondere Linien- und Kugel-complexe. Math ematische Annalen, vol. V, 1872, p. 164. LORIA. Remarques * sur la geometrie analytique des cercles du plan. Qu arterly Journal, vol. XXII, p. 44. MOBIUS. Uber eine besondere Art dualer Verhaltnisse zwischen Figuren im Raume. Crelle's Journal fiir Mathematik, vol. 10, p. 317. PASCH. Zur Theorie der linear Complexen. Crelle's Journal fiir Mathematik, vol. 75, p. 11. PLUCKER. Neue * Geometrie des Raumes. On* aNew Geometry of Space. Wissenschaftliche Abhandlungen, p. 469. STEPHANOS. Sur la geometrie des spheres. Comptes Rendus de l'Academie des Sciences, vol. XCII, 1881, p. 1195. Sur une configuration remarquable de cercles dans l'espace. Op. cit., vol. XCIII, p. 579. Sur une configuration de quinze cercles et sur les congruences lineares de cercles dans l'espace, vol. XCIII, p. 633. SMITH. * Solid Geometry. SNYDER. Die Lieschen Kugel-geometrie. STUDY. Geometrie der Dynamen. STURM. Line Geometry.