/<7 / " _J SOLID GEOMETRY. CAMBRIDGE : PRINTED BY W. METCALFE AND SON, TRINITY STREET. I SOLID GEOMETRY. PERCIVAL FROST, M.A., FORMERLY FELLOW OF ST. JOHN'S COLLEGE, MATHEMATICAL LECTURER OP KING'S COLLEGE, REVISED AND ENLARGED, OF THE TREATISE BY FKOST AND WOLSTENHOLME. ILontion : MACMILLAN AND CO. 1875. [7^« Rifffit of Tramlation ii reserved]. a,^ ^!«*<>' o^c PREFACE. It was with a feeling of great discouragement that I began the preparation of another Edition of tliis work, deprived, as I was, of the valuable assistance of my friend Mr. Wolstenholme, in working with whom I had had so much pleasure while writing the First Edition. Mr. Wolstenholme, who is now Professor of Mathematics in the Royal Indian Engineering College at Cooper's Hill, thought that there would be great difficulty in carrying on this work satisfactorily by correspondence, even if the important duties in which lie is engaged did not fully occupy his time ; I was, therefore, re- luctantly obliged to undertake the whole labour of remodelling our original work. As we contemplated making additions, and many alterations both in form and substance, my friend desired that his name might not appear in the Second Edition, and I have been compelled to alter tlie title of the work, and to take the responsibility of the changes which have been introduced. PREFACE. The problems which appeared in the former Edition were for the most part original, and a large proportion of them were due to Mr. Wolsten- holme; in this department, therefore, a most important one in my opinion, I have not lost the advantage of his valuable assistance. The present Edition is intended, in its complete form, to occupy two volumes, but for the con- venience of Students who may wish to have in one volume all those portions of Solid Geometry which would be useful to them in their studies of Physical subjects, I have endeavoured, as far as I could without material departure from the arrange- ment which I considered best for the proper treat- ment of the subject, to include in the first volume nearly all that will be required from their point of view; with this object, I have reserved for the second volume those parts which are chiefly in- teresting as Pure Geometry. The Student who desires to confine his reading to the more practical portions of the subject should omit Chapters VI., VII., VIII., IX., Art. 155—157, XV. and XVII., the Three-Plane system of Co- ordinates being employed exclusively in the remain- ing chapters. I feel bound to say a few words with respect to my persistence in retaining the word ' Conicoid ' PREFACE. vii to represent the locus for the equation of the second degree. It was natural that the distinguished analyst, who has done so much towards the inves- tigation of the properties of surfaces of higher degrees than the second, should seek a term for that of the second degree, wliich would connect it with those of higher degrees. But I cannot help thinking it unfortunate that the terms ' quadric' should have been selected, which had already a different meaning. I quote the words of the author of the well-known treatise on Higher Algebra: ''It '' is convenient to have a word to denote the "function itself without being obliged to speak of "the equation got by putting the function = 0. " The term ' quantic ' denotes, after Mr. Cayley, " a homogeneous function in general, using the " words ' quadric,' ' cubic,' ' quartic,' ' w-ic,' to "denote quantics of the 2"^, 3""^, 4''*, n'^, degrees." Now, ' quadric,' as used in the other sense, is not even the equation found, but it takes two steps and becomes the locus of the equation. I consider that the surface of the second degree at present, wliatever may be the case in some future development, stands on a platform of its own, on account of the services which it has rendered to all departments of Mathematical Science, and well deserves a distinctive name instead of being recog- PREFACE. nised only by its number, a mode of designation which, I am informed, a convict feels so acutely. Man might be always called a biped, because besides himself there exist a quadruped, an octopus, and a centipede, but, on account of his superiority, it is more complimentary to call him by some special name. The useful word 'conic' being well-established, the term 'conicoid' seems to suggest all that can be required, when it is employed to designate the locus of the equation of the second degree in tlu'ce dimensions, at least so long as the analogous words spheroid, ellipsoid, and hyperboloid arc in use, at all events it is not open to the great objection of being equally applicable to plane curves, as is tlie term ^quadric;' cublcs and quartics being actually so employed in Salmon's Higher Plane Curves, Chapters V. and VI. To the many excellent mathematicians, whose talent is shewn in the composition of tlie yearly College papers and the papers set for the Mathe- matical Tripos examination, I am indebted in the highest degree both for tlie problems which I liave added to the collection, and also for the hints derived from them in the treatment of the subject itself. I have also to make thankful acknowledgments for PREFACE. IX the valuable assistance received from my friends. Mr. Moulton, of Christ's College, has given me great help in parts of the subject which, except in the chapters on the general equation, do not appear in this volume. Mr. H. M. Taylor, Follow of Trinity College, was kind enough to look over many of the proof sheets ; and I am indebted to Mr. Ritchie and Mr. Main, of Trinity College, and Mr. Stearn, of King's College, for their kindness in testing a large number of the pro- blems, as well as in looking over the 2^i'<^c)f sheets throughout the process of publication. But especially I wish to thank Mr. Stearn for the great assistance which he has rendered in super- intending the work during my frequent absence from Cambridge, and also for his many valuable criticisms. Cambridge, Octoher, 1875. CONTENTS. CHAPTER I. - ON COORDIIfATE STSTBMS. Page Coordinate system of three plcones . . . . .2 Polar coordinate system ..... 3 CHAPTER n.' GENERAL DESCRIPTIOK OF LOCI OF EQUATIONS. SURFACES, CURVES. Locus of an equation Cylindrical surface defined Locus of the polar equation Equations of curves CHAPTER III." PROJECTIONS OF LINES AND AREAS. DIRECTION-COSINES AND DIRECTION-RATIOS. Projection of a limited straight line on a straight line . . .12 Algebraical projection . . . . ,13 Direction-cosines of a line . . . . . .1-1 Angle between two lines in terms of their direction-cosines . . 14 Direction-cosines of a line perpendicular to two lines whose direction-cosines are given ....... Direction-cosines of the two bisectors of two intersecting hues . 15 Angle between two lines whose direction-cosines are given by two homogeneous equations of the first and second degrees .... Direction-ratios of a line . . . . .17 Projection of a line on a plane . . . • .18 Projection of a plane area on a plane . ' . . . 19 The sum of the projections of the faces of a closed polyhedron on a plane is zero 20 Area of a plane figure in terms of the projections on rectangular coordinate planes 15 16 21 Plane on which the Bum of the projections of any given ai-eas is a maximiuu . 21 CONTENTS. CHAPTER IV. - DIVISIOX OF LINES IN A GIVEN RATIO. DISTANCES OP POINTS, EQUATIONS OF A STRAIGHT LINE. Coordinates of a point dividing in a given ratio the line joining two points Distance between two points, rectangular axes , oblique axes , polar coordinates Straight line ...... Symmetrical equations ..... Non-symmetrical equations ..... Number of independent constants in the equations Equations of a straight line parallel to a coordinate plane one of the coordinate axes Angle between two straight lines whose equations are given Conditions that two straight Unes may be parallel Condition of perpendicularity of two straight lines Condition that two straieht lines may intersect Equations of a straight line under conditions Passing through a given point .... Passing through a given point, and parallel to a given straight Une Intersecting a given straight line at right angles Intei-secting two given straight lines Parallel to a given plane, and intei-secting a given straight line Distance from a given point to a given straight line Equation of a circular cylinder .... cone .... Shortest distance between two straight lines is perpendicular to both whose equations are given Equations of the hne on which the shortest distance lies Simple form of equations of two straight lines rage 25 25 27 27 28 29 30 30 31 31 32 33 34 34 35 35 35 35 36 36 37 38 38 38 39 39 40 CHAPTER v." GENERAL EQUATION OF THE FIRST DEGREE. EQUATION OP A PLANE. Locus of the general equation of fii"st degi-ee is a plane . . .44 Plane at an infinite distance ..... 45 Number of conditions which a plane may satisfy . . .45 Equation of a plane in the form Ix + my + m =2? . . . 45 Greometrical intei-pretation of the expression j) - Ix - my - nz . . 46 Angle, between two planes, whose equations are given . . 47 Conditions of parallelism and pei-pendicularity ' . . . .47 ' Angle between a straight line and plane, whose equations are given - . 47 Pei-pendicular distance of a point from a plane . . . .48 Distance of a point from a plane, measured in a given direction . 48 Equation of a plane in the foi-m '- + ^^ + - = 1 . . . .49 X tJ z Geometrical iutci-prctatioa of the expression 1 - " - "r - ' • • 50 CONTENTS. Xm Pape Equation of a plane in the form z = px + qy + c . , .60 Polar equation of a plane ..... 51 Equations of planes under conditions . . . .61 Passing through a point whose coordinates are given . . 51 in which three planes intersect . . .52 Passing through two given points .... 52 Passing through the line of intei-section of two planes . . .53 Two planes forming an harmonic system with two given planes . 53 Plane passing through three given points . . . .53 Geometrical interpretation of its equation .... 54 Plane passing thi-ough a given point and parallel to a given plane . . 54 two given points and pai-allel to a given line . 55 ^ a given point and pai-allel to two given lines . . 55 ""■■■ Plane containing one line, and parallel to another not in the same plane . 56 Plane equidistant from two straight lines not in the same plane . . 56 Conditions that the general homogeneous equation of the second degree in X, y, z may represent two real or imaginary planes . , 57 Symmetrical equations of the line of intersection of the two planes . . 58 CHAPTER VI. QUADRIPLANAR AND TETRAHEDRAL COORDINATES. Description of the quadei planar system of coordinates . . 63 Relation of coordinates in the fom'-plane system ... 64 Tetrahedral coordinates . . . . . .65 Coordinates of a point dividing in a given ratio the lines joining two points 65 Distance of two points ...... 65 Equation of a sphere . . . . . ,66 Equation of a straight line, four-plane and tetrahedral coordinates . 66 Direction cosines of a straight line . . . , .67 Angle between two straight lines, tetrahedral coordinates , . 68 Condition of perpendicularity of two straight lines . . .68 Locus of the general equation of the first degree in tetrahedral or four-plane co- ordinates is a plane ...... 69 Greometrical interpretation of the constants in the equation of a plane in four- plane and tetrahedral coordinates . . . , .69 Equation of a plane at an infinite distance .... 70 Conditions of parallelism of two planes . . . .70 Perpendicular from a point on a plane, four-plane or tetrahedral coordinates 71 Angle between two planes ..... 72 Direction-cosines of the normal to a plane, tetrahedral coordinates , . 73 CHAPTER VII. FOUE-POINT COORDINATE SYSTEM. THE POINT. THE PLANB. Description of the four-point coordinate system ... 76 Distance of a point whose position relative to fixed points is known, from a plane whose distance from the fixed points is given . . .76 Equation of a point ...... 78 Interpretation of constants in the equation of a point , . .78 XIV CONTENTS. Page Equation of a point dividing in a given ratio the distance between two points, whose equations are given ..... 79 Equation of a point at an infinite distance . . . .79 Distance between two points, whose equations are given . . 79 Inclination of a plane to the faces of the fxmdamental tetrahedron . , 81 Relation between the point coordinates of a plane ... 82 Linear relation between the direction-cosines of a plane . . .83 Why the relation between the point-coordinates of a plane is of the second degree 83 Angle between two planes whose coordinates are given . . 84 Point coordinates of a plane passing through the intersections of two planes . 85 Equation of the first degree represents a straight line if the constants involve one variable in the first degree ; a plane if the constants involve two variables in the first degree ...... 85 CHAPTER VIII. LOCI OP HQTTATIONS. TANGENTIAL EQUATIONS OF SUEFACE8. TORSES. DUAL INTERPRETATION OP EQUATIONS. BOOTHIAN COORDINATES. Representation of surfaces by equations in four-point coordinates . . 88 Distinction between ordinary and developable surfaces . . 88 Analogy between torses and curves . . . . .89 Poles of simihtude of four spheres ..... 90 Dual interpretations of equations . . . . .92 Description of boothian coordinates .... 92 Cartesians are a particular case of tetrahedral and Boothian of four-point coordinates . , . . . . .93 CHAPTER IX. transformation of coordinates. Change of origin, the direction of the axes unchanged . . 96 Three terms may be made to disappear by change of origin . . .96 Transformation from one system of rectangular axes to another having the same origin ....... 97 Relation between the direction-cosines of the new axes . . .98 Equations of the lines bisecting the angles between two lines given by Jx + my + nz = 0, aa;2 + by- + cz^ =: 0. . , . . 99 Euler's formulas for transformation from one system of rectangular axes to another . . . . . , .100 Plane sections of a surface examined by transformation of axes . . 102 Circular sections of a surface ox- -f- %2 + (,-2 _ j_ .... 103 Transformation from one system of oblique axes to another . . 103 any one system of axes to any other . . . 104 Degree of an equation unaltered by transformation . . . 105 In\'ARIANT8 of a ternary quadric, axes rectangular . . . 105 oblique . . . 106 Transformation from rectangular to polar coordinates . . , 107 a four-plane to a three-plane sj-stera . . 107 one four-point system to another . . . 107 CONTENTS. CHAPTER X. - ON CERTAIN SURFACES OF THE SECOND DEGREE. Equation of a sphere ...,., Number of conditions which a sphere can satisfy . Greneral equation of a sphere passing through a given point passing through two given points in the axis of t Equations of spheres touching each of the coordinate axes Equation of a sphere touching the plane of xy in a given point Geometrical interpretation of (x - «)^ + Gy - 6)- + (3 - c)- — cP Power of a sphere .... Radical plane of two spheres The six radical planes of four spheres meet in a point Equation of an elliptic cylinder Conical surfaces defined Equations of a cone on an elHptic base, vertex in the orfgin any given point Two systems of circular sections of any oblique circular cone Spheroid, oblate and prolath defined Equation of a spheroid Ellipsoid defined .... Equation of an elhpsoid . , . ^1 yi z- Locus of the equation -5 + rj + -j = 1 Htperboloid of one sheet defined Equation of an hyperboloid of one sheet Locus of the equation — ^ + tj - -5 = 1 Conical asymptote of an hyperboloid of one sheet Hyperboloid of two sheets defined Equation of an hyperboloid of two sheets Locus of the equation — 2 + jw — 2 = 1 Conical asymptote of an hyperboloid of two sheets Elliptic paraboloid defined Equation of an elliptic paraboloid y- g* Locus of the equation j + -j, = x Hyperbolic paraboloid defined Equation of an hyperboUc paraboloid .... 128 Locus of the equation -r ~ y = a; . . • . . 129 Plane asymptotes of an hyperbolic paraboloid . . .130 EUiptic and hyperboHc paraboloids particular cases of eUipsoids and hyperboloida 132 On the general forms Ax^ + Bif + Cz" = Z> and Bij^ -^-Cz^^Ax . . 133 CoMCOiD defined ...... 134 Pago 110 111 111 112 112 112 113 113 113 114 114 115 115 115 116 117 117 118 118 119 120 120 121 122 123 , 123 124 , 125 127 . 127 128 CHAPTER XI. ON generation by straight links. Geometrical account of generation of hyperboloida of one ahoet by straight lines 137 XVI CONTENTS. Page Two systems of generating lines include all straight lines which lie entirely on an hyperbcloid ...... 139 Surface generated by a line meeting thi-ee non-intersecting lines . . 139 Conditions that a line drawn through a given point of a conicoid may lie entirely on the surface ...... 140 Points on an hyperboloid for which the generating lines are perpendicular . 141 Equations of the generating lines of an hyperboloid of one sheet . . 142 Projections of generating lines on the principal planes are tangents to the traces on those planes . . . . . . .143 Two generating lines of the same system do not intersect . . 143 Generating lines of opposite sj'stems intersect .... 144 Equations of the generating lines of an hyperbolic paraboloid . . 146 Lines of the same system on an hyperbolic paraboloid do not and those of opposite systems do intersect ..... 147 Projection of generating lines on the principal planes are tangents to the traces on those planes ...... 147 CHAPTER XII. SIMILAR SURFACES. PLANE SECTIONS OF CONIOOIDS. CYCLIC SECTIONS. Similar surfaces defined . . . . . .153 Surfaces similarly situated ..... 153 In what sense hyperboloids of one and two sheets may be similar . . 154 Sections of the same conicoid by parallel planes and sections of similar and similarly situated conicoids by the same plane are simUar and similarly situated conies . . . . . .154 Similar and similarly situated conicoids intersect the plane at infinity in the same conic ....... 156 Nature of a plane section of a conicoid examined by projections . 155 Locus of centres of sections of a central conicoid by parallel planes . .156 Position of the cutting plane when the section is a point-ellipse or line- hyperbola ...... 157 Locus of centres of sections of a paraboloid by parallel planes . . 157 Position of the plane for a point-ellipse or line-hyperbola . . 168 Magnitude and direction of axes of a plane central section of a conicoid . 168 Direction of the plane section whose axes arc of given magnitude . 159 Nature of a central plane section of a central conicoid . . . 160 Nature of any plane section of a central conicoid . . . 161 Angle between the real or imaginary asymptotes of a plane section . . 161 Area of an elliptic non-central section of a conicoid . . . 1G2 Volume of asymptotic cone cut o£E by a plane touching an hyperboloid of two sheets is constant . . . . . .163 Magnitude and direction of axes of any plane section of a paraboloid . 163 Nature of any plane section of a paraboloid .... 164 Cyclic sections of a conicoid ..... 165 Generation of a conicoid by the motion of a variable circle . . .166 Umbilics of a conicoid defined ..... 166 Any two cyclic sections of opposite systems lie on one spliere . . 167 Geometrical investigation of the direction of cyclic sections . . 168 CONTENTS. xvii CHAPTER XIII. TANGENTS. CONICAL AND CYLINDRICAL ENVELOPES. NORMALS. CONJUGATE DIAMETERS. Pa(?c Tangent LINE to a given central conicoid at a given point . .171 Tangent plane . . 172 Equation of a tangent plane to a conicoid drawn in a given direction . 172 Equation of a tangent plane of a cone .... 173 Equations of an asymptote to a central conicoid . . . ,173 Nature of the intersection of a central conicoid with a tangent plane : 173 Axes of the section by a plane through the centre of a conicoid . .174 Locus of points of contact of tangent planes passing through a given point 174 Polar plane of a point and pole of a plane with respect to a given conicoid defined •••.... 174 Fundamental property of poles and polar planes . . . 175 Conical envelope of a central conicoid . . . .175 Cylindrical envelope of a central conicoid . , .176 Corresponding results for a non-central conicoid . . . .177 Equations of the normal to a central conicoid at a given point . 177 Six normals to a central conicoid through a given point . , . 177 Locus of a point from which three normals to a conicoid have their feet in a given plane section . . . . , .178 Diametral planes of a conicoid defined . . . .180 Diametral plane of a central conicoid for a given system of parallel chords . 180 Conjcgate diameters and conjugate planes . . .181 Relations between the coordinates of the extremities of a system of conjugate diameters ....... 182 The sum of the squares of the projections of three conjugate diameters on any line or any plane is constant ..... 182 Relations between the lengths of three conjugate diameters and the angles between them . . . . . .182 Diametral plane of a paraboloid for given parallel chords , . . 185 Harmonic definition of a polar plane .... 186 CHAPTER XIV. CONFOCAL CONICOIDS. FOCAL CONICS. BIFOCAL CHORDS. CORRESPONDING POINTS. Confocal CONICOIDS defined . . . . .192 Two modes of representing a system of confocal conicoids . . 192 Through every point pass three conicoids confocal with a given conicoid . 193 If two parallel planes touch two confocals, the difference of the squares of the perpendiculars on the planes from the common centre will be constant 194 Locus of poles of a given plane with respect to a system of confocala . .194 Elliptic COORDINATES explained ..... 195 When three confocals pass through a point, each of the normals at that point is' perpendicular to the Other two .■ . , , .196 Axes of a central section of a conicoid expressed by means of two confocals to the conicoid . . . . . .197 If P be a point in the intersection of two confocala A, B, the diameter of A paraUel to the normal at P to jB is constant . . . .198 Lengths of perpendiculars on tangent planes at a point common to three confocala 198 C XVlll CONTENTS. Page pd constant for every point in the intersection of two confocals . . 199 Principal axes of a cone enveloping a given central conicoid , . 199 Equation of the enveloping cone referred to normals to the three confocals through the vertex as axea ..... 200 Equation of the enveloped conicoid referred to three normals , . 201 Properties of a line touching two confocals to a given conicoid . . 201 Two conicoids can be constructed which are confocal with a given conicoid and which touch a given straight Une ..... 202 Length of a chord of a conicoid touching two conicoids confocal with it . 202 Two confocals appear to cut at right angles .... 203 Gilbert's method of soh^ng problems in confocals . . . 203 Focal conics are particular confocals ..... 205 Locus of vertices of right cones envelopmg a conicoid . . . 207 Bifocal chords defined . . . . . .208 Properties of bifocal chords ..... 208 Construction for the four bifocal chords through a point in an ellipsoid . 209 Corresponding POINTS of two ellipsoids defined . . . 211 Relation between two points on one eUipsoid and the corresponding points on another . . . . . . .211 Correspondence of extremities of conjugate diameters of two ellipsoids . 212 Correspondence of curves of intersection of a series of confocal ellipsoids with a fixed confocal hyperboloid ..... 212 Umbihcs of confocal ellipsoids correspond .... 213 Plane^cui-ve con-esponding to the curve of intersection of confocal conicoids . 213 CHAPTER XV. MODULAE AND UMBILICAL GENERATION OP CONICOIDS. PROPERTI^ OF CONES AND SPHER0-C0NIC3. Mac Ccllagh and Salmon's modular and umbihcal generation of conicoids Locus of a point moving according to the modular method umbilical Chaslbs' definition of confocal conicoids Focal and dirigent conics for central conicoids Focal and dirigent conics reciprocals of each other Line joining the foot of a directrix to the corresponding focus is normal to focal conic ...... Property of a section by a plane perpendicular to that of a focal conic Focal and dirigent conics for paraboloids Surfaces coiTCsponding to different values of the modules Properties of Conicoids deduced by the modular and ixmbilical methods Cones and sphero-conics .... Focal conics of conical surfaces .... Cyclic sections of a cone ..... Cyclic planes defined ..... Conjugate diameters of a cone .... Reciprocal cones ...... Corresponding lines and planes Cyclic planes of a cone correspond to focal lines of the reciprocal cone Theorems relating to cones of the second degi-ec, and the reciprocal theorems the 217 218 219 220 221 221 221 222 223 225 226 229 229 230 231 231 231 232 232 233 CONTENTS. Spherical ellipse and hyperbola Reciprocal sphero-conics Theorems relating to sphero-conics, and the reciprocal theorems Example of the use of spherical trigonometry Page 236 237 237 238 CHAPTER XVI. DISCUSSION OP THE GEWEEAL EQUATION OP THE SECOND DEGREE. General equation in the form u=u^ + Ui + d = . , . 242 Discriminant of m is a square, when that of «*, vanishes . . 243 Centre or locus of centres of a surface of the second degree . . . 243 Classification according to the nature of the centre . . . 246 tio reduced to ax- + P)f + yz- by transformation .... 247 Discriminating cubic ...... 248 Roots of discriminating cubic separated ..... 248 Condition of equal roots . . ... • • 248 Equations of the coordinate axes for which the terms in yz, zx, xy disappear . 249 Directions . 250 Locus of middle points of parallel chords of a conicoid . . . 250 Principal planes of a conicoid mutually at right angles . . 250 Surfaces distinguished When the roots of the discriminating cubic are finite . . . 252 When one root vanishes, and the centre is single and at an infinite distance 253 When there is a line of centres at a finite distance . . . 253 an infinite distance . . 254 a plane of centres at an infinite distance . . . 254 General classification ...... 255 Processes for finding the locus of an equation • • ' " ' ^^^ Axes of a conical envelope of a central conicoid . . • 260 CHAPTER XVII. AND classes OP SURFACES. DEGREES OF CURVES AND TORSES. COMPLETE AND PARTIAL INTERSECTIONS OP SURFACES. Degree and class of a surface defined Degree of a curve and torse .... Number of points of intersection of three surfaces , Degree of the complete intersection of two surfaces Number of conditions which a surface of the n'^ degree can satisfy Surfaces of the n"" degree which have a common curve of intersection Cluster and base of a cluster of surfaces defined Theorem by catley on partial intersections of surfaces Surfaces of which a given curve is the partial intersection Surfaces of the n"' degree, which pass through {x, ?/, z) = 0. 10. As an illustration of tracing surfaces, we will take the case of the surface whose equation is {x + 7jY = az. If a; = 0, if = az^ therefore the trace upon the plane of yz is a parabola whose axis is Oz and vertex 0. Similarly the trace on zx is an equal parabola having the .same axis and vertex. \i z — h^ [x + yY = «/f, the latter is the equation of two planes parallel to Os, equally inclined to the planes yz^ zx, therefore the trace on the plane z = Jt is two straight lines equally inclined to the planes of y.:, zx. Hence, the surface may be generated by straight lines such b GENERAL DESCRIPTION OF LOCI OF EQUATIONS. as PQ, which move parallel to the plane a-j/, constantly passing through the traces OP, Q on the planes zx, yz^ and inclined to these planes at equal angles of 135°. The shape is therefore a cylindrical surface. Locus of the Polar Equation. 11. We shall examine In order the loci of equations in Polar coordinates which involve one or more of these coordinates. (1) If the equation be F[r) =0, this is equivalent to a series of equations r = a^ r=h^ ... any one of which being satisfied the original equation is satisfied ; >• = a is satisfied by all points at a distance a from the origin, measured in any direction ; therefore the locus of F[r) = is a series of concentric spheres, whose centre is the origin. (2) If the equation be i^(^) = 0, it is equivalent to ^ = a, 6 = ^^ ..., any one ^ = a is satisfied by every point of lines through inclined to Oz at angles equal to a ; therefore the locus F{6) = is a series of conical surfaces, whose common axis is Ozj common vertex 0, and vertical angles 2a, 2/8, .... (3) If the equation be i^(<^) = 0, it is equivalent to ^ = a, <^ = /3, ..., any one = a is satisfied by every point in a plane through Oz inclined at an angle a to the fixed plane zOx; therefore the locus of F[(j)) —0 is a series of planes through Oz inclined to zOx at angles a, /3, ... . (4) If the equation involve only r and 6, as F[r^ ^) = 0, since for all values of 4> the same relation exists between ?• and 6, the locus of the equation is the surface generated by the re- volution of a curve traced on a plane passing through 0~, as this plane revolves about Oz as an axis. (;3) If the equation involve only 6 and (/>, as F[6^ ^) = (^j for every value of <^, there is a series of values of ^, correspond- ing to which if straight lines be drawn through 0, every point in these lines will be such that its coordinates will satisfy the. equation, and as changes, or the plane through Oz revolves, these lines assume new positions relative to Oz, and generate, during the revolution of the plane, conical surfaces, a conical fiENERAL DESCRIPTION OF LOCI OF EQUATIOXS. 9 surface being defined to be a surface generated by a stralglit line moving in any manner with the restriction that it passes through a fixed point. (6) If the only coordinate involved be r, <^, as in F[r^ ^) = ^j for each position of the plane through 0.z inclined at any angle (f) to the plane zOjc, there is a series of values of ?• which are constant for all values of 0, i.e. there is a series of concentric circles iu the plane, the coordinates of each point in whieii satisfy the equation. The locus of the equation is therefore a surface generated by circles having their centres in (9, and varying in magnitude as their planes revolve about the line (9^ through which they pass. (7) . If the equations involve all the coordinates, as jp(?', 6, 0) = 0, let any value, as /3, be given to cf), then cor- responding to this value there is a plane through Oz^ and if tlie locus of F{r^ 6, ^) =0 be traced on this plane, and such curves be drawn upon all planes corresponding to values of h> c, represents two planes and shews that the curve of intersection is composed of two circles, which ai'c the intersection of the sphere and the two planes. 15. It is often convenient in practice to consider a curve as the intersection of two cylindrical surfaces, whose generating lines arc parallel to two of the coordinate axes. In this way l'lU)bLE.M;S. 11 of considering curves, the equations of the surfaces are of the form As a simple example of this method the straight line joining the points n^ N in the figure on page 2 is determined by the two plane surfaces whose equations are X z and f + - = 1. b c II. Trace the surfaces represented by the equations (1) x^ + y' = ax. (^) z^ = tfx + by. (^) x" + y^ -i- z"- - 2ax + 2by + 2cz. H) X' t ?y' = (12. (5) xz' = c'y. (G) xy = az. (') cy = {c-zria^-x% («) {X + yY = e{z- x). (9) Shew that the surfaces (X + yy = « (= - X), and x + y - s = 0, intersect in a parabolic cylinder. (10) Desci ibe the three surfaces ?• = a %\nO, r = « COS0, A0=2n ^ 77 sin 40. CHAPTER III. TROJECTIONS OF LINES AND AREAS. DIRECTION-COSINES AND DIRECtlON-RATIOS. IG. Def. The (jeometrical j^^'ojection of a straight line of limited length upon any other straight line given in position is the distance intercepted between the feet of the perpendiculars let fall from the extremities of the limited line upon the straight line on which it is to be projected. 17. The geometrical projection of a straight line of limited length on a given straight line is equal to the given length multi- plied hy the cosine of the acute angle contained between the lines. Let PQ be the line of limited length, AB the indefinite line upon which it is to be projected. Let QRN be a plane through Q perpendicular to AB meet- ing it in iV, PR parallel to AB meeting QRN in R. Therefore PR being parallel to AB is perpendicular to the plane QRN^ and therefore to RN and QR, and QN is perpen- dicular to AB; hence, if PM be drawn perpendicular to AB^ MN is the projection of PQ^ and OPR is the acute angle con- tained between PQ and AB^ and since PRNM is a rectangle, MN=PR = PQcosQPR. If PQ produced intersects ABy the proposition is obviously true. riiOJELTIuNS Dl" LINE.- 13 is. Def. The algebraical iirojiction of a line I'iJ upun an iiidcHiiltc line AB g-ivcu in position is tlic projection estimated in a given ilireetion, as AD. It' a be the angle tlirougli which PQ may be su]»posed to have revolved from Pii, drawn in the positive direction AB^ the algebraical projection o( FQ = PQ cosa. If J\' lies in the opposite direction with reference to J/, a is obtuse, and FQ cosa is negative. The algebraical projection of a limited straight line upon a line given in position measures the distance traversed in the direction of the latter line in passing from one extremity of the former to the other. This consideration shews that, if all the sides of a closed polygon taken in order be projected on any straight line given in position, the sum of the algebraical projections of these sides is zero ; since, in passing round the perimeter of the polygon from any point, the whole distance advanced In any direction is zero. Hence, the algebraical projection of any side AB of a closed polygon Is the sura of the algebraical projections of the remain- ing sides commencing from A and terminating in B. Xote. In future, when the term projection is used, the algebraical projection is to be understood. 19. Let PQ be any line, PM, JAV, 2\Q three straight lines drawn in any given directions so as to terminate In Q, and /, 7», n the cosines of the angles which PQ makes with these directions. Then PQ will be the sum of the projections of PJ/, JAV, and XQ on PQ ; therefore PQ = l. PM + m . MN-Y n . NQ- 14 J)I UECTlOX-COSINliS. Direction- Cosines. 20. The direction of a straight line in space is determined wlicn the angles which it makes with the coordinate axes are known. DtF. If the coordinate axes be perpendicular, the cosines of the inclinations to the three axes are called direction-cosines. 21. To find the relation hctiveen the direction-cosines of a straiylit line. If /, ?«, n be the direction-cosines of PQ^ and PJ/, JLV, NQ be parallel to the coordinate axes, PM=PQ.I, MN^PQ.vi, NQ^PQ.n. Join PN^ then, since QN is perpendicular to iVJ/, J/P, and thci-cfore to the plane PMN, PNQ is a right angle ; hence PQ' = PN' + NQ' = P2P + MN' + NQ' ; which is the relation required. Hence the three angles of in- clination cannot all be assumed arbitrarily. 22. To find the angle hcticecn tivo straifjht lines in terms of their direction-cosines. Let PQ^ P'Q' be two straight lines whose direction-cosines are (/, /«, n) and (?', vi', n') respectively. Let PM, MN, NQ be drawn parallel to the axes, connecting DIHECTION-fOSINES. 15 iny two points P, Q, and PP', QQ perpendicular to P Q\ and let 6 be the angle between Pi} and P' Q'. Then P' Q\ the projection of PQ on P' Q\ will be equal to the sum of the projections of PJ/, JAV, xY(> on F Q\ namely, therefore PQ cos 6 = PJ/. /' + JAY. m + A"(?. »', and since P.V^PQ.I, MN=PQ.m, NQ = PQ.n, .\ cos 9 = 11' + mvi + nil ; hence, sin"^ = [F + m^ + ?«''') (P + m"^ + ;i'^) - (Z^ + mm + «»')" = [mn - m'uY + (»/' - 7i'l)~ + (/m' - rmf. AYhch 6=l7r^ W + m'm' + nn' = 0j the condition that the two straight lines may be at right angles. 2ii. To Jind the direction-cosines of a straicjJd line perpen- dicular to two strai<]ht lines whose direct ion-cersines are given. • Let /, ?«, w, and l\ m\ n be the given direction-cosines, and X, /A, V the required cosines of the perpendicular. Then from the condition of perpendicularity, 1\ -f mfji + nv = 0, and I'X + m'fj, + n'v = 0, whence (In — I'n) X + [mn — m'n) /u. = 0, . X' fi V . and — ; —=—„'= — - z= -- — -- by symmctrv, ran — m n nl —nl Im - i m "^ = + J-^, (Art. 22), if 6 be the angle between the lines. 24. T'o find the direction-cosines of two straiejlit lines ichich lie in the plane containing two straight lines, tohose direction- cosines are given, and hisect the angles between them. Let AP, AQ be the two given lines, whose direction-cosines are ?, m, n and I', m', n. Take AP=AQ = r, join PQ and bisect it in 7?, AR is one of the bisecting lines, let its direction-cosines be \, /i, v, and if 2^ be the anisic between AP and AQ, AIl=r cos^. IG DIia:CTI0N-C08INLS If ylP, AQ, and AB be projected upon the axis Ox, the projection of E bisects that of FQ ; .'. 2r cos^,X = />• + /'/•, and X= , ^ , ' 2 cost/ and simihirly for yu, and v. Produce QA to (/ so that Aq = r, and bisect Pq in r, ylr is the other bisector, and since the direction-cosines of Aq are -l\-m\—n' and J/- = 2/- sin ^, if X', /i', i^' be the direction- cosines of Ar: i-r .-. 2r sin d.X' = h- + (- l) )• and V = , . . , 2 sine/ and siniihirlj for /x' and v', ^ being determined by the equation cos2^ = //' 4 vvH + nn\ (Art. 22), 25. To find the angle between the two stra/qhf lines ivhose (I irect ion-cosines are given hy two homogeneous equations of the first and second degrees respectively. Let the given equations be ar + hni' + en' + 2a' mn + 2h' nJ + 2c'Ini = 0, and aZ + /3m + yn = 0. That there are two lines may be seen by eliminating v from the two equations, whereby we obtain the equation giving two values of / : in, r {((T' + Im^ + 2c7/;0 - 27 [al + /3m) [h'l + am) + r (a/ 4 ^m)' = 0, or vr + 2w'Jm + nni' = 0, ^vhcrc V = 07"^ — 2h''y(x + ca", w' = c'7'^ - (a'a + h'^) 7 -f ra/3, w = c/3"' - 2o'/37 + hf. Isow, let /,, 7/?,, v^, and /.,, w^, 7/._, be the direction-cosines of tlic two straight lines, then, /, : »?, and /^ : m,^ being roots of the equation, //^ _ in^vi,^ _ /,w.,+ /.^?/J, _ j{l^r)}^ + I^m^)^ — iIJ,^m^m,y- DIRECTIOX-RATIUS. 17 Now, It can be shewn, by collecting the coefficients of the different powers of 7, that 'id'' - uv = i' [A(^ + Z?y3' + C'i' + 2 J'/37 + 2i?'7a + 2 C'a/3), where A = a"^ — he and A' = aa — h'c\ and similar expressions. We have, therefore, from symmetry, ?f V w 2aP 2iSP ~ 27P ' where P' is written for the symmetrical expression ^a'H-...+ 2^ri87. Therefore, If be the angle between the lines, COS0 _ sin<^ i:tVv + ^ ^ 2P{a^ + ^Uiy ' Cor. The conditions that two such equations may represent two perpendicular or two parallel directions arc u + v + iv = Oj and P=(), respectively. The condition of perpendicularity may be written [a + b-^c] («•-' + /3'^ + r ) -/(a, /3, 7) = 0, if /(/, 7>j, ?i) = be the equation of the second degree. The condition of parallelism may be expressed by the determinant j «, c', b', a c', b, a, y3 I &', a', c, 7 = 0. Direction-ratios. 26. Def. If the coordinate axes be not perpendicular to each other, the direction of a line PQ is fully determined, when the ratios of PJ/, MX, NQ to PQ are given, P.l/, MN, NQ being parallel to the axes. These ratios arc called (lirect ion-ratios. D 18 PROJECTIONS. 27. To fiuJ the relation between the direction-ratios of a straight line. In the figure on page U, let the angles yOz^ zOx, xOy be \, /i, v, and let a, /3, 7 be the angles between PQ and the axes, 7, 7», 71 the direction-ratios of PQ. Projecting the line PQ and the bent line PMNQ terminated in the same points on Ox^ PQ oosoL = PM+ MN cosv + NQ cos /a, .-. cosa= ? + ?« cosv4-ncos/i;' similarly cos^ = Z cos v + m + n cosX, j and COS7 = I cos//. + m cos\ + n. Also, projecting PMNQ on PQ^ PM cos a + i/iV cos /3 + NQ cos 7 = PQ, .-. Z cosa + ?7i C0S/3 + ?? cos7= 1, .'. 1 = Z' -f vf + 7i'' + 2mn cos\ + 2nl cosyu, + 2lm cosj/, which is the relation required. Projection of a Line on a Plane. 28. Def. The orthogonal projection of a line of limited Icngtli on a plane is the line intercepted between the perpen- diculars drawn from the extremities of the limited line upon the plane. 29. The orthogonal projection of a line upon a pi the length of the line multiplied hy the cosine of the angle of clination of the line to the plane. ane ts in- ^ Q P ^^^ X A / M / h / rROJECTIONS. 19 Let PQ be the given line, AD the plane, P.1/, (>-^" perpen- diculars upon the plane. Since PJ/, QN arc perpendicular to the plane AB^ PM is parallel to QN^ and the plane MPQX is perpendicular to the plane AB] join JAY, and draw PL parallel to MN] .'. lPLQ = l MNQ = a right angle ; .-. JAV= PL = PQ cos QPL, and MN is the projection of PQ on AB, L QPL = the inclination of PQ to the plane, whence the proposition. Projection of a Plane Area upon a Plane. 30. Def. The orthogonal projection of a closed plane area upon a fixed plane, is the area included within the line which is the locus of the feet of perpendiculars drawn from every point in the boundary of the plane area. If a series of planes be taken forming a closed polyhe- dron, the algebraical projections of the faces upon any plane are their areas multiplied by the cosines of the angles which their normals, draicn outwarcb, make with the normal to the plane. 31. T/ie orthogonal projection of any plane area on a given 2)lane is the area multiplied hy the cosine of the inclination of the plane of the area to the given plane. 20 PROJECTIONS. Let APB be any closed curve described upon a given plane, and A'P'B' the orthofjonal projection upon any other fixed plane, which is the locus of the feet of the perpendiculars drawn to the second plane from every point of the curve APB. The areas APB^ A'P'B' may have inscribed in them any number of parallelograms, such as PQ^ P'Q'-, whose sides are in planes PJ/P', Q2^Q' drawn perpendicular to the line of intersec- tion of the given planes, and parallel to that line, and these parallelograms are in the ratio of 1 : cosine of the inclination of the planes ; therefore the sums of the parallelograms are in the same ratio. Hence, proceeding to the limit when the breadths of these parallelograms are indefinitely diminished, the area of the pro- jection of ^Pi? = area of J-PP x cosine of the inclination of the planes. 32. If the faces of any closed polyhedron he projected on any plane^ the sum of the algebraical projections of the faces on any fixed plane loill he zero. One side of the fixed plane being selected as that to which the normal is drawn, the angle between this normal and the normal, drawn outwards, at any point of the closed polyhedron, is quite definite ; and the projection of any face will be positive or negative according as this angle is acute or obtuse. Now any straight line whatever (produced indefinitely both ways) will meet the polyhedron in 0, 2, 4, ... or some even number of points, since passing from outside to inside, or from inside to outside, necessitates crossing a face once. Draw a straight line parallel to the normal to the plane of projection meeting the, polyhedron in points P^, P^, P,, ..., Pj„, and round it an indefinitely small cylinder whose transverse section is a, then the projections of the sections of this cylinder made by the faces of the polyhedron which it meets will be alternately + a and — a, and since the number of them is even, their sum will always bei»i:o. This being true for every straight line per- pendicular to tlhe. plane of projection, will be true for the total ]>r()jeclion of the polyhedron ; and will also be true when the number of faces is indefinitely Increased, and the areas of some. PROJECTIONS. 21 or all of them, diminished indefinitely; that Is, the sum of the algebraical projections of all the elements of a closed surface on any fixed plane is zero.* 33. To Jind the area of any i^lane surface in terms of thelareas of the projections ujwn any rectangular coordinate planes. Let /, w, n be the dlrcctlon-coslnes of a normal to the plane on which the given area A lies, A^^ A^^ A^ the areas of the projections upon the coordinate planes oi yz^ zx^ xy. Then, since I is the cosine of the angle between Ox and the normal to the plane, which is the same as the angle between the plane of A and the plane of ?/.^, A^ = Al, and similarly, A^ = Am^ and A^ = An ; .-. A" = A' [r + m' + ri') = ^/ + yJ/ + A^. 34. To find the plane upon ichich the sum of the projections of any number of given plane areas is a maximum. Let A^ A\ A"... be any number of plane areas, 7, ?», w, l\ m\ n ... the direction-cosines of the normals to their planes, X, yii, V those of the normal to a plane upon which they are projected ; and let ^^, A^., A^ and A'^^ A\^ A\... be the areas of the projections of the given areas upon the coordinate planes. Then since l\ + ?»/* + nv is the cosine of the angle between the plane of-i-1, and the plane upon which It is projected, the projection of A is A [l\ + mix -\- nv) = A\ + A fji, + Ay] therefore the sum of the projections of all the areas upon the plane (X, /i, v) is \2 (JJ + /*:£ UJ + vS (^IJ which is to be a maximum by the variation of X., /j,, v, subject to the condition .: 2 [A J dX -\- 2 (yl J dfi+1. (^1.) dv = 0, and \dX + fid/x + nlv = 0, must be true for an Infinite number of values of d\ : dfx, : dv ; X _ H' _ V _ 1 • Sec Thomson and Tail's Elements of Natural Pliilosophy, Arts. 446-150. 22 PROBLEMS. which determine the direction of the plane of projection, in order that the sura of the projections of the areas may be a maximum. III. (1) Two straight lines are drawn in the planes of ry and yz, making angles a, 7 with the axes of x, z respectively ; the direction-cosines of the straight line perpendicular to the two are proportional to tann, - 1, tan'/. (2) If two straight lines he inclined at an angle of 60^, and their direction-cosines be /, tn, n, /', m', 71', there will be a straight line whose direction-cosines are I - I', m - m', and n - n', and this straight line will be inclined at angles of G0° and 120° to the former straight lines. (3) If the angles which a straight line tlirough the origin forms with the coordinate planes be in arithmetical progression, whose difference is 45°, the line must lie in one of the coordinate planes. If it form angles a, 2a, '3a with the coordinate axes, it must lie in one of the coordinate planes. (4) The angle between two faces of a regular tetrahedron is sec'' 3. (5) Find the angle between the two straight lines, whose direction- cosines are given by /* + wt' = «' and I + ?« + n = 0. (6) Shew, by projecting upon the base, that the area of the surface of a right cone is tto/, a being the radius of the base, and I the length of a slant side. (7) Shew a priori that the rational equation connecting the direction- cosines of a sti-aight line can only involve even powers of those quantities. (8) Three circles whose areas are in the ratio 3:4:5 lie in three perpendicular planes, shew that the plane on which the sum of the projec- tion is greatest is inclined at an angle 45° to the plane of one of the circles. (9) If a plane mirror be equally inclined to each of the three coordinate planes, and X, ;«, v be the direction-cosines of a ray incident on it, shew that those of the reflected ray will be J (2/t + 2i; - X), I {2t> + 2X - /.), and J (2\ f 2/i - i>). (10) If cO be the small angle between two lines, whose direction-cosines are respectively /, tn, n and / + f>l, m + ^m, n + 6n, prove that '60]* = I/]' 4 0^7/1' f 6^,]*- (11) DL-lerminc the plane and the area of the maximum projection of the hexagon formed by the six edges of a cube that do not meet a given diagonal. PROIJLEMS. 2;} (12) The sum of the three acute angles which a straight line forms with three rectangular coordinate axes is less than 120'^. (13) The sum of the acute angles which any straight line makes with rectangular coordinate axes can never be less than | scc''(- 3). (14) The direction-cosines of a straight line perpendicular to the two whose direction-cosines are proportional to /, tii, n and tn + n, n + /, / + wj, are proportional to m - 71, n - I, I - m. (15) The straight lines vhose direction-cosines are given by the equations al + bm + en =0, aP + /3m' + 7?j« = 0, will be perpendicular, if a' (/3 + 7) + b- (7 + o) + c^ {a ^ (i) ^ 0, and parallel, if — + — + - = 0. a 13 7 (16) The straight lines whose direction-consines are given by the equations fl^ t bin + en = 0, l m n will be perpendicular, if - + ^ -I- - = 0, a b c and parallel, if V(««) i V(^/3) ± V(c7) = 0. (17) The direction- cosines of a line making equal angles with three straight lines whose direction cosines are (/, m, n), (/', ?;»', n'), {I", m", n"), are proportional to m (n' - n") + m' («" - n) + m" (ti - n'), n (l' -l")^}i' {I" - 0-f n" (^l-l'), I (m' - »«") + I' (m" - m) + I" {m - »»'). If the given lines be mutually at right angles, the direction-cosines will be I + r+ t' TO 4 m' + jn" n -\- n' -¥ n" (18) If the direction-cosines of two straight lines be given by the equations amn + bnl f chn - 0, al + (im + 7?* = 0, prove that the tangent of the angle between the lines will be {(g* \ /3' + 7') C^"^' +•••- 2tc^7 -•••))^ , a/37 + 67a + Ca/3 24 PUOBLEMS. (19) Find the direction-cosines of the two straight lines which are equally inclined to the axis of z, and are perpendicular to each other and to the line which makes equal angles with the coordinate axes. (20) If A, B, C, D be four points in a plane, A', B', C, B' their projections on any other plane, the volumes of the tetrahedrons ^5 CD', ^'iy'C'i)willbe'equal. (21) If I, m, n be the cosines of the angles which a straight line makes with three oblique coordinate axes, and X, yu, v be the angles between the axes, P sin'X + m' sin*/* + ?»' sin'i' + 2mn (cos/x cos v- cosX) + 2nl (cos V cos X - cos /i) + 2hn (cos X cos /t - cos v) = 1 - cos*X - cos''/t - cos^ f + 2 cosX cos /t cos v. (22) If A, B, C, D be the areas of the faces of a tetrahedron ; a, h, c, a, /3, 7 the cosines of the dihedral angles (BC), {CA), (AB), {DA), {DB), (DC), respectively; then will A' ^ JB* ^ C^ 1 _ „« _ i« _ c» - 2abc 1 - a« - /3= - c- - 2fly3c ~ 1 - «* - i' - 7' - 2a&7 ^ n^ " 1 - a' - i' - c' - 2ahc ' U^• I \ i:i:srr V oi- I (JALIKUUMA. CHAPTER IV.. DIVISION OF LINES IN A GIVEN RATIO. DISTANCES OF POINTS. EQUATIONS OF A STRAIGHT LINE. 35. To find the coordinates of a point ivhich divides tlie straigJit line joining tioo given points in a given ratio. Let the given points be P{x^ y, z), and P' [x\ y\ z')^ and let Q divide PF in a given ratio, so that PQ : QP' :: X' : \. If M^ jV, M' be the feet of the ordinates of P, Q^ P' parallel to O2, and viQni parallel to JAVJ/' meet MP in ?», and M'P' in m\ Pm : 7n'P' :: PQ : ^P' :: V : A, ; .-. if |, 7;, ^be coordinates of Q^ and similarly for f and 77, When ^ lies in PP' produced in the direction of P', PQ and ^P' being measured in opposite directions are affected with opposite signs and X is negative. In like manner, when Q is in PP' produced in the direction of P, X' is negative. In all cases due regard being paid to the signs of X and X' have PQ^QP'^ PP' X' X X + X" Distance between tivo points. 36. To find the distance between two points whose coordinates are given^ referred to rectangular axes. Let (a:, 7/, s), (x', ?/', z) be two points P, (> whose coordi- nates are given referred to a rectangular system ; arid let a parallelepiped be constructed whose diagonal is PQ, and whose edges PM, MN, NQ are parallel to the coordinates axes O.v, Oi/, Oz- and join PN. E 26 DISTANCE HCTWEEN TWO I'OINTS. M N Then, since Q'S is perpendicular to the plane PMX, and tliereforc to PX, PQ' = PN'+QN% but PN' = P2P-\-MN'', .-. PQ' = PJ\P + MN" + NQ\ PM is tlie difference of the algebraical distances of Q and P from the plane i/Oz^ and similarly for 3/xV, NQ: .'. PCf = {x - xY + i'^' - yf + (^' - z)\ \i a, /9, 7 be the inclinations of PQ to the axes of coordinates, X —x = PQ cos a, 7j -y = PQco^^, z - z —PQ COS7, .•. 1 = cos'^a + cos'^yS + cos' 7. The double sign, which appears in the value of PQ^ may be interpreted in a manner similar to that adopted in the case of the radius'vcctor in polar coordinates in Plane Geometry. If the angles a, /S, 7 define the direction of measurement of the distance PQ of Q from P, the opposite direction is defined by TT + a, TT + /3, tt + 7, and therefore these angles with an algebraical distance — PQ equally determine the position of the point (J with reference to P. The distance of the point {x\ y\ z) from tho origin is DISTANCE BETWEEN TWO PuINTS. 27 37. lojiiid the distance between two points nfernd to oblique axes. Let A,, /i, V be the angles between the axes ; and {x, i/, s), {x'j y\ z) two points P and Q. Let a parallelepiped be constructed whose diagonal is FQ^ and whose edges i'J/, JAV, IS^Q are parallel to O.r, Oy^ Oz. Now, the projections on PJ/ of the line P^, and of the bent line PMXQ terminated in the same points, are equal. Therefore if a, /S, 7 be the angles which PQ makes with the axes, PQ cos a = PM + 3IX cosv + NQ cos/i, i similarly PQ cos/3 = JLY+ XQ cosX + PM cos v, > (1). and P() C0S7 = NQ + PM cos^ + MX cos?i, J Also PQ is the projection ofPMNQ on P(2; .-. P() = PJ/cosa + .lAVcos/S-f-iV(?cos7 (2). Therefore multiplying the equations (1) by PJ/, J/iV, iV(2 we have by (2), PQ' = PM' + JAV^ + NQ' + 2MN.NQ cosA, + 2^V^ . PJ/ cos /x + 27^J/. MN cos v, and PJ/ is the difference of the algebraical distances of Q and Pfrom yOz, and therefore =x' - a-, and similarly 2IN=y' - y, and NQ = z' — z; .-. P(^' = (.V-crr+(y-3/r+(^'-4^ + 2(3/'-.y)(^'-..)cos\ + 2 (2;' - z) [x —x] cos fi + 2 [x — x) [y — y) cos v, whence PQ is determined as required. 38. If /, 7??, n be the direction-ratios of PQ^ PM=.l.PQ, MN=m.PQ, NQ = n.PQ; .'. l = r + m^ 4 w"'^ + 2w7i cosX + 2nl cos^u, + 2//n cosv, which is the equation connecting the direction-ratios of any line referred to oblique axes. 39. To find the distance of two iwints ichose iwlar coordinates are given. Let (?•, 6, (f)) and (r', 6', cf)') be the given points P and Q. 28 THE STRAIGHT LINE. Join OP, OQ, QP, and let a spherical surface, whose centre is and radius unity, intersect OP, OQ, and OZ in ^), (7, and r. Then, rp = ^, ?7 = 0', and Z qrp = )} ; .-. P(2' = r' + )•"' - 2rr' {cos^ cos 6' + sin 6 sin ^' cos(<^' - 0)], whence the distance PQ is determined in terms of the polar coordinates of P and 0. 40. Tiie distance may be de- termined without Spherical Tri- gonometry as follows. Draw PJ/, QN perpendicular to the plane of xy^ join JAY, OM and OX, and draw PR perpen- dicular to QN] .-. PQ'=QK' + PB' = QR'-\-3IN'\ QR = r COS & — r cos ^, and J/A"' = OJP + O.V^- 20J/.O.VcosiI/OiV = r' sin^ 6 + r'^ sin' ^' - 2rr' sin ^ sin ^' cos (0' - 0) ; .-. P()' = r'i-r"-2rr' {cos^ cos^' + sin^ sin^' cos(0'- <^)}, which gives the required distance. T/tr Straifjht Line. 41. The general equations of the straight line wliich will be employed are of two forms : one form is symmetrical, and the equations are deduced from the consideration that tlic position of a straight line is completely determined, when one point in the line is given, and the direction in wliidi the straight line THE STKAIUIIT LINE. 29 is drawn. The symmetry of this form gives great advantages, and in all questions of a general nature the general symme- trical equations will be almost exclusively employed. The other form is unsymmetrical, and the equations are deduced from the consideration that a straight line is the intersection of two planes, and is completely determined when, the equa- tions of the two planes are given. These equations in their simplest forms are the equations of planes parallel to two of the coordinate axes, and are the same as the equations of the projections of the straight line parallel to these axes upon two of the coordinate planes. It will be seen that, in cases in which the elimination of the constants is an essential part of the solution of a problem, the unsymmetrical equations may be used with advantage. 42. To find the symmetrical equations of a straight line. Let A be a fixed point (a, 6, c) of a straight line, P any other point (x, ?/, z)^ /, ?«, n the direction-cosines of AP] and let AP= r. Then the projection of AP on the axis of cc is x — a^ and ^ ^ qi A it is also Ir. hence — j— = ^\ and, similarly, = ?•, and also = r. The equations of the straight line are therefore n X — a _y — h _z — c m n I - ,., - .. » X— a 71 — 1) z — c if Z, J/, ^Varc any quantities proportional to /, in^ n. It should be carefully remembered tlvat, when the former equations are used, each member of the equations is equal to the distance r of the current point {x^ y, z) from the fixed point (a, bj c). The equations of a straight line will be of the same form if the axes be oblique, the same interpretation being given to r, and I, ?», n being the direction-ratios. Tiie projections employed In the above proof will then be the intercepts on 30 THE STRAIGHT LINE. the axes made by planes through A and F parallel to the coordinate planes. 43. To find the non-syynmetrical equations of a straight line. If a straight line PQ be projected by straight lines parallel to the axes Oy^ Ox^ whether rectangular or oblique, on the two coordinate planes a-z, yz^ each projection will be a straight line, ^^ 2^1-) p q •) in these planes respectively. Hence, th 3/ coordinates x , -he coordinates a?, z of any point (.r, y^ z) in FQ being the same as those of the projection of the point in pq^ satisfy an equation of the form x=pz-\-h^ and the coordinates ?/, z similarly an equation of the form y = qz + k'y and, con- sequently, the equations of the line may be written x = 2}z + h, y = qz-\-]c. 44. On the numher of indc-pendent constants employed in the equation of a straiyht line. It may be noticed that the latter system of equations involves only four constants, whilst the symmetrical system involves six. Of the three /, ?«, n, however, we know that they are connected by the relation V -^ m"^ ■\- n^ = \ (Art. 21) or an equivalent relation (Art. 27) if the axes be oblique, which renders them equivalent to only two independent constants; and, if we take Z, ^7, N^ since they are only required to be thl: sTnAKiirr lini:. 31 proportional to I, m, n, one of these may be assumed arbitrarily, and they are still equivalent to two constants only. Also, of the three a, b, c, one may be assumed at pleasure ; for, since the straight line cannot be parallel to all the co- ordinate planes, let it not be parallel to that of t/z ; then at whatever distance a from i/z we take a parallel plane, the straight line will meet this plane, and we may take the point where they meet for the point [a, b, c), that is, we may give to a any value we please, and the three «, h, c are consequently equivalent to two independent constants only. 45. To find the equations of a straujlit line parallel to a coordinate plane. If a straight line be parallel to a coordinate plane, as that of yz^ every point in it is at a constant distance from this plane, and we have the equation x = /«, therefore the equations will be of the form Taking the symmetrical form, since the line will be per- pendicular to the axis of ar, Z=0, and therefore ^ = 0, and the equations of the line assume the form x — a__y-b_z-c_ m n ' X- a _y — b _z — c '''■ "IT ~~jr '"W' which form implies that x = a for every point in the line at a finite values. a finite distance, since the members are not infinite for sue 46. To find the equations of a straight line p)o.rallel to one of the coordinate axes. If the straight line be parallel to one of the coordinate axes, it will be parallel to the two coordinate planes passing through that axis, and consequently any point in it will be at an invariable distance from each of these planes. Hence, if a straight line be parallel to the axis of .~, the distances of any 32 THE STHAIGIIT LINE. point in it from the planes yz, xz -svill be constant, a fact expressed by the equations x = h^ ?/ = ^•, which will, therefore, be the equations of the line. As before, the symmetrical form is x — a_y — l>_z-c . ~ ~0~ ~ 'W ' 47. To find the angle hetween tioo straight lines lohose equa- tions are given. If the equations of a straight line be given in the form x—a_y—h_z—c then, if /, ?«, n be its direction-cosines, i _ m _ n _ ± V(^' + m' + ?i^) ±1 or the direction-cosines will be ±L ±M ±N If the equations be given in the form x=2yz + h, y = qz + kj since these may be written X — h y — k _z the direction-cosines of the line will be ±P ±q ±}^ In (1) and (2) the ambiguities have the same sign. Hence, if the equations of two straight lines be x — a _y~h _ z — G x — d _y — h' _ z — c' (I). (2). TIIE STRAIGHT LINE. 33 the angle between thoui will be LL' + MM' + NN' And, if the equations be x=^z +;., y = qz +/.-, x=j)z-rli, y = qz + h\ the angle between them will be _, pp + g, c) be the given point, we Iiave already seen that the symmetrical form x — a y — h _z — c will represent a straight line passing through that point. The unsymraetrical form is > x-a=^2^{z -c)^ g-h = q{z-c). 52. To find the equations ofi a straight line passing through a given point and parallel to a given straight line. The equations of a straight line passing through a given point [a^ h, c) are x — a y — l> z — c L ^' T/ " iV ' and if this be parallel to a straight line whose direction-cosines are I, //?, ??, ^ = -^'=-^\ (A.-t.4.S) t m 71 theretore the required equations will be x — a_y — h_ z - c. I m 11 53. To find the equations of a straight line passing through a given pointy and perpendicular to and intersecting a given straight line. Let (rt, bj c) be the given p(/int, and the ccpiatiiMis of the v,givcn straight line be x — a' y — l>' Z — r I ni n nc THE STRAIGHT LINE. will bo the required equations of the straight line, where the ratios L : M : N are to be determined from the equations LI + Mm + Nn = 0, (Art. 49) a — rt, /, L h' - h, m, M = 0. (Art. 50) 54. To find the equations of a straicjid line passing through a jyoint and intersecting two given straight lines. Let (a, h, c) be the given point, and let the equations of the two given straight lines be X- a y-y M' N' and L" M"~ N" and let the equations of the straight line satisfying the re- quired conditions being x — a y — l> g — c By the conditions of intersection given in Art. 50 Z«, il/, N satisfy the equations LF-^-MQ +NB' =0, LP" + MQ" + NB" = 0, wliore P'j (/, J?', &c., are the first minors of the two cor- responding determinants, whence the equations of the straight line become X -a _ y — h _ z — c Q'li" - Q'R 11 F'- R'F ~ FQ'- F" Q" ' ^h). To find the equations of a straight line imssing through a given pointy ^>ara//e/ to a given ^)7a«<', and intersecting a given straight line. Ijct [a, li, c) be the given point, /, m, n the direction-cosines of a iiormal to the pKine, which will therefore be perpendicular THE STRAIGHT LINE. 37 to the straight line -whose equations arc required, and let the equations of tlie given straight line be X — a y — h' z- c '^ *^ I m n The required equations will then be , x — a_y-h_ z-^^c ( ' /., ~Tr "'"17' " N' ' /y "'V where L\ M\ jVare determined by the equations-^'/- /' LI + Mm + Nn = 0, and a - rt, Z' L y — J, m^ M 0. X C - C, 7l\ iV 5G. To find the distance from a given point to a given straight Let A be the given point [x\ ?/', z')^ y—h _z—c m n X — a the equations of the given straight line, B being the point (rt, i, c) ] AP the pei-pendicular from A on the straight line ; then the projections of BA on the axes of a;, y, z are respectively x — a, y -h^ z —c] and the projections of these on the given line are l[x-a)^ m{y'-b), 7i{z' — c)^ but the sum of these projections is the projection of BA on the straight line, or BP= I [x -a)+ m {y -h)-\-n {z - c) ; hence, ^P-' = 7?yr-i?P-^ = {x - aY^ {ij'- hY^ {z-cY - {I{x- a) + m {y'-h) + n [z' - c)]% giving the required distance, which may be written V[(n(y-?.)-m(.'-c)}^+.[/(e'-c)-n(x'-a)r+{m(a;'-a)-/(y-&)n. If the equations of the line be x=2)z + h, y = Qz + kj which may be written x — h _ y — h_z 38 THE STRAIGHT LINE. the distance will be ■J p{x'-h)+q{y-h) ^ z'Y f + !2' + 1 ]■ 57. To find the equation of a circular cylinder^ the equations of ichose axis and the radius of a circular section of which are given. The circular cylinder being- the locus of a point whose distance from tlic axis is constant and equal to the given radius r, if the equations of the axis be X- a y - h z — c I m w ' the equation of the surface will be, by the preceding article, (a; - aY+ [y - hf^ {z - cf- {I [x - a) + m [y - h) + n {z - c)Y= r. 58. To find the equation of a circular cone, whose vertex, vertical ancjle, and the equations of whose axis are given. If V be the vertex, P any point of the cone, FQ perpen- dicular on the axis, and 2a the vertical angle, VQ'= TT'cos'^a; therefore, if («, b, c) be the vertex, the equations of the axis as before, the equation of the cone will be [f(_x-a) + vi{y-h) + n{z-c)Y-=cos'a{{x-ay + {y-hY+{z-cY}. 59. To shtic that the shortest distance between two straight lines which do not intersect is perpendicular to both. Let AP, BQ be the two straight lines, and let a plane be drawn through DQ parallel to AP, and BR be the orthogonal projection of ylPupon this plane, B being the projection of ^ ; therefore AB will be perpendicular to both straight lines, for it meets two parallel lines AP, BR, to one of which, BR, it is perpendicular, and it is also pcrpcndicuhir to BQ, since it is drawn perpendicular to the plane (jIUl. THE STliAUllIT LINE. 39 Let P, Q be any points in ylP, BQ^ join PQ^ draw PR perpendicular to BE^ and join (?/i; then PQ is greater than Pli^ being opposite to the greater angle, and PR = AB] there- fore AB is less than PQ^ or the distance which is perpendicular to both.straight lines is less than any other distance. 60. To find the shortest distance between two straight lines whose equatio7is are given. ^ Let the equations of the two straight lines be x — a y — h z — c ^ X — a y - b' z— c I m n ' I' ni n' ' and let X, /i, v be the direction-cosines of the straight line per- pendicular to each, then (Art. 23) \ _ /^ _ '' _ ^ mn — m'n nl' — n'l hn — I'm sin 6 ' 6 being the angle between the lines. Kow, if we suppose P, Q^ in the last figure, to be the points (rt, 5, c), («', h\ c'), the projection of PQ on AB^ which is AB itself, will be X(a — a') -f /x (i — Z*') + v (c-c'), hence (rt - a) [mn — m'n) + (& - h') [nl — n'l) + (c - c') [hn — I'm) AB = [[mn — m'n)'^ + [nl' — n'l)'^ + [hn — rvi)'^]^ 6L To find equations of the line on which lies the shortest distance between two straight lines ichose equations are given. Taking the equations of the last article, if (|^, ?;, f) be any point of the line considered, the equation of the line will be x-^ ^ y -y _ ^- K mn —m'n nl' n'l hi ii we have by Art. 50, since it meets each of the two given lines. mn' — rn'n^ l\ V nl'- , n n'l, hn' — I'jn h\ t- c = 0, 40 PKoDLli.MS. and, since (f , ?;, f ) is any point on the line these are equations of the line. If Ij 7», n and T, m, n be direction-cosines, since m [hn - I'm) — n [nl - nl) = I {mm + nn) — I' [m^ 4- n^) = I ill' + mm' + nn) — /', these equations may be written [I co%6-l'){x-a)^-{m cos9-m') {i/-h) + {n cos6-n')[z~c) = 0, (r cosd-l){x-a') + {m' cosB- m){ij-b') + {n' cosd-n){z-c')=0, where 6 Is the angle between the given lines. 62. A very simple form, lu which the equations of two straight lines can be presented, will be obtained by taking the middle point of the shortest distance between them for the origin, the line in which It lies for one of the axes, suppose that of z, and the two planes equally inclined to the two straight lines for those of zx, z>/. If 2a be the angle between the two straight lines, 2c the shortest distance between them, their equations will then become y = x tan a, z = Cj and y =- — x tan a, z — — c. IV. (1) The straight line given by the equations a; + 2y + 3r = 0, 3x + 2y + s = 0, makes equal angles with the axes of x and s, and an angle sin'' -;7- with Y'3 the axis of y. (2) Prove that the equations - = = represent seven X -\ y-l s - 1 straight lines which all pass through the same point. (3) Find the direction-cosines of the straight line determined by the equations Ix + my -f nz = jnx + tiy ( h = nx 4 ly + mz. (4) The angle between the two straight lines given by tbe equations X = y and xy ■{ yz -^ zx = U, is sec"' 3. PROBLEMS. 41 (5) Find the equations of the straight line passing through tlie points {b, c, a) {c, a, b), and shew that it is perpendicular to the, line passing through the origin and through the middle point of the line joining the two points, and also to each of the straight lines whose equations are y y h (6) Find the shortest distance between the axis of z and the straight r line —y- = '- — = - , a, r) is in the locus of the equation (1)^ Aa +Bb + Cc +B = 0, (4) similarly, Aa + Bh' + Cc + i) = 0, .-. A[a'-a)^B[h' -h)+ C{c-c)=Oj whence, the conditions (3) give Al -h Bni 4 On = 0. (5) Now the straight line (2) meets the locus of (I) in all points for which the equation in ?• A{a-\- h) + B{h + mr) + C (c + nr) +B = 0, is .satisfied, i.e. for all values of r, by (4) and (5) ; therefore, every point in the .straight line lies in the locus, and this is true wherever the two j)oints («, b, c), [a\-b\ c) are chosen. Hence, the locas is a plane. EQUATION OF A PLANE. 45 G4. The student will readily deduce the following special positions of the plane. (1) If i) = 0, the plane passes through the origin. (2) If A = 0, the plane Is parallel to the axis of x. (3) If -4 and i?= 0, the plane Is parallel to the plane of xi/. (4) 1{A,B and i) = 0, the plane Is that of xij. (5) If yl, i? and C=0, while D remains finite, the plane is at an infinite distance. For, the point in which the axis of x meets the plane is given by the equations y = 0, z = 0, Ax + B = 0-, hence, the distance from the origin being — j , if --'I be in- definitely diminished, while B is finite, the plane cuts the axis of X at an infinite distance from the origin, and the same being true for each axis. It follows that the plane is at an infinite distance from the origin. 65. It is important to observe that the existence of three arbitrary constants in the general equation of the first degree, viz. the three ratios A : B : C : B, shews that a plane may be made to satisfy three conditions, provided each condition is one which gives only one relation between A, B, C, B. Thus, passing through a given point at a finite or infinite distance is such a condition, but being parallel to a given plane is equivalent to two such conditions. Equation of a Plane. C6. To fncl the equation of a plane in the form Jx + my + nz= p, /// vhich p is the perpendicidar from the origin upon the plane ^ ami /, ;h, n its direction-cosines. A plane may be considered as the locus of a straight line which passes through a given point, and is perpendicular to a given straight line. Let OB=p be the perpendicular from the origin upon a 46 THE I'LANE. plane, ?, ?«, n Its direction-cosines, {x, ;/, z) any point P in the plane, then, by the definition, PD is perpendicular to OB, and OD is the sum of projections of the coordinates of P on OD ; .•. Ix + my + nz =p, ^vhich is the equation of the plane in the form required, in which, if the axes be rectangular, F + m^ + n' = 1. 67. Interpretation of the expression p — lx— my - nz. The equation p — Ix — my — nz = Qi represents a plane, in which 2^ is the perpendicular from the origin, and /, ;», n arc its dircction-cesincs. TllK PLANE, 47" Let ABC be this plane, and suppose (97), QR to be drawn perpendicular to it, in the direction defined by (/, ?/?, «), from the origin, and from the point Q (a-, ?/, z)^ and join IW, which will be perpendicular to OD. Let QE = q^ and project rr, ?/, .t; and q on OZ), then p = Ix + m?/ -\-nz-\-q] .-. q =^p - Ix — my — nz. Hence, the expression ^j — 7.r— ??«?/ — ??2: represents the per- pendicular drawn from (.r, y, z) upon the plane 21 — Ix — my — nz = 0, estimated positive in the direction defined by the cosines 7, m, and )i. G8. To Jind the angle between two j^Ia'^cs ichose equations oj-e given. Let Lx + My + Xz^D^ and L'x-\- M'y + N'z = D\ be the given equations ; then (Z-, il/, N) and {L\ M\ N') arc proportional respectively to the direction-cosines of the normals ; but the angle between two planes is equal to the angle between their normals, hence the angle between the planes is LL + MM' + NN' The conditions of parallelism and perpendicularity are there- fore respectively ^^ — ^^^^' ^V. (L _ ^ _ ^ \L'~ ]\l'~Wj and Z^r^rJ^fzin^xY' = 0. The student may also deduce the conditions of parallelism from the consideration that parallel planes intersect in a straight line at infinity, or directly from the parallelism of the normals. 69. To find the angle between a straight line and iilane ichose equations are given. j.x-a y-b z-c . L'x-\^M'y^N'z = D, (2) 48 THE PLANE. be the given equations. The angle between a straight line and a plane is tlie complement of the angle between the straight line and the normal to tlie plane ; hence the required angle is • LL'+MM'^NN' 70. To determine the perpendicular from a point {/, g, h) i(pon a jylane whose equation is Ax 4 Bi/ + Cz + D = 0. If we compare the equation Ax + Bi/+Cz + D = with the equation of the plane in the form Ix + m7/ -f nz —p) = ; 1 ,, I m n '^''''a = -b = -c A.=4- D~- ^J{A' + B'^C^')' where, if the ambiguous sign be so taken that p shall be an absolute length, ?, m^ n will be completely determined. Tlie perpendicular from ( /, //, li) upon the plane, estimated positive when drawn in the direction defined by these cosines, =P - V- ^"9 - "^^ - Af+Bg+Ch + D + s/{A' + B-'+ C^)' that sign being chosen Avhich is the same as that of D. 71. Tojind the distance from a given point to a given plane ^ measured in any given direction. Let the equation of the plane be Ax ■\- By + Cz + D = 0, and let (/, ^, h) be the given point, (/, ?>?, n) the given direction, /, ?;j, n being direction-cosines for rectangular axes, and direction- ratios for oblique. The equations of a line drawn through (/, g^ h) in the given direction arc / m n ' EQIIA.TION OF A PLANE. 49 and where this straight line meets the plane, A (/+ Ir] + B{g-^ vir) -\- C{h + «r) +D = 0', ,, • 1 V . • Af+Bg+Ch + D .•. the required distance is —n — '^ 7^ — . Hence, if the given direction be perpendicular to the plane, and tlie axes be rectangular, ]_ _m _ n _Al + Bm + Cn 1 A~B~~C~ A' + B'+C' ~ ^TI^TB^'+C^) ' and the perpendicular distance will be _ ' .. ./ — ^^ — r^^ , the sign being chosen so that — -,,--r^ — ^, — ;=^ is positive. •^ ^ + \/(^ +B'+ C) ^ 72. Tojind the equation of a plane in the form - + f + - = 1. a c Let OA = a, OB=b, OC=c be the intercepts on the axes o( X, y, z by *he plane ABC, and let PA, PB, PC, PO bo drawn from the point P [x, y, z) in the plane. y Draw PM parallel to xO, meeting yOz in M. Since the pyramids POBC, A OBC are on the same base, vol POBC : vol OABC :: PM : AO :: x : a- x vol POBC a vol OABC ' II 50 EQUATION OF A PLANE. c,. ., , y \o\FOCA z vol POAB c~ voXOABC and vol POBC + vol PO CA -f vol FOAB = \o\ OABC] a h c ^ which is the equation required. The student is recommended to investigate this equation by the employment of a figure in which P lies in another compartment, as xy'z^ of the coordinate planes, taking care to Interpret the geometrical Into algebraical distances. 73. If q be the perpendicular from a point Q {x^ ?/, z) on the plane ABC estimated In the direction of ^>j the perpendicular from on the plane, q_ ^ vol QABG f " vol OABG _ X y z a h c ' 74. The equation of Art. 72 may be obtained from the geneml equation of the first degree. For let «, bj c be the intercepts on the axes of a;, ?/, ^, Ax + Bi/+ Cz-\- D = Oj the equation of the plane. Since (a, 0, 0) is a point In the plane, — D= Aa^ and, similarly, =Bb= Cc. Ilencc, the equation of the plane Is -+ "I + - = I. ^ a c lb. To find the equation of the plane in the form z=2)x + qy + c. Consider the plane as a surface generated by a straight line which moves subject to the conditions that It always Intersects one given straight line and is parallel to another. PLANES UNDER TAKTICULAR CONDITIONS. 51 Let the equations of the line which it intersects be z=i)x + c^ y = 0; (I) and those of the line to which it is parallel z = qy, x = (), the equations of the moving line will therefore be of the form z=qi/ + ^, x=a, (2) and, since the two lines, whose equations are (1) and (2), intersect, therefore, for every point in th<3 plane, z - qi/=px + c; that is, the equation of the plane is z =2)X ^■qi/-\-c. In this form of the equation, c is the intercept on the axis of z cut otr by the plane, p, q are the tangents of the angles made respectively with the axes of x and y by the traces on the planes of zx, yz^ if the coordinates be rectangular ; and the ratios of the sines of the angles made with the axes in those planes, if the coordinates be oblique. 76. To find the polar equation of a plane. Let c, a, /3 be the polar coordinates of the foot of the per- pendicular from the origin on the plane ; r, ^, ^ those of any point in the plane, then if -v/r be the angle between the lines joining these points to the origin, c = r cos-v/r, and co3V^ = cos^ cosa + sin^ sina cos(0 -/3), (Art. 39) whence - =cos^ cosa + sin^ sin a cos((^-/3), the most convenient form of the equation of a plane when referred to polar coordinates. Planes under rarticular Conditions. 77. Equation of a jylane passing through a given point. Let a, J, c be the coordinates of the given point, and the equation of the plane Ix + my + nz =^;, then since (r/, />, c) is a point in this plane la-{-ml)-{- nc—p)^ or, eliminating ^>, / [x - a) + m {y-h)-\- n (2 - c) = 52 PLANES UNDER PARTICULAR CONDITIONS. is the generdl equatlou of a plane passing through the point (a, h, c). 78. Equation of a jJ^ane passing through a j^oint determined hy the intersection of three given jy^anes. If the point be given by the equations of three planes, ?< = 0, v = 0, w = 0, passing through it and not intersecting in one straight line, then \u + fMv+vw = will be the general equation of a plane passing through that point, for it is satisfied by tlie values of a:, y, z, which are given by the equations u = 0, v = 0, to = 0, taken simultaneously, and therefore passes through the inter- section of these planes, which is the given point ; and since this equation is of the first degree, and Involves two arbitrary constants, namely, the ratios \ : fi : y^ it is the general equation of a plane passing through the given point. If the three planes, u — 0, v = 0, ?t' = 0, intersect in a straight line, then these equations, and therefore the equation Xu + fiv + vio = 0, will be simultaneously satisfied for all points lying in that straight line. Hence, Xu -\- ^iv + vio<= Q cannot be the general equation of a plane passing through a given point. The position of a point is not, in this case, completely determined by the given equations, but only the fact that it lies on a certain straight line. 79. Equation of a plane passing through two given points. Let (a, ft, c), (a, h\ c) be the given points; the equation of a plane passing through (a, ft, c) is l[x — a) + m [y — h)-\- n {z- c) = 0. If this plane also pass through (a', ft', c'), we shall hav€ I {a -a) + m (ft' - ft) + n (c' - c) = 0, which Is tlie condition to which I : m : n are subject ; or, the equation of the plane may be written ^ X - a ?/ — ft z — c X - — +fjif, — y + V r— = 0,. a - a 0-0 e — G PLANES UXDER rARTICULAR CONDITIONS. 53 X, /i, V, being subject to the condition \ -f ^ + J/ = 0. It is easily seen that if the points be given by the two systems of planes, u = 0, v = 0, W = Oy and 11 = a, v = h, w = c, that the equation of the plane will be Xu + fiv + vw = 0, subject to the condition \a + /u?>-t- vc = 0. 80. Equation of a plane passing titroitgk the line of inter- section of two planes. If M = 0, y = be the equations of the two planes, the equation \u-\- fiv = will represent a plane passing through their line of intersection ; and since this equation involves one arbitrary constt'int (\ : /x), it will be the general equation of a plane passing through the straight line which is given by the two planes. 81. To find the equations of two ^>/a?2(?s lohich form an harmonic srjstem with two given planes. These two planes must pass through the line of intersection of the given planes, and divide the angles between them, so that the sines of the angles made by each with the given planes shall be in the same ratio. Let M = 0, i; = be the equations of the given planes, and let p, ", c"), we shall have I [a —a.)-^m [b' - h)-\-n{c' —c) — 0, I (a" - a) + m [h" - 5) -f w (c" - c) = 0, and ernnlnating /, wi, n between (1), (2), and (3), we obtain [x — a) [h [c — c") + h' [c" — c) + h" [c — c')] + {i/ — h) [c {a — a") + c (a" — a) + c" (a — a)] + [z- c) [a [b' - b") + a! [b" -b)+ a [b - b')] = 0, (1) as the equation of the plane passing through three given points. The coefficients of a;, ?/, z in this equation are the projections on the coordinate planes of the triangle formed by the three given points, call these A^^ A^, A^ ; then xA^ will be equal to three times the volume of the pyramid whose base is A^^ and vertex the point {x, i/, z). Hence, equation (1) asserts that the algebraical sura of the pyramids whose bases are the projections of any triangle on the coordinate planes, and common vertex any point in the plane of the triangle, is constant for all positions of this point. The equation here obtained becomes nugatory if b (c - c") + b' (c" - c) + b" (c - c) = 0, c [a - a") + c {a" — a) + c" [a — a) = 0, and a [b' - b") + a' {b" - b') + a [b - b') = 0, which are equivalent to {Jj -l')i^c" - c')- {c - c'){b" -b')=% (c - c) [a" - a) -{a- a) (c" - c] = 0, (a - a) [b" - b') -{b- b') {a - a) = 0, a — a b — b' c~c' or to -7— T, = ,, — .» = T, , a — a b—b o— c and these are the conditions that the three given points should lie in a straight line. 83. To find the equation of a iiilane passing through a given 2)oint and paraUcl to a given jt^ane. If («, b^ c) be the given point, and /, ?h, n the direction-cosines J'LANES UNDER PAKTICULAU CONDITIONS. 55 of a normal to the given plane, the equation of the proposed plane Avill be I [x - rt) + m iy-h)^- n (^ - c) = 0. 84. Tojind the equation of a plane wJtich passes through tivo given points and is parallel to a given straight line. Let (a, J, c) («', h\ c) be the given points, and ?, ?«, n the direction-cosines of the given line, the equation of the plane will be of the form \{x -a)+ti[g -h) + v[z -c) = o, where X (a - a) + /i (// -l) + v {c -c)=-0 ^ (1) and since its normal is perpendicular to the given line X/ + /im+ v?« = 0, (2) the equation is therefore A' — a, y — h^ z- c \ a — a^ b' — hj c —c =0. ?, m, n I This equation will become identical if = 77—7 = -1 » a —a b —0 c —c ' which are the conditions that the given straight line may be parallel to the line joining the two given, points. The equations (1) and (2) will in this case be coincident, or every plane passing through the two points will necessarily be parallel to the given straight line, as is otherwise evident. The required equation will then be the equation of any plane passing through the two given points. 85. To find the equation of a plane passing through a given point and parallel to tioo given straight lines. If the direction-cosines of the two straight lines be Ij ?«, n and /', m\ n'^ and the coordinates of the given point a, b, c, the equation of the plane will be {mn -m'n){x- a) + {ni: -n'l]{y- b) + {lm' - rid){z-c) = 0. {Ai't.2S). If 7, = — , = - , this equa'tion will be satisfied for all values / m n ' ^ of a;, ?/, Sj or, if the given straight liuca be parallel, there 56 PLANES UNDER PARTICULAR CONDITIONS win be an Infinite number of planes satisfying the given eon- dltions, the direction of the normal to the required plane being indeterminate. 86. To find the equation of a plane loMch contains one (jiven straight line, and is jjarallel to another, not in the same plane. Let the equations of the given straight lines be x — a _y — h _z — c <^ I m n ' , x — a y — l) z — c' ^i and — jT- = - — r- = — — • ^ L m n The plane contains the first line, and passes througli the point (a, h, c), also its normal is perpendicular to each of the lines, which properties are expressed by the equations \[x — a)+fi{y -h)-\-v[z — c) = Q, \l -f [xm + vn =0, \l' \- fini + vn' = 0, and the equation is {x — a) [mn - m'n) + {y - l^) [nl' — n'l) + {z — c) [hi — I'm) = 0. The equation of the plane containing the second and parallel to the first is {x — a) [mn - in'71) + {y — h') [nl - n'l) + {z — c) [Im — I'm) = 0. The shortest distance of the lines is the diflference of the perpendiculars from the origin, estimated in one direction, giving the same result as in Art. (60). 87. To find the equation of a plane equidistant from two given straight lines, not in the same plane. Let the equations of the two given straight lines be I ~ m ~ n ~'' ^^■^ /' ~ m' ~ n ~ ' ^'■> (a^.» 2/,i ^,) ^ Po>»t in (1), (.-r,, 7/^, .g a point in (2), (|, 7;, ^) the middle point of the line joining (a^,, y,, z^) and (.r_,, y,^, z,^. EQUATION OF TWO PLANES. 57 Then, 2^ = a;, + x^ = a -f a' + ^r + l'r\ '^1 = y 1+ y^ = h + h' + mr + m'r\ 2^ = s, + -^ = c + c' + trr + «';•', and eliminating r and r', we obtain, for the locus of (|, 77, ^), the equation (2| - rt - a') (w«' - m'n) + (277 - Z? - Z/) [nT - 7i7) + (2^- c - c') (?w' - I'm) = 0. (3) The plane represented by this equation bisects all lines joining any point of (1) to any point of (2), and therefore bisects the shortest distance between them ; and since the direction-cosines of the normal to (3) arc proportional to inn' — m'n, nV — n'l, hn — I'm, the normal Is parallel to the shortest distance between the lines (Art. 60). Ilence this plane bisects at right angles the shortest distance between the lines, which is clear from the geometry. 88. To determine the conditions necessary and sufficient in order that the general homogeneous equation of the second degree may represent two real or imaginary pla7ics. Let the general equation be written M.^ = ax^ + hy^ + cs'*' + 2a' yz + Ih'zx + 2c' xy = 0. If a be finite, the equation Is equivalent to [ax + c'y + h'zy = (g" - ah)f -V 2 [h'c - aa) yz + {h"' - ac) z\ But, if the equation represent two planes, x must be capable of being expressed as a linear function of y and z, and this can only happen when the second side of the above equation is a complete square, and therefore of the form {py + qz^, and the tw.o planes will have equations ax + cy ^Vz = ± {pg + qz) ; every point of the line of intersection of the two planes will therefore satisfy ax + eg + Vz = and pg + <2" — ^^- 58 KQUATION OF TWO PLANES. By solving with respect to y and 2, we obtain similar results. Hence, for every point in the line of intersection, ax + c'y + h'z = 0, c'x + by -f a'z = Oj ( 1 ) h'x + a'y+ C2 =0; therefore, by eliminating x, y, and z, a, c', h' c', J, a = 0, h'y a', c or //(j^.J = ^^^ + 2a'b'c' - aa"^ — hb"^ — cc"^ = 0, this is the necessary condition, which might also have been obtained from the condition for a perfect square, (b'c - aa'f = (c" - ab) {b"" - ca), or a (abc + 2a'b'c' - aa'^ - bb'^ - cc"') = 0, which, since a is finite, gives the same result. The symmetrical form of II(u^) shews that the result would have been obtained In this way whether a, ?>, or c were finite. If none of them be finite, it is easily seen that a', b'j or c must be zero, and the equation will still hold. In order that the planes may be real, it is necessary that c'^ — ((b, b"^ -«^, and, similarly, a'^ — bc shall not be negative. 89. When the gmeral equation of the second degree re- j)resents tioo jylanes, to find the equations of their hne of intersection in a symmetrical form. Any two of the equations (1) given in the last article are equations of the line of intersection. If we eliminate z from the first two of these equations, and x from the last two, we obtain the symmetrical equations of the line X [b'c — oa) = y {c'a — bb') = z [ab' — cc). riJOBL&Ms. 59 (1) The equation of a plane passing through the origin, and containing the straight line X - a _y - b z - c I VI n I \f/i H/ m \n I J 71 \c ml Hence, find the equations of the straight line passing through the origin, and intersecting two given straight lines ; and examine the case in which the straight lines are parallel. (2) Find the equation of the plane passing through the points (a, h, c), (6, c, a), (c, a, h), and the equations of the pianos, each of which passes through two of the points and is perpendicular to the former plane. (3) The equation of a plane passing through the origin, and containing the straight line whose equations are X + 2y -t- 3z + 4 = 2j: + 3y 4 43 f 1 = 3j: t 4y + 3 + 2, is a: + y - 2z = 0. (4) Find the condition that four planes whose equations arc given may pass through one point. (5) The equations of three planes are a: + 2y-3z = 1, 2j: - 3y + 02 = 3, 7x- y - z =2. Shew that the equation of a plane, equally inclined to the three axes, and passing through their common point, is z + y + z = 6. (6) Shew that the locus of a poirtt dividing the distance between any two points on the two straight lines X - a y -b _z - c X - a' y - V z - c t tn n ' I' m' n' in the ratio \' : \, is the plane whose equation is , / \fl + \'a'\ _ {mu - inn){x - — 1 + Sec. = 0. (7) Employ Art (35) to shew that the equation Ax \ By t Cz = D represents a plane, according to Euclid's definition. (8) The edges of a parallelepiped meeting in a point A arc a, b, c, and a plane is drawn cutting off parts a, b', c from these edges ; prove that the plane will cut the diagonal A li in a point B', such that -^^ = (~. C 60 PROBLEMS. (9) Find the coordinates of the centre of perpendiculars of the triangle, which the coordinate planes cut from the plane - + ? + - = !. a b c (10) The equation of a plane passing through two straight lines x-a_y-h_z-c x - a' _y -b' _ z - c' a' b' c' ' a b c is {be - b'c) X + {ca' - c'a) y + {ab' - a'b) z = 0. Give a geometrical interpretation of the equations. (11) Shew that the three planes Ix + my + nz = 0, (w + n) x + (n + /) y + (^ + rn) z = 0, z + r/ + z = 0, intersect in one straight line X y _ - m - n n - I I - m' (12) Shew that if the straight lines X y z X _ y _ z ^ _ V _'^_ a'^ ^~ ~{' aa~ b^" c^i' 1 ~ m~ n' lie in one plane, then - (6 - c) + - (c - a) + - (a - 6) = 0. (13) Determine the conditions necessary in order that the planes ax + c'y + b'z = 0, c'x i by ^ a'z = 0, b'x + o'y + cz = 0, may have a common line of intersection, and shew that the equations of that line are X {aa' - b'c') = y {bb' - c'a') = z {cc' - a'b'). Find the conditions necessary in order that the three planes may be coincident. (14) The equation of any plane containing the straight line x-a_y-b_z-c\ { x - a) /a- (y - b) v (z - c) _ -r— — IS ; 1 + — Vf I m n I m n X, /I, V being connected by the equations \ + /* + f = 0. Hence find the equation of a plane containing one given straight line and parallel to another. (15) A straight line is projected on a plane which always passes through a given straight line ; find the locus of the projections. (IG) The equations of two lines are X = y + 2a = G (z - o) and a: 4 c = 2y = - 12z. Find the two planes, each containing one line and parallel to the other, and thence shew that the shortest distance of the lines is 2a. I PUOnLEMS. Gl (17) The angular points of a tetrahedron are (1, 2, 3), (2, 3, 4), (3, 4, 1), and (4, 1, 2); find the equations of its faces, and shew that two of the dihedral angles are right angles, two supplementary and one 60^ Also, that the perpendiculars from the angular points on the opposite faces (18) The equation of a plane passing through the origin and containing the straight line a + mz- ny _ ft + nx - h _ <■/ + ly - mx I m n is (^ + m* + n') {ax -^ fty \ ^z) = {la- + mft + «-/) {Ix + my ^ nz), (19) If .£, A' ; B, B'\ C, C are fixed points in any three fixed straight lines passing through a point, the intersections of the planes ^^C, A'B'C; A'BC, ABC; ABC, ABC; and ABC, ABC are four straight lines lying in a plane dividing the fixed lines harmonically. (20) Find the equation of the plane which passes through the two parallel lines x-a_y-b_z-c^ xj^a' _ y -h ' _ z - c' I w» n ' I in n ' and explain the result when — ; — = = . / m n (21) The equation of the planes which pass through the straight line -^ L = - I 711 n' and make an angle a with the plane Fx + jn'y + n'z = 0, is {f {ny - mz) 4 m' {Iz - nx) + n' {mx - ly)}* = cos' a (/'* + ?>i''' + m'*) {{ny - wiz)' + {Iz - nx)* + {mx - ly)*]. What limitation is there to the value of a ? Shew that for the limiting values the two planes coincide. ^ + -= 1, a; = 0, and c parallel to the line --- = 1, y = 0, is?-^-- + l = 0; and, if 2d be a c a b c the shortest distance between the lines, shew that -„= -5+71+ -.. cr a* 6' c* (23) Shew that the line represented by the equations a + mz - ny b + ux - Iz c -i- ly - mx m - n n - I I - rm 62 PROBLEMS. is at an infinite distance in the plane X {m - n) + y (n - Z) + s (/ - m) = 0, unless la + mh + nc = 0, when it is indeterminate. (24) The equations ax + c'y H h'z _ c'x ^ by + a'z _ b'x + aiy + cz ~ X ~ y "" z represent in general three straight lines mutually at right angles ; but, if h'c' , c'a a'V a = b - -Y~ = c , a b c' ' they represent a plane and a straight line perpendicular to that plane. (25) A straight line moves parallel to a fixed plane, and intersects two fixed straight lines not in the same plane ; prove that the locus of a point, which divides the part intercepted in a constant ratio, is a straight line. (26) li,mi,n^; h, m^, Ui] l^, 7n3, n^, are the direction -cosines of three planes at right angles to one another, and pu po, p^ are the perpen- diculars from the origin upon these planes ; prove that the locus of a point equally distant from these three planes is the line x - (^1^1 -f hpz-^hPs) ^ y - ("iiPi + ffljP* + "'aPJ ^ g = (wijt>i + n op.; + w,pj CHAPTER VI. QUADRIPLAXAR AND TETRAHEDRAL COORDINATES. 90. We now proceed to describe other systems of co- ordinates, which are employed in cases in which it is an object to express the relations between lines, planes, surfaces and cm'ves by means of equations which are homogeneous in form, on account of the facilities which such forms present in the application of theorems of higher algebra. Four-Plane or Quadrtplanar System. 91. In the quadriplanar coordinate system, four planes are fixed upon as planes of reference, which form, by their inter- sections, a pyramid or tetrahedron ABCD. The position of a point is determined in this system by the algebraical distances a;, y, 2, lo from the four planes respectively opposite to the vertices A^ J5, C, i), these distances being all absolute distances when the point is within the tetrahedron. Hence, for a point in tlic conipartnunt between the plane 64 COORDINATES IN THE FOUR-PLANE SYSTEM. AC'D and the other three produced, y will be negative and X, z, to positive; between BAC^ CAD, and BAB, produced through A, x will be positive and ?/, z, w all negative. If a be positive, x = a is the equation of a plane parallel to BCD, at a distance a from it, on the side towards A] x = — a that of a plane on the opposite side at the same distance. 92. In this system of coordinates the following peculiarity must be observed, viz., that any three of the coordinates X, y, z, w are sufficient to determine the position of the point, since, when x, y, z are given, three planes are determined parallel to the faces opposite to A, B, C which intersect in tlic point, and so determine its position completely. Hence, when x, y, z are given, w ought to be known from the geometry of the figure, and we proceed to determine the relation between the coordinates in this system. Relation of Coordinates in the Four-Plane System. 93. Let V be the volume of the tetrahedron contained by the four fixed planes, A, B, C, D the areas of the triangular faces. If the point P, whose coordinates are x, y, z, to, be joined by straight lines to the angular points of the tetrahedron, four pyramids will be formed, whose vertices will be at P, and whose bases will be the faces of the tetrahedron. The algebraical sum of these four pyramids will make up the volume of the tetrahedron ; therefore, remembering that th6 volume of a pyramid is one-third of the base x the altitude. Ax + By + Cz+Dw = S F= Ap^ = Bq^ = Cr,, = Ds^, Pi>i Qai ^0) '■'o being the perpendiculars from the angular points on the opposite faces, whence, when any three of the coordinates of a point are given, the fourth may be found. The object of the introduction of a fourth coordinate, in this system, is the same as that for which trlllnear coordinates are employed in Plajic Geometry, viz. to obtain equations homo- geneous with reference to the coordinates, and thus to arrive at symmetrical results. By means of the equation given above, any equation which / TETRAIIEDRAL COOHDINATESi. ' ' "/ "&§ /^ red/iced to «ich ' / * a form imincdiately. ' -// /. * y. Thu3 the equation x = a of a plane may be redncea fq/the J- homogeneous form ^ «'" j . O ^a; y z w\ \ /. '^' -+-+- + -. ^x W Tetrahedral Coordinates. 94. It is evident that the relation between the coordinates given in the last article would be much simplitied if we were to select as coordinates Ax By Cz Dw x y z w 37' 3F' 3F' JV' ^^^7' ^' 7^' J/ to which these are equal. Such a system of coordinates is called a s^-^em of ktrahedral coordinates^ each coordinate being the ratio of the pyramid, whose base is a face of the tetrahedron and vertex the point considered, to the volume of the funda- mental tetrahedron, sign being of course always regarded. If (a:, ?/, 2r, u') represent a point in this system x-\-y-\-z-\-io = \^ and any giv(j.n equation involving four-plane coordinates may be transformed into the corresponding equation in tetrahedral coordinates by writing p^x^ q\i/^ i\z, s^w for a?, y, 2;, w. Since both these systems are never employed in the same discussions, it is unnecessary to adopt a different notation for the coordinates. 95. It may be shev/n, as in Art. 35, that the four-plane coordinates of a point which divides the^line joining two points ( ' (*j y? ~) ^^) and {x\ y\ z\ lo') in the ratio /* : A, are -^ — — , &c. ; and the same result will be true, if the coordinates be tetrahedral. 96. To find the distance of two given j^oints in tetrahedral ■ ' 'Ordinates. Let [x^ y, c, w) and {x\ y\ z\ w') be two given points P, (>. K 66 THE STRAIGHT LINE. The square of the distance between them is easily seen to be of the second degree in terms of cc — a-', y — y\z-z'^ w- lo. But x-\-y+z + ^o=l=x'^-y+z'-]r w', .'. {x - x] + 0/ -2/') + (2 - z') + {to - w'] = ; .-. {x-x'f = -{x-x)[y-y')-..., and similarly for [y — y')'\ &c. Hence, the square of the distance can be expressed in terms of the six products {x — x) [y — 3/'), &c. Let 7 be the coefficient of {x — x) [y - y'), and let us apply the expression to find the distance AB; now the coordinates of yl and B are 1, 0, 0, 0, and 0, 1, 0, 0, hence every product but one vanishes, .-. AB'^ = - 7? and - PQ' = AB' [x - x) [y-y) + AC'' [x-x) {z -z')+.... 97. Hence we may obtain the equation of a sphere, the coordinates of whose centre are/, ^, /?, A:, and whose radius is r, -r^ = a'{^-y)&-h) + h^z-h){x-f) + c^{x-f){y-g) + a" [x -/) {w - k) + h" {y -g) [w - h) + c" [z - h) {w - 1% «, h, c being the sides oi ABC opposite to A^ B, C) a, h', c the edges DA^ DB^ DC respectively opposite to «, 5, c. The Straight Line. 98. To find the equations of a straight line in four-plane coordinates. If {f g^ h, k) be a fixed point in a straight line, (.r, ?/, s, w) any other point, B the distance between them, X, yu,, v, p the cosines of the angles between the straight line and the normals to the corresponding faces of the tetrahedron, x—f—\B^ &c. Therefore the equations of the straight line are ^~i - y.ZLO _ ^i _ ^Hr.h - 7? \ fj, V p where, since two equations are sufficient to determine the line, there must be a relation between X, /it, v, p. „ x — f y — fj z — h IV - k , , Now — ^+-y—A + 4. =0 Art. 93 , Po !Zo »'o «o THE STRAICIIT LINE. 67 . . . X U V p . the relation is — + — + - + - = 0. 2'o Q. ^0 «% Another relation, not homogeneous, may be obtained fiom the value of ii, in Art. 96, changed to four-plane coordinates fiv -\ vA, -I Xa H Xp + — up H vp= — 1. J?,.**.) ^'o/'l 2\i^0 Pi>^i> ^Ji>^<> ^'""^. In this form of the equations, X, /x, v, p may be called the direction-cosines of the line. In tetrahedral coordinates the corresponding equations are, for the straight line X —f y — z — h w — h R X jx V p a ^ with the conditions \4/t4- j/4-p = 0, and a'fjLV -\- V'vX + c'Xfi + a'Xp + Vjxp -I- c'Vp = - o-^, ¥o M. ^"o P^ being the direction-cosines. 99. The general equations of the straight line may be written in tetrahedral coordinates X —f y — (J z — h tc — h where X, 3/, A'', i? satisfy the equation X + il/-f- A^+ i? = ; /v» _^ y 11 ^^ ft Z ^' ft and it may be observed that, if two equations —jr^ = ^r — yr be given, the fourth member may be derived from it, since each _ X — /+ y —g -\- z — h _ lo — k L + M+N " ~ir' If the straight line pass through one of the angular points, 88^,(1,0,0,0) x-l_y_z_to ~ir~M~N~R' If it join the middle points of AB^ CD, viz. (.^, h, 0, 0) and (0,0,^,^), r^ = *^ — -- = - = — ; or a; = ?/ and z = ic. (iS THE STRAIGHT LINE. 100. To find the angle between two straight lines whose equa- tions are given in tetrahedral coordinates. Let tlic equations be x—f y—g z — h_io — k_R X /i V pa'' x — f y —g z — h'_ w — k' _ R Take two straight lines parallel to these, passing through D and meeting ABC'm P, P'. The equation of DP is -,0, andZ>P=-- X y z u P 1 _ R and the coordinates of P are — and similarly for P' and DP'; X P' - P' - \p p )\p p J \p p J \p p 1 \p p J \p p J p p pp KP p J \p p J p P pp whence, substituting the values of a' and a'^ (Art. 98), we obtain ±2o(t' cos PDP = a:' [fiv + fjb'v) +...+ a'^\p' -]-\'p) +... . 101. As an example of the use of this formula, we will find the angle between AD and BG, whose lengths are a and a. For AD, y = and z = Oj and for BC, x = 0, w = 0, and the values of A, /x, ... , V, /u,', ... may be taken respectively, 1, 0, 0, - 1 and 0, 1, - 1, 0, .-. a' = a'' and a"' = d% and, if 6 be the acute angle between those lines, 2aa coBd={F + h")^{c' + c"). 102. The condition of perpendicularity of two straight lines given in tetrahedral coordinates x-f_y-g_z-h_w -k x-f _ y-g' _ L'~ M ~ 'k ~ R ' ^"'^ L' ~ M' — •' is a'{MN'-vM'N)+...+ a:'{LR + LR)-\-...= 0. THE PLANE. G9 It may be seen, as an example, that, if the lines joining the middle points of (a, a) and {h, h') be perpendicular, c and c will be equal. The Plane. 103. The general equation of the. first degree represents a playie. Let Ax+ Bg +Cz + Dio = be any equation of the first degree. If arbitrary points [f,g, h, k), (/', g\ h\ k') be taken, which satisfy the equation, any point P in the line joining them may be represented by i^ , -^ ^ , ... J , and since Af ^Bg +Ch +Bk =0, Af'-\-Bg' + Ch' + Bk' = 0; .-. A {\f+ fjf') +B{\g -^ fig')+...= 0', therefore the coordinates of P satisfy the equation, and the whole straight line joining any two arbitrary points lies in the locus of the equation, which is therefore a plane. 104. Geometrical interjyretation of the constants in the equa- tion of a plane Xx + fig + vz + pic = 0. Let E be the point in which the plane EFG cuts ABj its four-plane coordinates being x, g', 0, ; .'. \x' + fig' = 0. D Draw Ee^ Aa perpendicular to BCB^ Ee EB x' EB , . ., , y' ^^i ^^^ or P., ^i^^' ' -'/, Aa- AB' "' p - AB^ .^.u, .......a..,, -yy^|-^i^, THE PLANE. also, If 2h 1 be the perpendiculars from A^ B upon the given plane, estimated In the same direction, AE EB ' .-. — = i- and similarly each = — = — p q r s and the equation of the plane Is pars i- x-\-^ y + - z->r-io = 0. Po do ^n In tetrahedral coordinates the equation becomes px -\- q7/ + rz -\- sw = 0. 105. To find the equation of a plane at an infinite distance. If the plane be at an Infinite distance, 2) = q = r = s^ and the equation in tetrahedral coordinates becomes x-\- y + z-\- 20 = 0. 106. To find the conditions ofi parallelism of two planes^ tohose equations are (jiven. Let the equations of the two planes be \x -\- fxy ■\- vz -\- pw = and 'X!x + p!y + v'z 4- p'lo = 0, these planes intersect In the plane at an infinite distance, whose equation is x-{- y + z -]- w=0, Using tetrahedral coordinates. From the three equations we deduce {X-p)x+{fM-p)y-^{v-p)z==0, {\'-p')x+{p,'-p')y+{v'-p')z = 0, which are satisfied by an infinite number of values of the ratios x : y : Zj they must therefore be Identical equations ; . ^ ZLP = f^ ~P = ^ ~P " X- p fM'-p' v'-p" THE PLANE. 71 these arc the equations of condition, which may also be written ^') H'', V, p [ =0. 1, 1, h 1 I They also follow immediately from p— p =q^- 'l = ••' . 107. To jind the length of the perpendicular from a given point iqwn a j^lane given in four-plane or tetrahedral coordinates. Let the equation of the plane with four-plane coordinates be \X ■+■ /jL?/ + vz + pw = 0, and let (x', ?/', z\ to) be the given point. Suppose the whole system to be referred to rectangular Cartesian coordinate axes, and let the equations of the four planes of reference be K^ + ^'h'n + \K=1\ &c., then the equation of the given plane will become [\l^ + til^ + vl^ + pl^ ^ +...^\p, + fx2K + yp, + pp., hence, by Art. 70, the perpendicular required will be \x' + fjuy' + vz -I- pio being the perpendicular from [p>^^ 0, 0, 0), the expression (1) will give ,,= ^S,= 7",&c., (3), whence the equation of the plane in four-plane coordinate will be px qu rz sio 2\, ?,. ''u ^o 72 THE PLANE. and, in tctrahedral coordinates px + qi/ + rz + sw = 0^ where, by equations (2) and (3), (MV(iyW!l)V(£y_2^eos^5- = l; \iyj \qj \rj \sj p,q, so that the left side of the equation in either form represents the perpendicular from any point (rr, ?/, z^ lu). 108. The method which we have adopted in the last article shews that the quadric function X'^ + //.'^ + . . . - 2X/i cos(^-B)— ... is reducible by transformation to three squares, the condition of which is that the discriminant vanishes, or that -1, cos(^5), cps(J.C), cos(ylZ)) cos(^i5), -1, cos {BC), cos {BD) cos{AC)^ cos {BC), -1, cos{CD) cos{AD), cos{BD), cos{CD), -1 Also, since each of the three squares is positive, there is only one system of values which reduces the function to zero, viz. that which belongs to a plane at an infinite distance, for which 2) = q = r = s^ whence, by (3) Art. 107, Xp^ = /xq^ = vr^ = ps^. That the discriminant vanishes, may be shcAvn independently by projecting any three of the faces of the tetrahedron on the fourth, and obtaining the determinant from the four equations similar to A-B cos{AB) - C cos{A C) - D cos[AD) = 0. 0. 109. To find the angle hetwecn two planes ichose equations are given. Let the equations of the planes be , \x + fiy-\- vz->r pxo = 0^ and ^^x-^- fx'y -\- v'z + pio = 0^ using the same method as in Art. (107), if 6 be the angle between the planes aa' cos e = (\?, + fJil., + v/, -f p/J (/tV, + pH^ -|- v'l.^ + pVJ +. . . = XA,' 4- pp 4 vv + pp — [Xp + \'/i) cos (J B)—.... TKOBLEMS. 73 110. To jiiid the direction-cosines of the normxl to a plane equation is gicen in tetrahedral coordinates. Lct^j; + qij + rz A- sw = be the given equatlqn. The equations of the perpendicular from A on the opposite face are easily shewn to be .T — I _ // z to „ i co^{AJJ) ~ cos(xl6') ^ cosjAJJ) ~ ' therefore, where it meets the given plane, j^ ( p q cob{AB) _ r cos{AC) s co& (AD) ] _ ^'~ \l\~ 1. ''o ^0 J ~ ' and i> = i2cos(^>,7;J; p o coafAB) r cos( AC) s cos(AI)) ••• cos(;;, pj =f-- '—^ ^ ^ , Fn 9o 'o ^0 and similar expressions for the other direction-cosines. VI. I (1) Shew that for every point in a piano through the eilge yf Z? bisect- 'i ifig the angle between the planes CAB, DAB, z - u; = 0, if the angle be the internal angle, z i w = 0, external ■ (2) Shew that for every point in a plane drawn through the vertex A I parallel to the opposite face, i'y + Cs + Du> = ; ur, with tetrahedral coordinates, y -f = + M7 = 0. (3) If AO be drawn perpendicular to the opposite face BCD, then I for any point in A O, By _ ^2 _ Dw _ if\ ^COD ~ ADOB ~ ABO0~ ~ '^' (4) Every point in a plane through CD parallel to AB satisfies the ! equation, in tetrahedral coordinates, X + y = 0. <4 rUUliLEMS. (5) A point is determined in tetrahedral coordinates by the equations Ix = my = MS = rw. "What plane is represented by the equation 7ny = nz, and what straight line by the equations 7)ii/ = nz = rtv ? (6) At any point in the straight line joining the first points of triscction of AB and CD, the tetrahedral coordinates satisfy the equations x = 2!/, s = 2tc. (7) Shew that the coordinates (tetrahedral) of the centre of gravity of the fundamental tetrahedron are given hy x = y = z = w. (8) The three straight lines joining the middle points of opposite edges of a tetrahedron meet in its centre of gravity. (9) A plane cuts each of the six edges of a tetrahedron ; another point is taken in each edge, so as to cut it harmonically; prove that the six planes through these latter points and the opposite edges of the tetrahedron intersect in one point. (10) If the equations of a point be - = 1 = - = !f / 7u 11 r and AO, BO, CO, DO be joined and produced to A', B', C, £>', such that O bisects the lines AA', Sec, the tetrahedral coordinates of the point A' will satisfy the equations 2x ^_==_»^_ 2 I - 711 - n - r VI n r I \ m -v n -i^ r ' and similarly for B', C, D. (11) The line joining the centres of the two spheres which touch the faces of the tetrahedron A BCD opposite to A, B respectively, and the other faces produced, will intersect the edge CD in a point P, such that CP : PD :: aACB : AADB, and the edge A B (produced) in a point Q, such that ^g : BQ:: ^CAD : ACBD. (12) If two opposite edges of a tetrahedron be trisected, and the points oflrisection be joined by two lines in either order, shew that the line which bisects these lines will also bisect two other opposite edges. (13) I, I' arc the lengths of two of the lines joining the middle points of opposite edges of a tetrahedon, w the angle between these lines, a, a' those edges of the tetrahedron which are not met by either of the lines, ■ilf i ' PROBLEMS. 75 (11) A point O is taken within a tetrahedron ABCD, so as to be tlie ! centre of gravity of the feet of the perpendiculars let fall from O on the faces ; prove that the distances of O from the several faces are proportional respectively to those faces. (15) Shew that the reciprocals of the radii of the spheres which can be drawn to touch the four faces of a tetrahedron, are the positive values of the expression Po '/o ^ «0 ?o' ^u> *o b> THE rOlNT. 114. To find the equation of a point in foxir-point coordinates. If the four points lie in ), Xy> -I- /x*^ = 0,") (ad), \p-[ps=0,\ (be), /i7+vr = 0,] (cd), vr + ps = the straight lines joining these pairs of points meet BD In. a point {h'd') whose equation is /xq = ps, and the equation of [bd] is /x^ + ps = ; therefore b'd', bd divide BD harmonically. Sirailarlj, the line joining (ab), (ac) and [bd), (cd) intersect BC in {b'c'), for which ^iq = vr; and the straight line [ab), {cd) meets the plane passing through A and the points {b'c), {cd') in the point whose equation is 2A^; + 2yu,^ - vr — ps = 0, since this equation may be written 2\p + {fiq - vr) + {fiq - ps) = 0. Again, the equation 2\p + fjLq + vr = 0, being of the form Lj) -i- Ms + N{\2) + fJ'q-i vr + ps) = 0, represents a point in the plane ADE, and, being of the form L {\p + fiq) + M {\p + vr) = 0, the point lies on the line joining [ab), {ac), and is obviously in the plane ABC. Let the straight lines AE, BE meet the opposite faces in A', B' ; the equations of these points are fxq + vr + ps = 0, and X/j) + vr + ps = 0, and therefore A'B' intersects AB in the point \p - p.q = 0, the same point In which {ac) {be), {ad) {bd), meet AB. 122. ]\Iany of the results of the following articles have been obtained in the preceding chapter, but the independent processes adopted here will serve to illustrate the subject on which we arc engaged. 123. Tojind the inclinations of a plane, lohose coordinates are giverij to the faces of the fundamental tetrahedron. Let p, q, r, s be the coordinates of a plane, \, fi, v, p the cosines of the angles between the direction in which the coor- dinates are measured, and the poipcndlculars from A, B, C, D 82 THE I'LANE. on the opposite faces, and lot Aa = i\^ be the perpendicular from A on BCD. c The tetrahedral coordinates of a are aACD aCD B cos(AB) « , . „, 2 = ^" cos {A 6'), w = ^' cos (^i)) . ' The equation of the point a is therefore ^^ cos (yl^j 57 +^'' cos {A C) r + ^-" cos {AD)s^ 0, and, the coefficients being tetrahedral coordinates, the first member is the perpendicular from a on the plane (p, q^ r, s), and therefore =2^—p^i .'. \=^^ -^coH{AB)-'-co?i{AC)-^^co!i{AD); see Art. 110. i Similar values may be obtained for yu., v, p. 124. 7o find tlie relation heftreen the four-jmnt coordinates of a plane. The tetrahedral coordinates of P, the foot of the perpendicular from A on the plane (;?, (7, r, .s), arc L(,, _,,x) -^ -^- -^-^• ( THE I'LANi:. 83 thcrcforc, tVom the equation of the point P, we have p. q. '-0 h P ^ '■ *' .-. ^—\ + ^ yLt4 -V + -p=\. A 7o '■„ *■. Hence, substituting the values of X, /i, v, p found above, avo obtain the relation between the coordinates ii! 4- -^ + -^ + 4 - — cos (AB) -...= 1, see Art. 107. 125. If the plane be at an infinite distance the left side of the above equation will vanish, the only system of values for which this will be the case being ^; = ^^ = ?' = 5, (Art. 108). Since the coordinates are equal and of infinite magnitude, the expression for \ in Art. 123 gives = - - - cos (AB) - - cos (A C)-- cos iAD\ Po % ''O ^M or O^A-B cos [xiB] - C cos (^1 C) - D cos {AD). "We have also from Art. 124 X /Li V p ^ - + -+- +- = 0, I'i, la *"() *o a linear relation between the direction cosines of any plane. 126. The equation which connects the four-point coordinates of a plane is of the second degree, whereas the corresponding equation for four-plane or tetrahcdral coordinates is of the firat degree. The reason of this equation being of the second degree should be explained. If three four-point coordinates of a plane be given, suppose I?, q^ r, this plane must touch three spheres M'hose radii arc Ptq^r^ centres J, J5, C; and, if we suppose the most general case, there will be eight such planes, two for which the three spheres lie on the same side, in which cases ;>, q^ r will be of the same sign, and six for which one of the spheres lies on the opposite side to the other two, in which cases two of the coordinates p^ «y, /• will be of opposite sign to the third. 84 THE PLANE. Whether ^;, q^ r be of the same sign, or p be of opposite sign to q and ?*, there are two positions of the touching plane ; tliat is, there are two values of s, viz. the perpendiculars in these two positions; the equation must, therefore, be of the second degree in 5, and similarly for p^ ^, and r. Hence, although, when three tetrahedral coordinates are given, the fourth is fully determined by the equation of con- dition ; this is not the case in four-point coordinates. 127. To find the angle between two planes whose coordinates are given. Let (y, q\ r\ s] and [p^ q\ r", s") be two given planes denoted by Z7', U'\ and let the line drawn perpendicular to V from the fundamental point A meet JJ" in the point ^^ so that AN=^'S7=p" sec(Z7', U"). Let V, /*', v', p be the cosines of the angles between the normal to the plane U\ and the normals to the corresponding faces of the fundamental tetrahedron. The tetrahedral coordinates of N are P^-'utX —■utijl — -ctv' — CT/)' hence the equation of N IS ;X — otV VJll THV TSp ^_9 V '^ q r s = : Po ?o ^ ^''o and, since the plane U" is a particular plane through N^ wc have, since - =cos(Z7', Z7"), , y^, ^_„, Xp" uq" v'r" p's" cos C/ ', U') = -^ + ^^ + + '^- . Po ?o '-0 *o But V=^- ^-- cos (yl^) -- cos (vl (7)-- cos (ylZ)) (Art.123); J n Jo '0 _ P£±1L± codAB) -..., sec Art. 109. Mo THE PLANE. 85 If the planes be parallel, since the perpendicular distance between them is constant, 'p - p" = q — q = r — r" = s' — s". As in Arts. GS and 22 in the corresponding case with Cartesian coordinates, these conditions can be deduced from the value of cos (f/', U"). For, since {V, f7") = 0, 2 cos(f7', U") = 2=K, +...- ^^-^^ cos(AB)-... + 1' +..._ -^-^-cos(AB)-... (Art. 124), whence ^^'-fl +...- ^ iP 'P") W - = Ijy' + mp'\ where f + 2hn cos ( U\ U") + ni' = 1, and similarly for q, ?•, s. 129. If Lp + Mq 4- Nr -\-lls — be an equation involving one variable t in the first degree, it may be considered as the equation of a line, since it may be put into the form ?< + tv = 0, and by varying t we may obtain every point in the line joining the points » = 0, v = 0. l.']0. If Lp-\- Mq-ir Nr->r Iis = Q be an equation involving two variables /, t' in tiie first degree, it may be considered as SG PROBLEMS. the equation of a plane, since it may be pnt in the form ti + tv + t'lo = 0, and by varying f, t' wc may obtain every point in the plane passing through the three points u = 0, r = 0, and w; = 0. VII. (1) The equation of the centre ol" gravity of the face ABC h 7? + <7 + r = 0. Hence, shew that the lines joining the vertices with the centres of gravity of the opposite faces meet in a point. (2) The equation of the centre of the circle circumscribing the triangle ABCh p sin2^ + J sin25 + r sin2C = 0. (3) The coordinates of the plane passing through the centres of gravity of the faces ACD, ADB, and ABC, are given by the equations (4) If P be any point in BD, Q, R points in A C, such that AQ-.QC:: BP : PB :: CR : RA, then PQ and PR will intersect the lines joining the middle points of BC, AD, and AB, CD respectively, and divide tliem in the same ratio as2£C. (5) If through the middle points of the edges BC, CD, DB straight lines be drawn parallel respectively to the opposite edges, these straight lines will meet in a point; and the line joining tliis point with A will pass througli the centre of gravity of the pyramid. (G) The equation of the centre of gravity of the surface of the tetra- hedron is [A \ B ^ C ^ D)(p \ q \^ r ^ s) = Ap ^ Bq ^ Cr + Ds. (7) Shew that the equation of the centres of the eight spheres which touch the faces or the laces produced of the fundamental tetrahedron are Ap ± Bq i Cr ± Ds = 0. (8) The centre C of the inscribed sphere lies on tlie linejoiiiing G, JI the centres of gravity of tiic volume and of the surface of the tetrahedron; also, shew that CG = '6GJI. I'KOBLIiMS. 87 (9) The points B, C, D are joined to the centres of gravity of the opposite faces, and the joining lines produced to points h, c, d, so that JB, b, &c., are equidistant from the corresponding faces, prove that the coordinates of the plane bed are given by the equations - 2p = q = r = s, and that this plane divides the edges AB, AC, AD in the ratio 1 : 2. (10) If points be taken in the lines joining B, C, D to the centres of gravity of the opposite faces, dividing them in the ratio m : n, the plane containing these points will divide the edges AB, AC, AD in the ratio m : 2m 4 3«. (11) If through any point P straight lines AP, BP, CP, DP be drawn meeting the opposite faces in a, b, c, d, the straight lines AB, ab ■will intersect, and their point of intersection and the point in which Cd meets AB will divide AB harmonically. (12) The straight lines joining D to the intersection of AB, ah, and A to the intersection of DB, db, will intersect in a point lying on Be. (13) If through any point three straight lines be drawn, each meeting two opposite edges of a tetrahedron ABCD; and if rt, a ; b, ft; c, y be the points where these straight lines meet the edges BC, AD; CA, BD; AB, CD; then will Ba.Cy.Dft = Bft.Ca.D'^{, Cb.Aa.D'/=C-{.Ab.Da, Ae.Bft.Da = Aa.Be.Dft, Ab.Bc.Ca = Ac.Ba. Cb. ( ) ]•■ CHAPTER VIII. LOCI OF EQUATIONS. TANGENTIAL EQUATIONS OF SURFACES, TORSES. DUAL INTERPRETATION OF EQUATIONS. BOOTHIAN COORDINATES. Tangential Equations of Surfaces and Torses. 131. If 2^1 9.1 ''5 ^ Ijc the coordinates of a plane referred to a four-point system, to find what Is represented by the general equation i^(p, 5-, r, s) = 0, we observe that there are generally an infinite number of planes, the coordinates of each of which satisfy the equation, and that these planes envelope a surface of which the equation is called the tangential equation, from the circumstance that each plane of the system is a tangent plane to the surface, and the surface may be called the envelope locus of the equation. As in a quadrlplanar or tetrahedral system, if we take three points on the locus of ^ (a;, ?/, 2:, w) — 0, the plane containing these three points will ultimately be a tangent plane to the locus, if tlie three points become ultimately coincident, so, if we take three planes touching the envelope locus of i^(^), <7, r, s) = 0, the point in which these three planes intersect will be ultimately the point of contact of these planes when they become coincident. Thus, p=/ is the tangential equation of a sphere whose centre is the fundamental point A. In the case of the linear equation Xp + /u.<7 + vr + ps = 0, the envelope degenerates into a point, through which all the planes pass which correspond to the various solutions of the equation. 1.32. Again, if we take two surfaces represented in the tetrahedral system by (a:, ?/, z^ 10) = 0, and <^j (.r, y, .t, ic) — 0, the coordinates of every point in their curve of intersection will satisfy both equations, and the two equations will determine the TANGEXTIAL EQUATIONS OF SURFACES. , 7, ?•, a) = represent two surfaces in the four-point system, the coordinates of every plane which touches both surfaces satisfy the two equations, and the series of phines so determined have for their envelope a particular kind of surface, called a Devehpahh Surface, or, according to Caylcy, a Torse, which touches the two envelope loci of the equations, the reason of the term developable being as follows : If we take three consecutive planes P, Q, ii, each of which touches the two loci S, S^, of the equations -F(p, q, >', s) =0, F^i]), q, r, s) = 0, Q will be intersected by P and R in two straight lines (P, Q\ and {Q, R), and in the limit the portion of Q intercepted between these lines will ultimately be a portion of the envelope of the planes ; similar portions of P and R will, with the former, constitute three elements of the envelope, each of which will touch both S and S^ ; and this envelope is called a developable surface, because the three elements can be de- veloped into one plane by turning them about the lines (P, Q), (^, 22) ; and the same is true for all the elements. According to this interpretation, the developable surface touching two %\A\q,yg%, p=f, q=g, is a system of two cones, of which one will be imaginary if the spheres intersect. two equations, as well as one, in the four-point system determine a surface, the case of this system is not analogous to that of the three or four-plane system. But the two kinds of surfaces in the point system are really as distinct from one another as the surface and curve in the plane system. For, as in the four-plane system one equation represents the limitation that the current point must remain on a surface, and the second equation confines the motion on that surface to posi- tions such that it also remains on a second surface ; so in the four-point system one equation limits the motion of a plane to positions in which it touches a surface, and the second equation allows the plane to touch that surface only in such a manner that it also touches a second surface. ►Stated in this way the analogy is complete, the point moving 90 TANGENTIAL EQUATIONS OF SUKFACES. in the direction of a lino, tlic plane turning round a line, to gain the consecutive position. 134. As a simple example of the use of the tangential equation of a surface, we will consider the properties of the Poles of Similitude of four spheres; the poles of simUltude of two spheres being the vertices of the two cones which envelope both spheres, points from which the lengths of the tangents to the spheres are proportional to their radii. These poles of similitude are called internal or ejcternal poles, according as they lie on the line joining the centres, or on this line produced. 135. To find the relative positions of the internal or external 2)oles of similitude of four sjjheres. Let the centres of the spheres be taken for the fundamental points A^ B, C, and D, and let their radii be ?•,, »•„, r^^ r^. The tangential equations of the spheres are 2> = ?•„ q = ?-^, r = r,,s = r^. The external and internal poles of similitude of the spheres (^1) and [B] have equations P :r 1 0, and similarly for the rest. (1) The external poles of (AB), (AC), and [AD] He in a plane whose coordinates are connected by the equations p _ q _ r s '\ ~ '-. ~ '\ " '-4 ' which evidently contains also the external poles of {BC), [CB) and [DB). (2) The coordinates of the plane containing the external poles of [AB) and {A C) and the internal pole of [AD) satisfy the equations p q r s ^1 ''a '"n '". ' and the same plane evidently contains the external pole of [BG) and the internal poles of (5/>) and [CD). TAXGEN^TIAL EQUATIOXS OF SURFACES;, 91 (3) The coordinates of tlie plane containing the external pole of [AB] and the internal poles of [AC) and [AD) satisfy the equations p q r s r, ~ r^ ~ r^ " r^ ' and this plane evidently contains the external pole of {CD) and the internal poles of [BC) and {BD). Hence one plane contains the six external poles, four planes contain each three external and three internal poles, and three contain each two external and four internal poles. The poles of similitude lie in eight planes, which are called planes of similitude, each of which passes through six poles of similitude situated three and three in four straight lines. Thus for the six external poles therefore the external pole of [BC) lies In the line joining those o^AB) and {AC). Similarly it lies in the lines joining those of {DD) and {DC). Hence, the six external poles lie in the sides of a plane quad- rilateral, as in the figure. 92 BOOTHIAX COOKDINATES. Dual Interpretations of Equations. 136. By ^vhat has preceded, we see that all homogeneous equations and systems of two equations in four variables a, /3, 7, 8, admit of a dual Interpretation, according as we conceive the four variables to be tetrahedral or four-point coordinates. Thus Xa 4- /i/S + V7 -I- pS = is the equation, in these two methods of viewing it, of a plane or of a point. So if a, /8, 7, h be tetrahedral coordinates, the equation i^(a, /3, 7, 8) = gives a surface on which every point lies, whose tetrahedral coordinates satisfy the equation, while if they be point coordinates, the equation gives a surface touched by every plane whose coordinates satisfy the equation. Two such equations may in like manner be interpreted to define (1) a curve, which is the intersection of the two surfaces represented by the separate equations ; or (2j a torse enveloped by all planes which touch both surfaces represented by the separate equations. Thus, the dual results given by the method of lleciprocal Polars, which will be seen to apply to three as well as to two dimensions, may be obtained by giving this dual inter- pretations to all our equations. We cannot, however, in a volume of moderate compass, pretend to include all the dual results to which our equations miglit give rise, but must confine ourselves to a development of the methods most generally useful. Boothian Coordinates. T37. There is another system of coordinates which bears the same relation to four-point coordinates as the Cartesian to the tetrahedral system ; these coordinates have been called Boothian from Dr. Booth, who first published the method in Dublin,* We have seen that the equation of a plane in one form is aa; + /3^4-72 = 1, where a, /S, 7 arc the reciprocals of the inter- * Both Cliasles and Pliicker seem to have conceived the idea previously. Briot and Bouquet, Giom, Anal. p. 3S8. BOOTIIIAN roOKDI NATES. 93 cepts on the coordinate axes, if this plane pass through a point P, (/, g^ h), then af-\- ^g + 7A = 1 is an equation between a, yS, and 7, which will be true for all planes passing through P. a, /3, 7 are called Boothian coordinates of a plane, and any equa- tion of the first degree in a, /:?, 7 expresses that the plane passes through a certain fixed point, and niaj be considered the equation of that point. Any equation whatever between a, /3, 7 will express that the plane touches a certain fixed surtace, and may be considered the equation of that surface. Thus, we know that the equation of a sphere is x^->r ?/+ z^ = r\ and that the distance of the plane Ix -\ my -\- nz = r from the origin is ?•; the plane therefore touches the sphere, and if the equation of the plane be written a^c + /3_j/ + 7^ = 1, then we have a" + /3'" + 7" = T^ = -7^ , which is the Boothian equation of the sphere and is of the same form as the Cartesian. 138. Cartesian coordinates are a particular case of tetrahedral^ and Boothian of four-point coordinates. If we imagine the plane ABC of the tetrahedron of reference to move oif to infinity and make the corresponding changes in our equations, any equation between a;, y, 2;, lo will become one between ^, ?;, ^ ordinary Cartesian coordinates, and any equation between p, q^ r, s will become one between a, /3, 7 Boothian coordinates of a plane. Thus take the equation px + qg + rz + sw = 0, where (a;, ?/, 2, xo) are tetrahcdral coordinates of any point, and (^>, (7, r, s) four- point coordinates of any plane tin-ough (.r, y, 2, w). If the plane meet Z>J, DB, DC in o, />, c, and ^, 77, ^be the Cartesian coordinates of (a;, ?/, z, w) referred to the planes meeting in D, ftena: = X, !/ = iljj, - = /^,, ''' = ^- ^A' DB' TJU' whence the equation becomes JJA ' s ^ i)B' s '^ DC i 94 PROBLEMS. „ « Aa s-p DA s-p 1 s Da^ s Da ' s.X'^l Ua and the equation of tbc plane becomes a^ + l3i] i-y^^l. 189. Any equation in which (;r, y, 2r, ?r) are involved will generally have the coefficients of the different terms functions of DA, BC, ... edges of the tetrahedron of reference, so that although for any finite point (^, 77, ^) we have, when ABC moves off to infinity, a: = 0, ?/ = 0, z = 0, w=l, yet we shall get a limiting equation between ^, 77, f when we have made the sul)stitutions above and take DA, DB, DC, ... all infinite. So, for any equation in /?, q, r, s, the coordinates of a plane at a finite distance from D, although in the limit - , - , - are all equal and zero, yet by making 2)^s{\-a.DA), q = s[l-l3.DB), r = si\-y.DC\ we shall obtain a finite resulting equation In a, yS, 7. But such transformation is very seldom of much advantage. It is, however, frequently convenient to render Cartesian or Boothian equations homogeneous by multiplying such terms as require it by 10, iv\ ... or by h, 8"'', ..., these being at some subsequent stage put equal to unity. VIII. (1) State some properties of the loci of the following equations, -vvhetlier ") ft) 7) ^ be regarded as tetrahedral or four-point coordinates. (1) lalS = m-jS. (2) (a 4 ftY = in^. a l-i 7 S (2) If M = be the equation of a surface of tlic second degree, v = that of a plane, referred to tetrahedral coordinates, r* = yiu is the equation of a surface touching the surface « = along the section made by r = 0. Give the interpretation wlu-n the tetrahedral arc replaced by four-point coordinates. I'ROIJLE.MS. 95 (3) Prove that. if the fundamental points be in the angles of a regular tetrahedron, the tangential equations of the spheres circumscribing and inscribed in the tetrahedron will be respectively p* + q' 4 » * i s' - qr - rp - j^q - sp - sq - sr = 0, and qr + rp + pq -^ sp ■{ sq -\ sr = 0. (4) Shew that in the siime case the envelope locus of the equation p' 4 j' + r' -I «' - 2qr - 2rp - 2pq - 2sp - 2sq - 2sr = will touch each of the edges of the fundamental tetrahedron. (5) In the last problem, find the two tangent planes parallel to one of the faces of the tetrahedron, and shew that their distances from that face will be 7— ,77r> {V(3) ± 1}, a being the length of an edge. (6) If two surfaces given by tetrahcdral coordinates intersect in two plane curves, what is the corresponding property of the torse in the dual interpretation ? (7) Prove that the Booth ian equation of a sphere passing through the origin of rectangular axes is of the form a^ 4 /3' + 7^ = [I - I- - n>{3 - ny'j, I, m, n being the direction-cosines of the radius r drawn through the origin. (8) Two spheres of radii r, s, pass through the origin, and have thtir centres on the axes of a: and y respectively; shew that the torse or develop- able surface enveloping both is a cone whose vertex has the Boothian equation CHAPTER IX. TRANSFORMATION OF COORDINATES. 140. The investigation of the properties of a surface repre- sented by a given equation is often rendered more convenient by referring It to a different system of coordinate axes, in the choice of which we must be guided by the nature of the in- vestigation proposed. We proceed to obtain the working forraulje by which such a transformation may be effected. 141. To change the origin of coordinates from one point to another J without altering the direction of the axes. Let (/, y, h) be the new origin referred to the primary system. If (cc, _?/, z)^ [x\ y\ z) represent the position of the same point P referred to the first and second systems respec- tively, since the algebraical distance of the plane of y'z from that of yz is /, and a-, x are the algebraical distances of P from the planes of yz and y'z\ we have x^x +/, and similarly, y = y + g, z = z' + h] or, suppressing the accents, the transformation Is effected by writing x+f y -{■ g, z + h, for x, y, z. 142. Since the formulse thus obtained involve three arbi- trary constants, we can generally by this transformation make the coefficients of three terms in the resulting equation vanish, but, since the coefficients of the terms of highest dimensions are unaltered, none of the three terms, so eliminated, can he of the same dimensions as the degree of the equation. Thus, in an equation of the second degree, we can generally destroy TRANSFORMATION OF COORDINATES. 97 the terras of one dimension in x^ y-, ^'i in an equation of the third degree three of the terms of two dimensions, and so on with equations of higher degrees. If, however, the terms, ■whose coefficients we desire to destroy, differ by more than one dimension from the degree of the equation, the equations for determining /", ^, h in order to effect this result will rise to second, third, or higher degrees. For, if F{x^y^ z) = be , an equation of the n^'^ degree, the transformed equation will be I F[x +/, y -f <7, z + A) = 0, and the terms of the [n — r)^^ degree in the expansion can be represented in the form j the coefficients of any term in which will involve F[f^ rjy h) I differentiated n-r times with respect to the quantities/,^,//; hence the resulting equation for destroying any such term will i be of the r^^ degree. If three terms are to be destroyed, it is necessary that the three corresponding equations should be consistent ; it may happen that these equations are not in- I dependent, in which case if two terms are made to disappear ' the third term will disappear at the same time, and we shall I be able to get rid of a fourth terra. 143. To transform from one system of coordinates to another system having the same origin^ both systems being rectangular. Let Ox^ Oy^ Oz be the first system, Ox'^ Oy\ Oz the j second; let «,, 6„ c, ; a^-,b,^^c^\ a^,, ?>3, Cj, be the direction-cosines of Ox ^ Oy\ Oz\ referred to Ox^ Oy^ Oz ; and x^y^z\ x\ y\ z coordinates of the same point in the two systems. Then the algebraic distance of the point from the plane of yz is a;; but, projecting the broken line x -\-y -\-z\ this same distance is a^x -i- a.^' + o^z . Hence x = a^x -\-a,;y' ^■a^z^^ and similarly, y = h^x + h,jj' + h^z\ \ (1) z =c^x' + r.y + c/J the formula; required. The nine constants introduced in these results arc connected by iux equations of condition, expressing that the two systems of 98 TRANSFORMATION OF COORDINATES. coordinates arc rectangular, for since 0.v, Ot/, Oz arc each two at right angles, we have the system of equations < + K'rr: = \, [A) and by reason of Ox'^ Oy\ Oz being also at right angles, the system «2«3 + ¥9+^,03 = 0, 1 «,«., + ?'/., + CjC^ = 0. ' The number of disposable constants in this transformation is therefore only three. The relations (^1), [B] subsisting among the nine constants involved in these formulai may be replaced by c,a, + c,a, + C3a3 = 0, r (i?') if wc consider Ox ^ Oy\ Oz the primary system of axes, in which case the direction-cosines of Ox^ Oy^ Oz, will be «,, a,,, o^; ^15^25^3 5 ^n^-ii^s' The equations [A') and {B') obtained from the same facts as the equations [A] and (7?), are of course dcduciblc from them. Either system may be obtained from the identical equation x' + y^ + z^ = x"^ + y"^ -^ z"\ by substituting for x, y, z their equivalents given in equations (1), or similarly for x\ y\ z. 144. Tiie relations between these constants may also be ex- pressed in the following convenient form. From the equations «,«, + ^i^i + ^.^2 = ^) ^'s"^! + ^/^ + ^3^ = *^> Ave obtain immediately a, _ ft, _ Cj ^ TRANSFORMATION OF COOltUlNATES. 99 each member of these equations is therefore equal to - {{a: + V + c-:) {a,^ + b.: + c/) - («,«3 + V'. + ^/sii* by equations [A), [B). In a similar manner, we obtain ^^ = _A = ^_ =+1 o, k c„ ±1, V2-V, c.«2-c,«, aJ>,-, x'Ox^=ylr, zOz' = 6, which is the same as the angle between the planes of xi/, xij . The transformations may be effected by successive trans- formations, each in one plane — (1) through an angle ^, in the plane of xy, from Ox^ Oy to Ox^, Oy, ; (2) through an angle B, in the plane of ^Z,-) ^o™ ^i/,) Oz \oOy^,Oz', (3) through an angle i/r, in the plane of //.,.r,, from Ox,, Oy.^ to Ox, Oy. The formulae for these transformations are, using the same suffix for any one of the coordinates as for the corresponding axis, x = x, coscf)—y, sin^,] y = x^ sin^-f ^, cos(/),j y^ = y^ cos 6 -z sin 6^ z=y.^ sin ^-1-2;' cos^,j x, = x cos>/r — ?/' sini/r,] y^ = x' sin^/r + y cosi/r,] 102 TKANSFUUMATION OF COUliDlNATES. from which we obtahi, by successive substitutions, a; = a;'(co3^ cos\lr — sincf) sini/r cos^) — y' (cos (f) sin yjr + sin cf) cos yjr cos 6) + z' sin ^ sin 6^ y = x'{ii'm(}) cos ■«//- + cos s'nwlr cos 6) - y (sin <\> sin i/r — cos cos'\/r cos &) — z cos sin ^, 2 = x' sin i/r sin Q -^-y cos -v/r sin Q •\-z' cos ^. These formulje might be established without successive trans- formation by Spherical Trigonometry, but this is left for the exercise of the student. 147. These formulae are too complicated and unsymmctrical to be generally employed. A modification of them, however, may be useful in determining the nature of any proposed plane section of a surface. We may in that case, by using the first two transformations, make the plane of x^y^ coincide with the proposed plane section, and then, making 2!' = 0, obtain the equation of the section in that plane. Tiie results may be at once derived from the final equations of the last article by making '«/^ = 0, 5;' = 0, or directly by geometrical considerations, and wc have the formulae x = x' cos 4>- y' cos 6 sin 0, y = x' sin<^ +y' cos^ cos<^, z =y' sin^, by efl'ecting these substitutions wc may obtaiji the equation of the curve which is the intersection of a surface with a given plane. If the equation of the plane be Ix + my + nz = 0, and the curve of intersection with the surface /(a;, y, ^) = be required, we shall have n , sind) cosii )sd= -TTT, 5 i^ , and — T = — - ^/{l' + in' -k- ii') ' -I m \/{l'' + -m') ' and the equation of the curve of intersection will be f{x CO3 - y cos 6 sin (p, x sin + ?/ cos 6 cos0, y sin 6) = 0. TRANSFORMATION OF COORDINATES. 103 148. As an example of this method, we will examine the position of a plane passing through the origin, when its inter- section with the surface ax^ 4 hy' + c^:' = 1 is a circle. Let the intersection of the plane with the plane of xt/ make an angle (f) with O.c, and let 6 be its inclination to the plane of xy. The equation of the section will be a {x cos (p -y cos 6 sin 6)'^+ h {x sin + ?/ cos ^ cos 0)"''+ cy' sin' ^= 1 , if this be a circle, the coefficient of xy = 0, and those of x^ and y^ will be equal and positive ; .*. (a -?») cos^ sin2(^ = 0, and a coa'cf) + b sin^'c^ = [a sm'cfy + h cos*^^) cos"'^ + c sin'^^ > 0, we shall therefore obtain the following systems of solutions if «, J, c be unequal : I. cos^ = 0, a cos''<^ + b sin'''(^ = c> 0, cos^(^ _ sin^<^ _ 1 b — c c — a b — a' II. 0, cos''^ _^^^^d _ 1 c -a a—b c -b' III. (^ = i7r, ^- = a cos'^-l-csin"''^>0, cos'^ sin'^^ 1 c — b b — a c — a If rt, Z>, c be of the same sign and in order of magnitude, III will be the only admissible solution, and the cutting plane must pass through the axis corresponding to the mean coefficient. If a and b be positive, c negative, b> a, II will be the only Bolntion. If a be positive, b and c negative, there will be no plane circular section through the origin. 149. Trans/or mat ion from one system of coordinates to an- other having the same origin^ both systems being oblique. Let O.c, 0/y, Oz and Ox\ ()y\ Oz be the two systems ; On^ On\ On" the normals respectively to yz^ zx^ and .ry, and let nx 104 TRANSFORMATION OF COORDINATES. denote the angle n Ox^ and so for the others. Then the alge- braical distance of a point whose coordinates in the two systems are respectively a;, ?/, z and x\ y\ z from the plane of yz^ is X cosnx +7/' cos???/ +z cos nz . "Whence x cosnx = x cos7ix' + ?/' cos ??_?/' + z' cos nz\ and similarly, y cos??'?/ =x' cos7ix' + y' cosn'i/' + z' cosn'z'j z cosii'z =x cosn'x -Yy cosn'y' + z cos7i"z\ the required formulaB, Involving in this form twelve constants, but, as they may be written in the form x = a^x'-]-a,^y' + a/, y = h^x' + h^' + h/, z = c^x •+ c.^y' + c/j where a, — ^-^ , and similarly for the others, we sec that ' cos??a; really only nine constants arc involved, and these arc connected by three equations on account of the angles between the original axes being fixed, so that again there arc only six disposable constants, 150. Transformatio7i fi-om any one system of axes to any other. If we wish in any of the above transformations of the direc- tions of the axes also to remove the origin, we may first remove the origin to the point (/, /y, /?), retaining the directions of the axes. This will give a^ = ^.+/? I/ = 'J:+.'7i ^ = ^, + ^', ajj, y^^ 2, being the coordinates of a point (.r, ?/, z) referred to the system of axes through tlic new origin parallel to the primary system. Now changing the direction by transformations of the form x^ = a^x + a^y' + a^z\ &c., TKANSFOKMATION OF COORDINATES. lOo we see that the most general transformation possible is obtained by formula} of the form x=f-\- a^x + a.,y' + a/.\ z =h + c^x -f c^Tj' + c/. 151. To shew that the degree of an equation cannot he changed hy transformation of coordinates. We can now prove the important proposition, that the degree of an equation cannot be altered by any transformation of coordinates : the degree of an equation meaning the greatest number which can be obtained by adding the indices of the I coordinates involved in any term. For let Asfxfz be a term in an equation of the ?i"' degree, such that ^; -f 5- + ?• = » : this I will be a type of all the terms of the n''* degree involved in the equation, any one of which may be obtained by assigning to li^p^ ^, r suitable values. Now on any transformation this term becomes I A (/+ a^x + ajj + a^zY [g + h^x + hj + h^z')' {h 4 c^x +c^y' + c/)\ I and no term in this product rises beyond the degree p + q + r i or 7J. Hence the degree of an equation cannot be raised by I transformation of coordinates ; nor can it be depressed, for if ; by any transformation the degree be depressed, then on re- : transformation, the degree of the equation so depressed would ! be raised to its original value, which we have seen to be I impossible. j 152. Relations between coefficients of a ternary quadric hcfore and after transformation of coordinates. I We notice here that in the case of quadric functions, relations between the coefficients in the original and transformed functions may be obtained without the use of the formulae of transforma- tion. The method of obtaining these relations depends upon the consideration that, if a quadric be the product of two linear factors, it will still be so after any transformation of coordinates has been effected . r 106 T15AXSF0IJMATI0N OF COORDINATES. The square of the distance of any point (cc, ?/, z) from the origin being x^ + i/^ + z\ If the axes be rectangular, this ex- pression will be unaltered in form when a change is made from one set of rectangular axes to another having the same origin. Let u = ax^ + bjf + cz^ + 2a' yz 4- "ih'zx + 2c xy be any ternary qnadric, and let h be supposed so chosen that h {x' + ?/ + s") - k shall be the product of two linear functions; the condition that this shall be the case is (Art. 88) (/, _ f,) (/, _ I) [h - c) - a" {h - a) - h" [h -h)- c" {It -c)- 2ab'c = 0, shewing that there are generally three such values of /^ Suppose now that, on transformation to another system of rectangular axes, u becomes V = ax^ + /3y^ + yz^ + 2ot.'yz + 2^'zx + 2j-hf-\-[z-cY = d\ This equation may be written in the general form x' + ^' + z' + Ax + ni/+ Cz 4- X> = 0, the equation required. THE S['HEl{r. 1 11 IGO. Since the general equation of the sphere contains four arbitrary constants, the sphere may be made to satisfy four specific conditions. It may be seen from geometrical considerations that, when I four conditions are given, there may be only one sphere, or a limited number, or an infinite number of spheres, which satisfy the equations ; at the same time the four conditions must be consistent with the nature of a sphere, and if this be the case, and the conditions be independent, there must be a limited number of spheres satisfying those conditions. For example, if four points be given through which a sphere is to pass, no three j points can lie in one straight line ; and if four points lie in one I plane, they must also lie in a circle, otherwise no sphere could j pass through them, and if such a condition be satisfied, an infinite i number of spheres can be constructed, each of which contains the I circle in which Jhe four points lie ; if the four points do not lie in a plane, so that the four conditions to be satisfied are inde- pendent, the sphere is complet-ely determined. i Again, if four planes be given, each of which is to be touched I by the sphere, no three of these must have one line of intersec- [ tion, and the four cannot pass through one point, except under a condition, and in that case an infinite number of spheres can be drawn, touching the four planes. In other cases, eight spheres can be drawn satisfying the conditions. Equation of a sphere under specific conditions. 161. To find the equation of a sphere iiassing tliroiigh a given point. Let (o, ^, c) be the given point, and the equation of the sphere x' + ?/ -V z' + Ax ^- By -^ Cz + D = ; .-. d' ^h'-^6'-^Aa + Bh^Cc + D = 0; .'. x'^y' + z' + A{x-a) + n{g-h) + C{z-c)=d' + bUc"- 18 the equation required. If tlic given point be the origin, the equation will become ■c" 4- f + z- + Ax + Bi/ + Cz = 0, and the sphere may be made to satisfy three more conditions. 112 THE SPHERE. 162. To find the equation of a sphere lohich passes through > two given points in the axis of z. Let c„ c^ be the distances of the given points from ; when a; = and y = 0, the equation must become [z — c,) (2; — c^) = ; therefore the equation of the sphere is a;' + / + (^ - cj [z - cj + ^cc + % = 0. If the sphere touch the axis of z^ c^ = c^ = 7, .-. a;' + / + z' ^Ax^Btj- 27^ + 7^ = 0. 163. To find the equations of spheres which touch the three \ axes. ■ I Let the equation of the sphere be I cc' + ?/' + z' + Ax + B?/+Cz + D = 0. "I Since it touches the axis of a;, let a be the distance from the origin ; therefore Avhen ?/ = and 2 = 0, x' + Ax + I) = 0, the roots of which arc each equal to + a ; .-. A = ± 2a and D = a\ Similarly, y^ + Bf/ + a^ is a complete square ; .-. B=±2a and C=±2a, and the equations of the spheres which satisfy the given con- ditions are x^ + y + z'^ ± 2ax ± 2ay ± 2az + ci^ = 0, which arc ciglit in number for any given value of a, corre- sponding to the different compartments of the coordinate planes. 164. To find the equation of a sphere touching the j)^(ine of xy in a given p)oint. Since the sphere meets the plane of xy only in the given point (a, //, 0), when 2 = 0, the equation must reduce to [x-aY + {y-hy = 0; therefore the required equation of the sphere is {x-aY+{y-hY + z^ + Cz = 0. THE SPHERE. 113 1G.3. Intciyretation of the crjyression [x-a)U{rj-hf+[^-cy-d' in the equation of a sphere. Let the equation of the sphere be [:c-af + {y-hY-\-{z-cY-d'' = Q, ami [x\ y\ z) be any point (?, G the centre of the sphere, and k't a straight line through Q intersect the sphere in tlie points P, P', and have for its equations x-x _y — y _ 2: - s' _ I VI n ^ therefore at the points Pand P' (/,. + ^' _ ay + [mr + y'- hf + {nr ^z - c)' - cV = 0, if r„ 1\ be the roots of this equation, therefore the left side of the equation for any point ix^ y\ z) is QF.QF, or -QF.QF, -according as Q, is without or within the sphere. If Q be without the sphere it will be the square of a tangent drawn from Q to the sphere. If Q be within, it will be the square of the radius of the small circle on the sphere whose centre is Q. Def. The product of the segments ()P, QF is called the power of the sphere with respect to Q. Cor. All tangents drawn from an external point to the sphere are equal. On the Relations of two or more Spheres. 16G. To find the equation of the radical plane of two spheres. Def. The radical 2)lane of two spheres is tlie locus of points, the powers of the two spheres with respect to which arc equal. Let the equations of the two spheres be {x-aY ^ {y -hy -^ [z - cY -d:"- u = 0, and {x - a'Y + {y - h'f + (2 - cf - d" s u' = 0. The equation of the radical plane is therefore u - 11 - 0. Q 114 CYLINDRICAL SURFACES. 167. To shew that the six radical j^^cines of four spheres intersect in one point. Let w = 0, u = 0, w" = 0, u" = be the equations, in this form, of the four spheres. The equations of the six radical pLanes are given by u = u = u = xi'\ which intersect in one point determined by these equations. Def. The point of intersection of the six radical phincs is called the radical centre of the four spheres. Cylindrical Surfaces. 168. It has been seen that the locus of an equation F{x^y) — ^i which involves only two of the coordinates, is a cylindrical surface, of which the generating lines are parallel to the axis of the omitted coordinate. We shall now shew how to obtain the equation of certain cylindrical surfaces in which the generating lines arc in a general direction. 169. To find the equation of the cylindrical surface^ whose generating lines are in a given direction and guiding curve an ellipse traced on the plane of xy. Let the equations of the guiding ellipse be X a + fj=l, and z = (1), and (7, m, n) the direction of the generating lines. Let the equations of any generating line be nx=h^as mj = mz + /3] ^ ' At the point of intersection of the generating line with the guiding curve, the values of x, ?/, z in (1) and (2) being the same, we obtain as a general equation, after eliminating '•''"=«', (3), V CONICAL SURFACES. 115 and since this is true for all positions of the generating line, eliminating a, ^ between (2) and (3), (^--P - ^^Y . ("2/ - "'f )' _ „2 a is true for every point in the cylindrical surface, and is therefore its equation. Conical Surfaces. 170. Def. a conical surface is a surface generated by a straight line which constantly passes through a given point, called the vertex, and is subject to some other condition. 171. To find the equations of a conical surface.^ loliose vertex is the origin^ (jenerated hy a straight line^ of tchich a guiding curve is an ellij)se, ichose centre is in the axis of z^ and jjlane parallel to the plane ofxy. Let the equations of the guiding ellipse be | + ^, = 1, and z = c, (1), those of a generating line in any position, x = a.z^ y = ^z. (2). Eliminating o", y, z^ the coordinates of the point in which the generating line meets the guiding curve, satisfying (1) and (2) simultaneously, V + -i-=^' (3). Since this equation is true for every position of the generating line, eliminating a, /3 from (2) and (3), /V.2 „,'i ~t X y z Ta "■ 72 ~ ~ij J a u c which is the required equation of the surface. 172. To find the equation of the conical surface, ivhose vertex is any given point, and of which the section hy the plane of xy is an elli^yse whose axes are in the axes of x and y. Let the coordinates of the vertex hcfg, h, and the equations of the elliptic section be . = 0, r,„d-j;+-^=,; 116 CONICAL SUKFACES. and let the equations of any generating line be x-f=o.{z-h), and 7j-fj = /3[z-h) (1), ■where this straight line meets the ellipse • • ~^r~ + i,^ -^ ^^^' eliminating a and yS from (I) and (2), avc obtain for every point in the surface {fz - hxf {gz - hjY _ which is the equation required. Cor. 1. If /, m, n be the direction-cosines of any generating line ifn-hir , {gn-7imy _ ci' + Ij' "''• Cor, 2. The equation of an oblique circular cone is {fz-hxY + {gz-/^yY = a^z-h)\ if a be the radius of the circle in the plane of xi/. 173. To shew that the^e are two sT/stons of circular scctiona of any oblique circular cone. When the circle which guides the motion of the generating line has equations s = 0, x^->rif = d\ the cone will be perfectly general, if we take the vertex In the plane of zx^ and therefore f 0, h for the coordinates of the vertex. The equation of the cone will then be, as in the last article, {f,-hxY^K'f = d\z-h)\ this may be written in the form /,' (a;-^ J, ,f + 2^ - a') = ;3 [2fhx - [f - W - d') z - 2hcr] . ' Ilejice, if the conical surface be cut by cither of the planes 2 = a, or 2fhx - [f - h' -a')z- 2ha' =^, the points of intersection will satisfy an equation of the form x'+f + z' + Ax + Bi/+C=0, THE SPHEROIDS. \ r\ f' WJ*' for all values of a and /9, and the sections will thcrefrf^/bp plali^ / . sections of a sphere. ' r , ^'j. Therefore, there are two series of circular sections mad^W r. two systems of parallel planes. .' '^ ' 174. The trace of the cone on the plane of zx^ putting y = 0, -— ^ has for Its equation being the two generating lines which He in that plane ; and the equation of two planes in opposite systems, giving circular .Kctions, is [z - a) [2fhx - (J'' - W - ci:) z - 2/ia' - /3} = ; by adding these equations we obtain W {x^ 4 z' + ^a; + i?2 + C) = 0, which shews that the four points, in which these generating lines meet the two circular sections. He in a circle ; hence, the first system of planes makes the same angle witii one generating Jinc which the second system does with the other. The Spheroids. 175. Def. a spheroid may be generated by the revolution of an ellipse about either axis. If the axis of revolution be the minor axis, the surface is called an Oblate Spheroid, and if the major axis, a Prohte Spheroid. 176. To find the equation of a spheroid. Let the centre be taken as origin, the axis of revolution that of ?, and let P be a point [x, ij, 2) in the ellipse GPA, which ig the position of tlic revolvjng ellipse, when Inclined at any anglg to the plane of zx, OM=x, iViV=i/, NP=z, OA = a, 00 = ON' iV^P" ^ a* c' ~ ' xU-jI z^ _ tt* c' ~ ' ■ c 118 THE ELLIPSOID. This is the equation of an oblate or prolate spheroid according c is less or greater than a. The EUq)soid. 177. Def. An ellipsoid may be generated by the motion of a variable ellipse, which moves so that its plane is always parallel to a fixed plane, and Avhicli changes its form so that its vertices lie in two ellipses having a common axis traced on planes perpendicular to each other, and to the fixed plane. 178. To find the equation of an ellipsoid. » Let QPiNhc a variable ellipse in any position, Q, E being its vertices lying in two ellipses AC\ BCj traced on perpendicular THE ELLIPSOID. 119 planes, taken for those of zx and yz ; the plane of xy^ to which the variable ellipse is parallel, being the plane containing the M mi-axes OA^ OB. Let o, c, and h^ c, be the scnii-axes of AG and i>C, and (x, y, z) any point P in QR^ PM perpendicular to QN. Then, ^.,+ ^^^,, = 1, and, since Q is a point in the ellipse A C, = 1 - z' "6' si mi arly, = 1- ' ? x' 4. z\ = 1, which is the equation required. 179. To construct the surface whose equation is x' y' ^' , a b c Lei the surface be cut by a plane whose equation is 2 = 7; the projection of the curve of intersection on the plane of xy has the equation a c therefore, the curve is an ellipse whose semi-axes a, ^ are given by the equations a c hence, the vertices lie in the two ellipses which are the traces of the surface on the planes of zx and yz. i9 Also, since - = t , the variable ellipse remains always similar ' a /) ' '■ to a given ellipse, which is the trace on the plane of xy. The suiface may therefore be generated by the motion of a variable ellipse, whose plane, tic. (See Uef.) 120 THE HYPERBOLOID OF ONE SHEET. The Hyperlohid of one Sheet. 180. Def. The hjperhohid of one sheet may be generated by the motion of a variable ellipse, which moves so that its plane is always parallel to a fixed plane, and which changes its form so that its vertices always lie in two hyperbolas traced on planes perpendicular to each other and to the fixed plane, these hyperbolas having a common conjugate axis. 181. To find the equation of an hyperholoid of one sheet. Let A Q, BR be the hyperbolas traced on the two perpendi- cular planes taken for the planes of zxj yz^ OC their common semi-conjugate axis, being the direction of the axis of z. Let QPll be the variable ellipse in any position, point (cc, y, z) in it, QN^ RN its semi-axes. Draw PM perpendicular to QN. Then MN=x, PM=y, ON=z^ any and X y W "^ RN 1; also, since Q^ R are points in the liypcrbolas, if OA = a^ OB=h, and OC-. QN" _z' , ^' THE IIYPEHBOLOID OF ONE SHEET. 121 z' x' y- or -, + ^ a b z' which is tlic equation of the hyperbolold of one sheet. 18S. To construct the surface loliich is the locus of the equation ^' 4. ^' _ i' _ 1 a*"*" b' c'~ Let the surfiice be cut by a plane whose equation is 2 = 7, then the projection of the curve of intersection upon the plane of xy has for its equation x^ ir , 7" a b c ' which is the equation of an ellipse, whose semi-axes a, /3 are given by the equations therefore the vertices of the ellipse lie r^sj^ectlvely on the hyper- bolas which are the traces of the surface on the planes of zx, yz. Also, since - = ^ , this ellipse is always similar to the ellipse which is the trace of the surface on the plane of xy. Hence the locus may be generated by the motion of a variable ellipse which moves, &c. (See Def.) 183. The locus may also be generated by the motion of an iiypcrbola, for, if the surface be cut by a plane parallel to the plane of ?/2, whose equation is x = a, the curve of intersection will be an hyperbola, the equation of whose projection on the plane of ^2 will be ■'Z- _ i _ 1 _ ^ b' c'~' a' If a < flTj the semi-axes /?, 7 will satisfy the cquat ion c' -^ rt, we shall have tt, =\ — —,- \\ hence the extremities of the transverse axis 27 will He on the hyperbola, which is the trace on the plane of zx. 184. To find the form of the surface at an infinite distance. If z be increased indefinitely, x"^ v" p'' ( c\ z^ -+j7.=^(l+-A=-i ultimately. Let this surface and the hyperboloid be cut by a straight line drawn parallel to Oz through a point (a:', ?/', 0), and let 2,, z^ be the corresponding values of z, , x" y"' < then — + TT" = 4- J a h c ' , x"^ ip 2: '■' and — + 7, = 4 + 1, a h c ' z"^ — z" ' ^. -I- ^„ if x\ or ?/', or both, and therefore z^ and z^^ be indefinitely in- creased, z^ - z^ will diminish indefinitely, and ultimately vanish ; x^ y'' _ z} •'• «^ "^ P ~ ? is the equation of an asymptotic surface, which lies further from the plane of xy than the hyperboloid. This asymptotic surface is a cone, for, if it be cut by any plane whose equation is - = - cos^, all the points of intersection will lie in the planes *{ = + - sin Q. The surface is thcr<;fore * b c capable of being generated by a straight line which passes through the origin, and is guided by the ellipse whose equations arc — 4 75 = 1, and z=^c. a h THE IIYPEHUOLOID OF TWO SHEET: 123 The figure shews the position of the conical asymptote rela- tive to the hypcrboloid. ABab is the principal elliptic section, A'B'a'h\ A"B"a"h" the sections of the hyperboloid and cone made by a plane parallel to that principal section, at a distance OC = c. The Hyperboloid of two Sheets. 185. Def. The hyperboloid of two sheets may be generated by the motion of a variable ellipse, which moves so that its plane Is always parallel to a fixed plane, and which changes its form, so that its vertices lie always on two hyperbolas having a common transverse axis, traced upon two planes perpendicular to each other and to the fixed plane. 186. To find the equation of the hyperboloid of two sheets. Let A Q^ AR be the hyperbolas traced on two perpendicular planes, taken for the planes of zx,, a-?/, and having the common semi-transverse axis OA^ and let QPR be the variable ellipse in any position, whose axes are QM^ RM^ parallel to the plane oiyz. Take P any point [x^ ?/, z) in the ellipse, and draw PN per- pendicular to iiW, then OM=x^ 3IN=y, and XP=z; there- fore, since P is a point in the ellipse. •^ - -L = 1 RM 124 THE IIYPEKBOLOID OF TWO SHEETS. and if a, c and a, h be the semi-axes of tlic two hyperbolas A Q, ABs Ji3P _ x' QM' ?/' Z^ X' , • • ^,-. + .^ .'^ h c 1/ & 1, which is the equation of the hyperboloid of two sheets. 187. To construct the locus of the surface whose equation is Let the surface be cut by a plane whose equation is £c = a ; the equation of the projection on the plane of yz of the curve of intersection is b'^ c^ d^ which, if a > «, is the equation of an ellipse whose semi-axes /3, y are given by the equations y ~ a' &' THE HYPEKBOLOID OF TWO SHEETS. 125 therefore the vertices of the ellipse lie in two hyperbolas, whose equations arc -^-•f, = ], and -- ^,=1, a b a c which are the traces of the surface on the planes of xt/, zx^ having a common transverse axis in the line Ox : and since -r = - , this ' be ellipse is always similar to a given ellipse, axes 2Z>, 2c. Hence the locus may be constructed by the motion of a variable ellipse which, &c. (See Def.) 188. The locus may also be generated by the motion of a hyperbola ; for, if the surface be cut by a plane parallel to the plane of ic?/, whose equation is 2^ = 7, the curve of intersection will be an hyperbola, the equations of whose projection on the plane of xy will be a b c x"^ ■?/" which may be written —, — ~ = I. whose transverse and con- •' a' 13' ^ jugate semi-axes will satisfy the equations «- = , + i* = ^. a' c^ b^ Hence the tranverse axis will have its extremities in the hyper- bola, which is the trace on the plane of zx, and the hyperbolic section will be similar to the trace on the plane of xy. 189. To find the form of the hyperholoid of tioo sheets at an infinite distance. If x be increased indefinitely, the equation — ^ = ; 2 + ~ + 1 shews that ?/, or 2;, or both, will also be increased indefinitely, and the equation becomes a' \V c'l y z" \ li' ^ d' " z" ^ + ^ ultimately. h c ■' 126 THE HYPERBOLOID OF TWO SHEETS. Let the hjperbolold, and the surface represented by this equation, be cut by a straight line parallel to the axis of a-, drawn through the point (0, y\ z\ ic„ a;.^, the corresponding values of x are given by the equations x; ?/ ' z' a b c and ±1=4+ "-,+.1 = 1, and x.^ a-, + it-, ' therefore x,^ — x^ diminishes indefinitely, and ultimately vanishes as y, or z', or both, increase indefinitely ; hence the liyporboloid of two sheets continually approximates to the form of the surface whose equation is — , = ^ + ^ , which is therefore called an asymptotic surface. Also, if this surface be cut by a plane whose equation is J- = - cos 6, all the points of intersection will lie in the two planes - = ±-sin^; and the surface can therefore be gene- rated by straight lines drawn through the origin, which inter- sect the ellipse, whose equations are '4 + -^ = l, x = a. THE ELLirnc paraboloid. 12; This asymptotic surface is therefore a cone on an elliptic base, and lies nearer to the plane of yz than the hyperboloid, since x^ < x.^'. Its position relative to the hyperboloid is shewn in the figure, in which BC \s the section made by a plane parallel to yz through the extremity of the transverse axis, and BE, de are sections of the hyperboloid and conical asymptote, made by a plane parallel to yz. The Elliptic Paraboloid. 190. Def. The elliptic paraholoid may be generated by the motion of a })arabola, whose vertex lies in a parabola traced upon a fixed plane, to which its plane is always perpendicular, the axes of the two parabolas being parallel, and the concavities turned in the same direction. 191. To find the equation of the elliiytic paraloloid. Let xOy be the plane on which the fixed parabola 0(2 is traced, Ox the axis of OQ', QR the axis of the moveable parabola QP, P any point {x, y, z) in the parabola. Draw Py perpendicular to QP, and QU, NM to Ox, then 128 THE HYPERBOLIC PARABOLOID, since P is a point in QP^ if /, l be the latera recta oi OQ and (?P, PN' = l'.QN, and QlP=I.OU', ... t + ^:l=OU+QN=OM=x, ■which is the equation of the elHptic paraboloid. 192. To construct the locus of the equation 2 2 Let the locus be cut by a plane, whose equation is y = ^i the projection of the curve of intersection upon the plane of zx has for its equation which represents a parabola whose axis is parallel to the axis of x^ the coordinates of whose vertex are ^, /3, 0; therefore the vertex of the parabolic section lies in the parabola whose equation is y'^ = ?x, which is the trace on the plane oi xy] there- fore the locus may be constructed by the motion of a parabola, whose vertex, &c. (See Def.) The HyperhoUc Paraboloid. 193. Def. l^ho, hyperhoJic paraloJoid may be generated by the motion of a parabola, whose vertex lies in a parabola traced upon a fixed plane, to which its plane is perpendicular, the axes of the two parabolas being parallel, and the concavities turned in opposite directions. 194. To find the equation of the hyperbolic paraboloid. Let xOy be the plane upon which the fixed parabola is drawn. Ox the direction of the axis of the parabola ; let QB be the axis of the moveable parabola QP^ parallel to Ox^ measured in the direction contrary to Ox. THE IIYPERBOLTC PARABOLOID. 129 Draw PN perpendicular to QR, and QU, NM to Ox] then, if P be any point (a;, y^ z) in QP^ OM = Xj MN=y^ and NP=z. Let /, I' be the latera recta of OQ^ QP; therefore, PIP=l'.QN, and QTT' = l.OU, OTT^ PN^ and ^^- - £p= 0Z7- QN= OM; 2' 7 = ^, which is the equation of the hyperbolic paraboloid. 195. To construct the locus of the equation Let the locus of the equation be cut by the pLane, whose equation is y = (3j the projection of the curve of Intersection upon the plane of sx has for Its equation -'■(T-^) which represents a parabola, whose axis is measured in the direction contrary to Oxj and the algebraical distance of whose vertex from the plane i/Oz Is -, ; therefore the sccliou by the l'>0 THE IIYrEKMOLIC I'AKAHOLOID. plane j/ = /3 Is a parabola, whose latus rectum is /' and the co- i ordinates of whose vertex arc -y , 13, 0; or, the vertex lies In a parabola traced upon the plane of xi/, whose equation is i/^ = Ix. llcncc the locus may be generated by the motion of a para- bola, whose vertex, &c. (See Def.) 196. The locus may also be generated by the motion of an hyperbola; for if it be cut by a plane parallel to that of yz on the positive side, whose equation is a? = a, the equation of the projection of the curve of intersection on the plane of yz will be ~- y = a, whose tranverse and conjugate semi-axes, yS, 7, will satisfy the equations /3' = Ioi. and ry'^ = — I'a, the extremi- ties of the transverse axis will lie in the trace on the plane of ary, and the conjugate axis will be equal to the double ordinate of the trace on the plane of zx corresponding to iC = - a. If it be cut by a plane parallel to i/z on the negative side, the section will be an hyperbola whose transverse axis will be in the direction of Oz. If a = 0, the hyperbolas will degenerate into two straight lines, which is the intermediate form in the transition. 197. To fjid the form of the h]iperhoUc 2)araholoid at an infinite distance. If y and z be indefinitely increased while x : z remains finite, //' z' [ l'x\ s"" , . , ■L = --li + —j=^ ultunatcly ; and if these planes and the hyperbolic paraboloid be cut by a straight line parallel to Oy, drawn through a point [x\ 0, s'}, ?/,, v/„ the corresponding values of y will be given by the equations I = n ^^"^ / - ^=^5 y.,^ - y ,' , ^^' • ■ - --^ or I/.,-!/, = I ' -"■' •" !/., + !/, THE inriCKBOLlC rAHAr.OI.OID. i:]l Tlicrctbrc, if a:' remain finite or small compared with ?/, or ?/.,, !l-~y\ ^^''^' diminish as ;:' increases and will ultimately vanish; and the two planes, whose equations arc "^^ = ± -y^ , will give the V ' V ' torni of the surface at an infinite distance for finite values of x^ or for values of a; which are small compared with %j or z. These planes will not form an asymptotic surface, except r points at which x vanishes compared with y or s, since -y, will not ultimately vanish in any other case, and simi- larlv for .?., -z^. B The figure is intended to slicw the position of the asymptotic planes with reference to the hyperbolic paraboloid. Ox is parallel to the axis of the generating parabola, of which OB is one position in the plane of zx. PAp^ FA'p are opposite branches of a hyperbolic section perpendicular to (9.c, the asymptotes of which IiCR\ rCr are sections of the asymptotic surface, A A' the transverse axis being parallel to Oy. 132 THE PARABOLOIDS. LL\ W arc the traces on the plane of yz of both the para- boloid and its asymptotic surface. QBq is a branch of a hyberbolic section on the negative side of Ox^ the two asymptotes of which SC'S'j sG's are sec- tions of the asymptotic surface, and the transverse axis BO' is parallel to Oz. 198. To skew that the elliptic and hyperbolic paraboloid are particular cases of the ellipsoid^ and the hyperboloid resptctively. a o c be the equation of an ellipsoid or hyperboloid, and let the origin be removed to the point (—a, 0, 0). The transformed equation is x^ y' z'^ _ 2x a c a Let a, b. c become infinite, while - , — remain finite quan- ' ' ' a ' a ^ titles, and denote these by I and l'. The equation may then be written ^" f ^' o all ' which has for its limit, when a becomes infinite, which is the equation of an elliptic or hyperbolic paraboloid. h' c' The assumption that - and - remain finite is the same thing as assuming that tiie latcra recta of the traces on the planes xy^ zx^ respectively, remain finite when the axes become in- finite, and the corresponding ellipses or hyperbolas become parabolas. It is obvious from the above, that the elliptic paraboloid is a limiting case of either the ellipsoid or the hyperboloid of two sheets, and the hyperbolic paraboloid of the hyperboloid of one sheet. SUia"ACES OF THE SECOND DEGREE. 133 199. The surfaces of the second degree, which we have been discussing, have equations of the two forms, Ax'-^By'-vGz'^D, (1) j and B7f-^Cz' = Ax', (2) and it will be shewn in a succeeding chapter that the equations of all surfaces of the second degree may by transformation of coordinates be reduced to one of these two forms. The first form of equation includes all surfaces which have a centre at a finite distance, and the second those which have a centre at an infinite distance. In the equation (1), if —a:, -?/, —z be written respectively for a;, ?/, 2, the equation will not be altered ; therefore if (a;, ?/, z) be a point in the surface, (-ar, — ?/, — z) also will be a point in it, so that [(FOP' be any chord through the origin 0, the chord will be bisected in 0, and will be a centre of the surface. Also, for any values of i/ and z, the values of x are equal and of opposite signs, therefore the plane of i/z bisects the chords which are drawn perpendicular to it ; and a plane which bisects the chords drawn perpendicular to it is called a prin- cipal plane of the surface. Hence the planes a-?/, yz and zx are principal planes of the surface. It is evident that the planes of zx^ xy are principal planes of the surfaces whose equations are of the form (2). The sections made by the principal planes are called prin- cipal sections. That the surface represented by (2) may be considered to have a centre at an infinite distance may be shewn by con- sidering this equation as the limiting form of (1) when the origin is transferred to a point (— a, 0, 0), a being determined by the equation Ao^ = D. The equation (I) will then assume the form Ax' + By' + Cz'=^2Aax, and this surface has a centre on the axis of a-, at distance a from the origin. iJow, if we suppose A to vanish, while Aa. remains finite, an equation of the form (2) is the result. But to satisfy these 134 PROBLEMS. conditions a must be Infinitely great ; hence a surface repre- sented by (2) lias a centre at an infinite distance on the axis of X, and also a third principal section, parallel to the plane of v/^, at an infinite distance. 200. Considering the peculiar importance of the properties of surfaces of the second degree, and their frequent occurrence in the solution of problems, and the establishment of theorems, in all departments of physical science, we have adopted a special term derived from the term Conic^ invented by Salmon for the locus of the equation of the second degree in Plane Geometry. Def. The locus of the general equation of the second degree is called a Conkoid.'^ \ X. (1) A straight line is drawn through a fixed point O, meeting a fixed plane in Q, and in this straight line is taken a point P such that OP. OQ is equal to a given quantity ; shew that P lies on a sphere passing through O, whose centre lies on the perpendicular from O upon the plane. (2) Investigate the equation of a sphere conceived to be generated hy the motion of a variable circle, whose diameter is one of a system ot parallel chords of a given circle, to which the plane of the variable circle is perpendicular. (3) Construct the sphere Avhose ])olar equation is ?• = a sinf cos. (4) A straight line moves with three fixed points A, B, C in the three ' coordinate planes ; shew that any other fixed point P of the straight line will lie on an ellipsoid whose semi-axes are equal to PA, PB, and PC. (5) Find the locus of a point whose distance from a given point bears a constant ratio to its distance, (1) from a fixed plane, (2) from a fixed straight line, (G) Find the locus of a point which is equidistant from two fixed liii' which do not intersect. • The reasons for not adopting the term Quadric, which is employed by Salmon and approved of by many writers, arc given in the Preface. I i'K(jnLi:.M.s. 135 (7) The locus of a point, wliose distance IVom a Hxl'cI plane is always equal to its distance I'rom a fixed line, is a cone. (8) Shew that the elliptic paraboloid may be generated by a variable ellipse, the extremities of whose axes lie on two parabolas having a common axis, and whose planes are at right angles to each other. (9) Shew that an hyperbolold of one or two sheets degenerates into a right elliptic cone, when its axes become indefinitely small, and preserve a I finite ratio to each other. (10) Three straight lines, mutually at riglit angles, are drawn from the x' ;/' z' '. [origin to meet the ellipsoid - + V; + i= 1> shew that, if their lengths be r„ r„ r^ I 111111 '•l ^2 ^3 « ^ (^ (11) The curve traced out on the surfoce4 + - = a: by the extremities I ^ ' be'' \ of the latera recta of sections made by planes through the axis of x, lies on i the cone j^" + z* = 4j:*. (12) The locus of the line of intersection of two planes at right angles to each other, each of which passes through one of two straight lines, inclined at an angle 2a, and whose shortest distance is 2c, is a hyperboloid of one sheet, one of whose axes is 2c, and the others are as cos a : sin a. (13) The surface generated by a straight line, revolving about a fixed straight line, with which it is supposed rigidly connected, will be a cone, or a hyperboloid, according as the straight lines do or do not intersect. (14) Find the equation of the locus of a line which always intersects two given lines, and is perpendicular to one of them. Interpret the result when the two given lines are at right angles to each other. (15) The locus of the middle points of all straight lines passing ihrou"h i a fixed point and terminated by two fixed planes is a hyperbolic cylinder. ! (16) Find the locus of straight lines which meet the two lines x = «, y = 0, and x = - a, 3 = 0, and touch the sphere a;' + j/" + z* = c' ; and shew j that the locus reduces to two central conicoids when c = a. (17) The ellipse, whose equations are ^i + W = ^» ^"^ ' = "'-^j rotates about the axis of s, prove that it always lies on the surface x' ^if - (a' - W) -^^ - h\ vi'ir 136 PROBLEMS. (18) Prove that the cones on the elliptic base — + Vn = 1. z = 0, whose vertices are on the hyperbola -; — tj ~ u= ^> y = 0, are right circular. (19) Of two equal circles, one is fixed and the other moves parallel to a given plane and intersects the former in two points ; prove that the locus of the moving circle is two elliptic cylinders. (20) If A, B, C be the extremities of the axes of an ellipsoid, and AC BC the sections containing the least axis, find the equations of the two cones whose vertices are A, B, and bases BC, AC resjjectively ; shew that the cones have a common parabolic section, and if / be the latus-rectum of this parabola, and /j, ^2 those of the sections AC, BC, then i; = y-, + r* • (21) Find the locus of a point through which three straight lines can be drawn mutually at right angles, and passing through the perimeter of a curve whose equations are 2 = 0, and ax^ + bi/* = \. (22) The trace of an ellipsoid on the plane of xy ia AB; shew that a cone which has AB for a guiding curve will intersect the ellipsoid in another plane curve, and that this plane intersects the plane of AB in the polar with respect to AB of the projection of the vertex on that plane. CHAPTER XL ON GENERATION BY STRAIGHT LINES. 201. In the preceding chapter avc have shewn how certain surfaces of the second degree may be generated by the motion of ellipses, hyperbolas and parabolas. In the case of the cylinder and cone we have Investigated the equations by sup- posing them to be generated by the motion of a straight line subject to certain conditions. "VVe shall in this chapter shew that the hyperboloid of one sheet, and the hyperbolic paraboloid, as well as the cone and j cylinder, are capable of being generated by the motion of a I straight line. ! But, before giving the analytical representations of the mode [ of generation by straight lines, a general geometrical discussion may be found useful. j 202. Since a surface of the second degree can be inter- I sectcd by a straight line in two points only, unless it should I turn out that the line lies entirely in the surface, as In the 1 case of a cylinder, it follows that no straight line can intersect a plane section of the surface in more than two points, and : that every plane section must therefore be a conic. Now, if a plane be drawn containing a tangent to the prin- j cipal elliptic section of the hyperboloid of one sheet and per- I pendlcular to its plane, the curve of intersection with the surface will, in consequence of the flexure of the surftice being in opposite directions, be a conic which crosses itself at the point of contact, and the only conic having this property is two intersecting straight lines. 138 GENERATION BY STRAIGHT LINES. Hence, through every point of the elliptic section two straight lines can be drawn which lie entirely in the surface, and by making the plane travel round the ellipse, such straight lines sweep round the whole surface, which can therefore be generated in two ways by the motion of a straight line. 203. If we take any two positions of the plane through tangents PT^ P'T to the principal elliptic section APP'A\ their line of intersection, wlilch will be perpendicular to the principal plane, will Intersect the hypcrboloid in two points only Q^ Q', and the two pairs of generating lines will be PQ, PQ and P' Q^ P'Q'i since no straight line common to the plane PQQ and the surfiice can meet QQ except in Q or Q. Thus, the orthogonal projections of generating linos on this principal plane are tangents to the principal elliptic section ; and similarly for the principal hyperbolic sections. Also, through every point such as Q^ two straight lines can be drawn which lie entirely in the surface; and it is evidentl that generating lines of the same system, such as PQ and PQ^ ' do not intersect. If a plane be drawn In any direction containing a generating line /••, g + mr^ h -f nr ; hence the equation « (/+ ^^f + ^ Cy + -'n rf + c (7i + nrY = 1, must be satisfied for all values of r ; .-. ar-^hn'^c£ = Q, (1) afl + hgm + chn = 0, (2) i^nA af^+lf + cK'=\. (3) (I) shews that one or two of the quantities «, &, c must be negative; let c be negative; then since, by (1), (2), and (3), [aV + hm') {af + hg') - [afl + hgnif = (l - cF) (- en') - cVi'71% ah{gl-fmy^cn'^ = 0- (4) I hence, unless «, h, or c = 0, which are cases of cylindrical sur- \ faces, qb must be positive, and therefore both a and h will be J)ositive, sinc'e all three cannot be negative. Thus the central surface, on which a straight line can lie entirely, must be the hyperboloid of one sheet. y' z" If the surface be non-central and its equation be y + — =^? the equation corresponding to (1) will be in' n' -7- + - = 0, b c ' which shews that the surface must be the hyperbolic paraboloid, since b and c must have opposite signs. GENERATION BY STRAIGHT LINES. 141 In this case, for every position of the point (/, sin^ sec0 z — c tan^ a sin (^ ± 0) ~ - ^ cos (^ ± ^) ~ ±c ^ OEXEKATION BY STRAIGHT LINES. 143 which shew tliat they meet the principal elliptic section in points whose eccentric angles arc ^ ± ^. 213. The projections of the generating lines upon the principal planes are tangents to the traces on those planes. The equation of the trace on the plane of zx is ^ _ ^' _ 1 a' e ~ ' and that of the projection of a generating line on the same plane - = cos Q ±" sin Q. a c and the points of Intersection are given by the equation ^ + 1- fcos^i- sin^j =0, z'^ 2z or —, cos'^ + — cos ^ sin 6 + sln^6^ = 0, which, giving equal values of Zj shews that the projection Is a tangent to the trace upon the plane of zx. Similarly, the projection on the plane of .t?/, and the trace on that plane, intersect in points given by the equations -cos^ + -|sIn^=l, t,+f, = l, a a whence f - sin^-^ cos^J =0. Hence the points of intersection coincide, or the projection is a tangent to the trace on the plane of xg. 214. To shew that two generating lines of the same system do not intersect. The equations of two generating lines of the same system are - = cos 6 +- sin 6. •,-= sin 6 +- cos 6. and - = cos^' + - sin^', ' =sln^' + - cos^' ; a c '• ' h 144 GENERATION BY STRAIGHT LINES. if the two lines meet, we shall have at the points of intersection, = cos d — cos & + ~ (sin 6 — sin 6'). ~ C ^ /J and = sin ^ - sin 6*' + - (cos 6 — cos 9') ; and the condition of intersection will be (cos e - cos ey + (sin e - sin ey = O ; which cannot be satisfied unless 6 = 6'. Hence, generating lines of the same system do not intersect. 215. To sJieio that cjenerating lines of oj)posite si/stems mitst intersect. The equations of two generating lines of opposite systems are - = cos 6? + - sm 6/ , "f = sm C7 + - cos 6/ ; a CO c and - = cos & + - sin B\ 7 = sin & ±"- cos Q' . a CO c If the two lines meet, we shall have at the points of intersection, = cos ^ — cos d' ±- (sin 6 + sin &). and = sin ^ — sin 0' + - (cos 6 + cos 0')^ and the condition that they may intersect will be cos' 6 - cos'^' + sin'^ - sin'^' = 0, which, being identically true, shews that any two generating lines of opposite systems intersect. 216. To find tlie locus of the intersection of tico (jencrating lines of opposite systems^ drawn through points in the ji^'if^cipal elliptic section^ whose eccentric angles difier by a constant angle. Let + a, and - a, be the eccentric angles of two points in the principal elliptic section, differing by a constant angle 2a. The equations of the generating lines of opposite systems arc - = cos (^ + a) ± - sin (0 + a), | = sin (0 + ot) T- - cos (0 + a), and - = cos(0- a) + ^ sin(0 — a), '. = sjn (0- a) ±*- cos(0- a). OENEl^VTION V,Y STUAIOUT LINES^ /^ l-i^j At the points ot Intersection, %. * / ■%' /^ O = sin0 slna + - sin0 cosa, .*. -=±tanW Vi ^'t. c ' c X ^/ , ' Also - = cos0 cosa± - cosO sina = cosC/ scca, / ' = sln0 cosa + - sin0 sina^sinO scca ; •-«• h "— + ',-, = sec" a, and - = ± tan a. (f b' c Therefore the locus required is the two elliptic sections, parallel to the plane of .r?/, which intersect the traces on the planes of ^.v, ?/-, at points whose eccentric angles arc + a. 217. The accompanying figure is meant to be a repre- sentation of the positions of sixteen generating lines of each system, corresponding to eccentric angles differing by ^tt. ABab is the principal elliptic section, A'B'a'b' and A"B"a'b" are the parallel elliptic sections which intersect the conjugate axis of the hypcrboloid at its extremities C", (7", the axes of which sections arc in the ratio v'2 '- 1 to the axes of the principal sections. The generating lines through the extremities of the axes Aa^ Bb intersect these two ellipses at points L\ A'', and Z", K'\ whose eccentric angles are — and ' — , i.o. at the extremities of -1 i equi-conjugate diameters; and those through Z, 7v, the ex- tremities of equi-conjugate diameters of the principal elliptic sec- tion pass through the extremities of the axes of the two ellipses. The two ellipses A'B'a and A"B"a" are the loci of the inter- sections of opposite systems of generating lines drawn through the extremities of conjugate diameters of the principal elliptic section. The figure serves to represent that the intersections of gene- rating lines of opposite systems drawn through points in the principal elliptic section, whose eccentric angles differ by a constant angle, lie in an ellipse, the plane of which is parallel to the principal plane. As, for example, such pairs of gene- rating lines as LB\ P'D^ and BL\ PP'. i: 146 GENERATION BY STRAIGHT LINES. V V^ K/ )fN)^' ^"^-•^OZ _\/ V .3^^^ B" "' >' 218. 'lofind The equation ile generating lines of a hjperlolic paraboloid of the hyperbolic paraboloid, is satisfied by the values of x, ?/, z for every point in the lines whose equations arc V X ^ - a 0) = f^. whatever be the value of a. GENERATION BY STRAIGHT LINES. 147 Therefore by giving a all values, we obtain two scries of straight lines, all of which He entirely In the surface ; and these are the two systems of lines M'lilch arc rectilinear generators of the paraboloid. The equation (I) shews that In the two systems all the gene- rators arc parallel respectively to the two asymptotic planes, whose equations are V - z V/ ^Jl' 219. To shew that generating lines of a hTjperhoUc parahohid of the same systetii do not intersect^ and that those of oj^posite systems do intersect. Let the equations of two generating lines of the same system be y J z a ^l ± V^' ~ ^ "" and If the two lines could intersect, these equations could be simultaneous, therefore a — ,S = 0, which is Impossible, since the two generating lines are distinct ; hence they do not intersect. Changing the order of the signs in the ambiguities in the second set of equations, we have the equations of a line in the system opposite to that of the first. If then the straight lines intersect, and the consistency of these equations proves that two generating straight lines of opposite systems will always intersect. 220, To shew that the projections of the generating lines on the principal planes are tangents to tho principal sections. Since the equations of the generating lines are y^ _ 2 _ a y z _ \'l' 71'' nV'-vT' s'l- ,/'- a-*"' 148 GEXEKATION JiY STRAIGHT LINES. the equations of their projections on the phanc of zx^ 2z \/l' ±" //' = — ^- \U a ■which, bciiia" of the form z = mx— -- vr gents to the parabohi on the phmc of xi/. -- , are the equations of tan- I'x] similarly for the projections 221. The accompanying figure is intended to represent the manner in which the hyperbolic paraboloid is generated by straight lines. IIAKj Il'A'K' arc portions of the branches of a hyperbolic section made by a plane parallel to that o{ yz^ cutting Ox on the positive side; KCE\ DCU are the asym])totes of the section. FBF\ GB' Cr are portions of the branches of a hyperbolic section parallel to yz on the negative side of Ox. A OA' and BOB' are the traces on the planes of .77 and zx. TKOBLEMS. 149 Tlie two sections arc so clioscn tliat the generating lines through i?, an extremity of the transverse axis of one section, pass through J, A\ the extremities of the transverse axis of the j other. I dOj (I Oo are tlie traces of tlie paraboloid on the plane yr, where the hyperbolic section degenerates into two straight lines. XI. (1) The equations of the generating lines of the surface i/z -^ zx + xi/ + a' --- 0, drawn througli the point f 0, am, J , are X (1 ± ?«) = a»t - J/ = + (tnz 4 a). (2) At any point where the planes x + i/^z = ±n meet the surface ary + 2/s + sr + n' = 0, the two generating lines of the surface are at right I angles to each other. j (3) The eccentric angles of the points in which the principal hyperbolic I sections are met by any generating line are complementary, and that of I the point in which it meets the principal elliptic section is equal to one of I these. -(^) Prove that the points at a finite distance on a hyperbolic paraboloid, at which the generating lines are at rij;ht angles to each other, lie in a plane. (5) Shew, by geometrical considerations, that the locus of intersection of two generating lines drawn through two points in the principal elliptic section of an hyperboloid of two sheets, whose eccentric angles differ by a constant quantity, is two ellipses parallel to the principal plane, at equal distances Irom it. I (6) Prove that, if any straight line intersect three straight lines which are all parallel to the same plane without intersecting each other, the j intersecting straight line uill in all positions be jjarallel to another fixed I plane. I i (7) Shew that there are two straight lines, and two only, which intersect ' four straight lines, no three of ^Yhich are parallel to the same plane, and f no two of which intersect. I (8) Find at what points of the principal elliptic section of an hypcrbo- jloid the generating lines can be at right angles^and shew tliat the I diameter parallel to the tangent at that point is equal to the length of the iniuginary axis. 150 PKOBLEMS. (9) If four generating lines intersect so as to form a quadrilateral, whose angular points taken in order are (^i0i), {O^fpi), (O^fp^), {,0^(pi), (see Art. 212), prove that 01 + 03 = ^2 + 6^, and 0, + 0s = 02 + 04- (10) A straight line moves so as to intersect the parabolas y^ = ax, z = 0; z' = -ix, »/ = 0; 1/ z and to be always parallel to one of the planes ^—=±~j; shew that its locus is the paraboloid - - - = x. (11) The equation of the locus of a straight line constrained to move so as to intersect three straight lines, which do not intersect each other, and are not parallel to the same plane, is, when referred to axes parallel to the straight lines, 6c ca (lb a c X V z (12) The generating lines of the hyperboloid -j + y-j — ;= 1, at any point x^ iP- s* ■where it is met by the cone -7 + ,7 — s = 0, are both perpendicular to some a^ U" c^ '■ * other generating line. If the generating lines be themselves at right angles, the point will lie also on the sphere z* + »/* + 2* = a* + 6* - c*. Shew that these conditions 1 1 1 cannot coexist unless — + - = — . «* b"" c' (13) If three generating lines of the same system on an hyperboloid be mutually at riglit angles, the shortest distance between any two will lie on a generating line. (14) If three generating lines of the same system, mutually at right angles, be made the edges of a rectangular parallelepiped, siiew tliat tlie angular points of the parallelepiped which are not on the hyperboloid will lie on the surface x" -t j/' + z' = a* + 6' - c*, and on the surface whose equation ■would be obtained by eliminating h between the equations h' + a^ h' + b' h' - c' ' ^./^■ > «')' (/i'-f i")' (/*» - c7 (15) If two planes be drawn, passing respectively through two gene- rating lines of the same system at the extremities of the major axis of tlie principal elliptic section, and intersecting in a tiiird generating line, tiie traces of these planes on either of two fixed planes will be at rigiit angks to each other. (16) If a ray of light be reflected between two plane mirrors, inclined at any finite angle, shew that all the reflected rays \\\[[ lie on an liyperboloid of revolution ; and find its position. PROBLEMS. 151 (17) The perpendiculars from the origin on the generating lines of the paraboloid -7 - r? = - lie upon the cones ( - ± '- ) {ax ± iy) + 2z' = 0. a 6' c \a 0/ (18) The perpendiculars from the origin upon the generating lines of the hvperboloid -; + Vi — ^, = 1 He upon the cone rt* b* c' ^J6'+c7 + J^,(c' + «'i' = _- i«'-i',». (19) The angle between two planes, each passing through the centre, and through one of the generating lines at any point of an hyperboloid, is given by the equation 2rcot0 1111 - ; =-»--.- r« +-» ' aoc p a c r being the distance of the point, and j} that of the plane containing the generating lines, from the centre of the hyperboloid. (20) If be the acute angle between the perpendiculars from the centre on the generating lines of an hyperboloid which pass through the point (a cosa, 6 sin a, 0), then will c' (g' - by ^ «^(/>ltc^)' h' (c' 4 g')* tan*i0 sin* a ■*" cos* a (21) If be the angle between the generating lines of the hyperboloid «' 6' c' ' ■which pass through a point at a distance r from the origin, and if/) be the perpendicular from the origin upon the plane passing through them, shew that 2alc cot0 =/; (r' - g' - i* + c'). (22) The tangent of the angle between the generating lines of the surface which pass through the point (/, ff, h), is , a - b h + — 4 (23) Prove that if r be the distance of any point of the surface ys + z^ + ry + 2g' = 0, from the origin, the angle between the two genera- ting lines at that point will be .. r* - Gg' 152 PROBLEMS. (24) The angle between the generating lines through the point {xj/z) of 3 hyperboloid of the equation a;* w' 2* \, + X. the hyperboloid — fVi — = lis cos'' .-- — r^ , where X„ X, are the roots a c X, - X.. I a (rt t Aj 6 (6 ^ X) c (c + X) (25) Generating lines of the hyperboloid x^ «' s« , «■' i'^ c' are drawn through points in the plane of xi/, whose eccentric angles are a, /3; shew that their points of intersection are given by the equations X y z 1 a + H~ , . a ^ fi ~ . a - li~ a - j:i ' a cos — —■ sin — ; — ± c sm — ^^^ cos also that the shortest distance c between two of the same system is given by the equation 4 sin* — - sm* — _ - cos*— ^-.^ cos* — ~ i* " a* "*" /.* "*" c* (26) The straight line which is orthogonal to each of two non-intersect- ing generators of the hyperboloid x- 4 ?/* - s* = a*, becomes a generator o! the opposite system when the two non-intersecting generators become consecutive. (27) Shew that the shortest distances between generating lines of the same system drawn at tl'.e extremities of diameters of the principal elliptic section of the hyperboloid whose equation is lie on the surfaces, whose equations are cri/ _ ahz a:* + y* ~ a* - i* * (28) Shew that, if an eye observe the generating lines of an hyper- boloid of one sheet, every generating line will appear to lie on another. If the eye be placed upon the surface of the hyperboloid whose equation is az* + hf + cz' = 1, ]>rove that the points through which the generating lines appear to be perpendicular lie on a plane whose equation is [a^b^-c) {abx + bgy + c7« - 1) = 2 {a*fx + ¥gu + cViz), where (/, g, h) is the position of the eye. CHAPTER XII. SIMILAR SURFACES. PLANE SECTIONS OF CONICOIDS. CYCLIC SECTIONS. 222. ^YE shall now consider the nature of the curves in ; which a plane intersects central and non-central conicoids, and ■we shall at present consider these surfaces as given by cqua- I tions in the simplest form, such as have been discussed in the i tenth chapter. We shall examine the special cases In which the section made hy a plane is circular, called a ci/cJic section, and the generation of the central conicoids and of the elliptic paraboloid by the motion of a variable circle, the plane of which is parallel to a given plane. Similar Surfaces. 223. Def. Two surfaces are similar, say U and U\ when for a7ii/ point determined with regard to U^ and a??^ two radii OP, OQ, another point 0', and two radii 0'P\ O Q can be found for V\ such that lPOQ = lP'OQ'^ and OT': O'Q -.'. OP: OQ. 224. From the definition it follows that if OA, OB^ OC ho three arbitrary radii at right angles to one another in U, three radii 0'A\ 0'B\ O'C" can be found also at right angles satis- fying the above proportion, and If the direction cosines of radii OP and O'P' referred to these as axes, in U and U' respec- tively, be equal, OP : O'P' : : OA : O'A'. The surfaces will be similarly situated when the lines OA, OB, OC are parallel to O'A', O'B', O'C, and In this case may always be chosen so that O and 0' coincide, in which 154 SIMILAR CONICOIDS. case the surfaces are said to be similarly situated with respect to 0. 225. The analytical expression of this statement is that, \( f{x, 1/, z) =0 be the equation of any surface, that of any similar and similarly situated surface will be f[X{x-a),\{^j-/3),X{z-y)]=0, where OP=XO'P'. It is easily seen that the number of conditions, which the coefficients of the equations of two surfaces of the 7i^ degree ^ . . . (n + l)(n + 2)(n + 3) ^ . . . ^ . must satisfy, is -^ 5, m order that they may be similar and similarly situated. Also, that the terms of the highest degree in the two equa- tions must be the same, except for a constant factor. Thus, in the case of the liyperboloids, they are similar if they have similar conical asymptotes. It will be seen that, according to the definition, hypcrbololds of one and two sheets may be similar, as ax^ + Inf - cz' = 1 — ax^ — hf + cz' = 1 , for imaginary radii of one drawn In the same direction as real radii of the other will be in the same ratio. 226. Sections of the same conicoid hj imraJkl 2>Janes an shailar and simiJarhj situated conies. Sections of similar and siniilarli/ situated couicoids h>/ tin same ^;?a?ie are similar and similarhj situated conies. These propositions are easily proved by transforming the axes of coordinates, so that the plane of xy is parallel to tic cutting plane, wlien the projection of any section, fouud by making z constant, will be represented by an equation in ./• and y, for which the terms of the second degree will be the same. Hence, we can deduce that a plane section of a hypcrboloid is a hyperbola if the parallel plane through the centre intersects the conical asymptote in two of Its generating lines. SIMILAR CONICOIDS. 155 227. It is of great importance to observe tliat, when two conicoidd are similar and similarly situated, the condition, that the terms of the second degree are the same in each exc(])t for a constant factor, or, in geometrical language, that their real or imaginary asymptotes have their sheets parallel, may be stated as follows: "similar and similarly situated conicoids intersect the plane at infinity in the same real or imaginary conic." A particular case of this is that " all spheres pass through the same imaginary circle at infinity/' 228. To dttermine tlie nature of tlie section of a conicvid made hy any gmn plane. This may of course be done by the substitutions of Art. 11 7, but for surfaces of the second degree the plane sections -vv III be curves of the second degree, so that simpler methods may advantageously be employed. If It be required only to dis- cover the species of conic to which the section belongs, we may effect this immediately, taking any orthogonal projection of the curve of section, since an ellipse, parabola, or hyperbola, will be projected Into a curve of the same species, though in general of different eccentricity. The only exception Is when the plane of section is perpendicular to the plane of projection, but as no plane can be perpendicular to all the coordinate planes, there is at least one of the coordinate planes which may, in any proposed case, be taken as the plane of projection, and which will not be perpendicular to the plane of section. As an example of this method, we may take the section of the paraboloid Inf -\-cz^ = x made by the plane Ix + my + nz = 0. The equation of the projection of the curve of section on the plane of yz is I [hy- -'t cz'^)-'rwy -\- nz = i)^ which is always an ellipse, or always an hyperbola, according as h and c have like or unlike signs. If ? = 0, the exceptional case above mentioned arises, and taking the projection on zx we have the equation [li'^h + m^c] z^ = ni^x^ or the section is parabolic, unless 7t'i4 m^c = 0^ when it reduces to a straight line, the other straight line completing the curve of intersection being at an infinite distance. Ilcncc, for the 156 PLANE SECTIONS. paraboloids, all sections parallel to the axis of the principal sections are parabolas, and all other sections ellipses for the elliptic paraboloid, and hyperbolas for the hyperbolic paraboloid. If, however, a more exact determination is required, it will be convenient to deal with the problem in the manner wc propose. Plane Sections. 229 To find the locus of the centres of all sections of a central conicoid made by parallel ^??a/?e.9. Let ax^ + h}f-^cz^=\ be the equation of the surface, and Ix'-'r my + nz=p that of one of the parallel planes. Any straight line drawn in this plane through the centre of the section will be bisected at that point. Let (f, 17, ^) be the centre, r any radius of the section drawn in the direction (A,, /x, v), therefore the values of r are given by a[^ + \rY-^h{'n + lirY + c[^+vrY=^\, (1), and, since the values of r are equal and of opposite signs, a^\ + htjiM + c^v = 0, (2) also, since the direction of r lies in the plane, we have l\ + 7/i/i 4- ?IJ/ =r 0, which equations being true for an infinite number of values of X : /A : V, we have a^_hji_cX^ I ~ m ~ n ' ^'^^ therefore the equations of the locus of centres of sections made by planes whose direction-cosines arc /, ?», n arc ax _ by _ cz I VI n 2'50. The equation for determining r being [aX' -f V + c^l r" = I - "f- - bif - c^\ (4) shews that the parallel plane sections are similar, for, if (X', /x', v) be the direction of another radius r', i-'^ : r"^ is independent of ^), or constant for all the parallel sections, which are thcicforc simil.ir and similarly situated. PLANE SECTIONS. 157 2."51. To Jind the jwsitioii of tin cuttiiiij plave. ichn the curve of intersection hecomts a jwint-ellipse or line-hijperhoJa. Since eaoh member of (3) boeoiiics ,\, I = ,: ., . , and i^ + m'r] + ng=2h a h c equation (-1) becomes [oX^ -f hfju^ -f cv") r' = 1 — .^ ., ^, . - + ^ +- a c If CT be the value of p when the section becomes a point- ellipse or line-hyperbola, ?- = for any direction which makes a>^ + hu.^ -\- cv^ finite : therefore ct" = — f- -7- + - . a b c The point-ellipse is when the values of X, /a, v given by aX' + h^Jl,^ + cv'^ = 0, and IX + w/x + j^i/ = are Impossible, and the line-hyperbola when they are real. [imiX — IbfiY + obcv^'ST'^ = 0, hence, tlie section degenerates to a point-ellipse when ale is positive, or for an ellipsoid and hyperbolold of two sheets, and to a line-hyperbola when abc is negative, or for an hyper- bolold of one sheet. 232. To Jind the locus of the centres of all sections of an ' ^Uptic or hyperbolic paraboloid made by parallel iilanes. Let hf -\- cz^ = 'lx be the equation of a non-central conicoid, and let the e([uatIon of one of the parallel planes be lx-\^ my + nz =p^ then using the same notation as in the la.st article, we obtain the equation &(77 + /z;f + r(r+v;f = 2(^ + Xr), (l) and deduce for an infinite number of values of \ : yn : i/ br]/j,-tc^v- X = 0, [-2) and vifj. + iiv +l\ = 0; therefore ^^ = ^"^= -A (3) nt 11 I ' ^ ^ 158 TLVNE SECTIONS. thus, tlie locus of centres of sections made by pl.-xncs whos( direction-cosines are /, in, n is a straight line parallel to the axis of the paraboloid. 233. 7 find the position of the itlane for xchkh the section is a point-ellipse or line-hyperhola. Since /^ + 7«77 + n^=p, and the equation for determining r is = J + [f -^-^J f^J^^-"^' suppose. The sections are ellipses, when r cannot be infinite, or when h and c are of the same sign ; point-ellipses \vhen^? = ct. They are hyperbolas where b and c are of opposite signs, and the directions of the asymptotes are given by i/*"^ + cv"^ = 0, which shews that the asymptotes are parallel to the same two planes for all values of l, m, and n. 234. To find the magnitude and direction of the axes of any plane central section of a central conicoid, and the area ichen the section is elliptic. The equations which connect the direction of any radius of, the section by a plane, whose equation is Ix + viy + nz = 0, are, i as in Art. 229, {aX' + hfj:' + cr) r' = 1 = V + /*' + v"\ and lX + 7nfj,+ 7iv = 0, (l) .-. n'[{ar'-\)\'+{hr'-l)f^'} + [rr'-l){l\ + mfiy = 0, (2) is an equation which, for a given length r, gives generally two values of \ : /i ; but if the given length be that of either semi- PLANE SECTIONS. 159 axis, the two values of X : fi arc equal, the condition for which is Kar" -1) n' + {cr'-l) l'] {{hr''-l) r' + {cr'- l) m'] = {cr - 1)' Pm\ .: l'(br'-l){cr'-l)+...= 0, (3) or —r, — - +; 2 -,-+.,— = ; (4) this quadratic in ^-'^ gives the squares of the semi-axes of the section. If 2a, '2/3 be the axes of the section, and - + — = r [b -I- c) + in' (c + a)-^- »' (a + b). a p When the section is elliptic its area TT = 7ra;S: \lirbc + vi'ca + 7i"''oJ) ' Again, the coefficient of V in (2) is n^ {ar'' - 1) + Z'(or'- 1), ,.,,,,. ., 1 T m' icr' — \) iar'' — \) . which, by (3), is easily reduced to ., -, hence the equation (2) becomes and, if we write a and /3 for ?-, these equations, with (1), will determine completely the directions of the two axes. The equations (4; and (o) might have been formed by raaking r' a maximum or minimum, but we leave this to the student, the process adopted being more instructive. 235. To find the direction of the plane section ichose axes are of a (jiven mafjnitude. The equation giving the axes in terms of the direction of the plane section is _l'_ m' n' 160 PLANE SECTIONS. whence, If 7, 8 be the reciprocals of the squares of the given axes, fhc + in^ca + n^ab = 7S, r{b + c)-]-m'{c + a) + n'{a + h) = y+8, r + vi" + n' = 1 ; multiplying the second and third equations by —a, a', and add- ing, we obtain, for the determination of I, w, and ??, P (a -h){a-c) = {a- 7) (a - 8) ; similarly, m^ {b - c) [b — a) = {b - y)[b - 8), and n^ [c - a){c - I) = [c — 7) (c — 8) ; the second equation shews also that if a, &, c be in order of magnitude, b must be intermediate between 7 and S, Hence, for a circular section, 7 = ^ = 8 .•> m = 0, and ^ = = -^ - . a- a — c — c 236. To determine the nature of a central 2^1(t^^& section of a central conicoid. The nature of the section may be determined from the dis- cussion of the roots of the equation C3) obtained in 2\rt. 23-1, viz. {Vbc\m\aWab) r*-[r [b+c)+vi- (c+a)+m'(a+Z»)} ?''-f-l=0, (l) observing that the discriminant {/■•' {b-{-c) + 07i' {c + a)+ n' {a + b)Y - 4 {I'bc + m'ca + n'^ab) can be reduced to the form [r [b - c) + m' {c-a)- n' {a - b)Y + AlW {a -c){b-c). (2). (1) For a hyperbolic section, the values of r'^ obtained from (1) must be of opposite signs; therefore I'bc + tii^ca + n^ab is negative, in which case the values of r' must be real. (2) For an elliptic section, the values of ?•"'', which by (2) are real if «7, A, c be in order of magnitude, must be both ])ositive j .•. rbc -\- m'ca 4 n\d) is positive. (3) For a rectangular hyperbolic section, the values of r* must be equal and of contrary signs ; .-. I' {b + r) -F ni' (c + rt) + n' [a + b) = 0, m = 0, and -— r = ,-— = -— - , PLANE SECTIONS. Hi I since uo real or finite values of a, b, c will, at the same time, admit of l^bc -f in^ca + n^ab = as a consistent equation. (4) For a circular section, the two values of r^ are equal and of the same sign ; therefore the expression (2) is zero, hence a- b b — c a — c or / = 0, and , = = -, , a- b c — a c-b^ of which only one is possible. (5) When I'bc + vfca -{ n'ab = 0, one of the roots is infinite, and the section becomes two parallel generating lines, the distance between which is 2r, where r is given by [r (J + c) + m' [c-Ya)+ if (a + b)} r'=l. From this analysis it appears that all sections of ellipsoids are ellipses, but that for the hyperboloids wc may have ellipses, hyperbolas, or parallel straight lines. 237. We have in this discussion considered only central sections, but the nature of any section and the magnitude of the axes may be found at once by considering the similarity of parallel sections, the sections corresponding to parallel straight 'J ,2 lines being parabolic ; or, by writing 7 ' ., instead of ?•', the argument may be earned out in the same manner as above. 238. To find the angle between the real or imaginary asymp' totes of a plane section. From the equation (a V + b^i^ + cv") ?•-=!, when >• = CO , a^^ + bfi^ + cv'^ — 0, and l\ + miM -|- wj/ = 0, it w be the angle between the asymptotes of the section, supposed liyperbollc, it may be shewn by the method of Art. 25 that entr, - ^' (^ + ^) + ^^^' (g + ^) + n' [a + b) ■ 2 sj[-{rbc + in'ca^n\ib]] ' or it may be obtained directly from the quadratic In r' (Art. 231), aiuce tan"' 7 = ^^- , and therefore cot'-'a) = ^^' t ^2 • 2 a' ' -4a-/3- 102 PLANE SECTIONS. Tills gives the condition that coto) will be real, hifiuite, or impossible as rbc + m''ca-\- )i\ih is negative, zero, or positive, thus determining the condition that the section may be hyper- bolic, parabolic, or elliptic. 239. To find tlie area of any elliptic section of a central conicoid made hy a 2>lcine not 2^<^ssing through the centre. Let the equation of the plane be Ix + my + nz =p ; the area of a central cHlptlc section has been shewn to be ^{Fbc + m'ca + n'ab) ' and any radius vector of the section considered is given by 1)1 n + ^ + — a c hence, ct being the value of p when the section vanishes. r m n + -T- + - a b ' c ^ and, since Pbc + m^ca + n\ib is positive, ct is real only when abc j is positive, or for the ellipsoid and hyperboloid of two sheets. ! But if we take ct for the value of p when the section of the hyperboloid of two sheets, which is conjugate to that of one sheet, vanishes, since in this case «, Z*, c have their signs ij changed 72 2 2 .^ r m n CT = . abc Hence, if A be the area of the parallel central section of the ellipsoid and hyperboloid of one sheet, and of the hyper- boloid conjugate to the hyperboloid of two sheets; and A' be the area of the section by the given plane, since they are In tli' duplicate ratio of homologous lines, for the ellipsoid A' = A H - ^ j , for the hyperboloid of two sheets A' = A{~2—\\ ^ £ ■st' for the hyj)erboloid of one sheet A' = A(i-\-£, PLANE SECTIONS. 103 If wc take two conjugate hyperbololtls ax* -\- hi/^ + cz' = ± I , and the asymptotic cone to both, ax^ 4 hi/'* + cz* = 0, the area of the section of the latter may be found, from those of the former, by making a, />, c infinitely large, preserving their ratios. Hence if A^, A., J A^ be the areas of the sections of the three surfaces made by any plane cutting them all in ellipses, and A that of the parallel central section of the hyperboloid of one sheet, we shall have ■whence A^ + A^=2A^, or the section of the cone is an arithmetic mean between the sections of the two hyperboloids. Also, if V be the volume of the cone cut off by a plane touching the hyperboloid of two sheets, we shall have 72 2 2 ,, . TT , „ r m n iSow A = .. „, , — 2 ; — tTn j ^^^ ^' = - + -r-^ ~i '^[loc + ra ca + 71 ao) a b c which is constant for all positions of the cutting plane. 240. To find the magnitude and direction of the axes of any plane section of an elliptic or hyperbolic paraboloid. The equation of the paraboloid being hf + cz* = 2x^ and that of the cutting plane Ix + my + nz = pj the equations connecting any central radius of the section with its direction (A, /*, v) is 2 (V + cv') >•* =- {p-zT) = M suppose (Art. 233) and IX + m^i + vn = ; .-. r (V + cv'O r^ = M[{m,i. + nvY + I' {f^' + v'j], (1) 164 PLANE SECTIONS. and if r be the length of either of the semi-axes, this equation will give equal values of yu, : v, {V [hr' - M) - m'M] [F [cr'' - M) - n'M] = M''m'n\ or r {br' - M) {C7-' - M) - 3fm' {cr' -M)- Mri' {br' -M)= 0, or rbc7-' - {r (& + c) + m'c + ii'b] Mr' + M' = 0. (2) This equation gives the magnitude of the axes 2a, 2/3, and the area of the section when elliptic = 7ra/3= , ,,, - = >^ ,,, X [j^ — '^j- The coefficient of /*'"' in equation (1), is and the equation becomes br' - if , , „ , cr' - i/ , , ^ . ^^;,-^ .nV - 2rnn^v + ^^^^^ m v = ; /. {br^ - M) ^^ = [cr^ -M)1=-m\, by (2), which, writing a, yS for r, completely determine the directlona of the corresponding axes of the section. 241. To determine the nature of any plane section of a paraboloid. Take the equation Vbcr' - [r (& + c) + m'c + n'b] Mr' -|- J/* = 0, and observe that the discriminant [l" [b + c] + m\ + in^ - irbc {r + m' + n'\ (1) is reducible to [r (b-c)- m'c + ii'by + ^ni'n'bc. (2) The two forms (1) and (2) of the discriminant shew that it is positive whether be be positive or negative, so that the values of r'^ : M are always real. (1) For an elliptic section, be is positive and M positive, therefore p — vr has the same sign as I. CYCLIC SECTIONS. 1G5 (2) For a hyperbolic section, he is negative, since the values of r* must be of opposite signs. (3) For a rectangular hyperbola, he is negative, and r [b 4- c) + m^c + n^b = 0. (4) For a circular section, be Is positive, and by (2) m = and r = ; = - ^ c- c ^ , ^' "'' 1 or n = and - = ^ = 7 , c h-c 6 ' only one of which gives a real position. (5) The condition ? = 0, which makes one value of r'^ infinite and the other finite, corresponds to the case of a parabolic section, since in this case 17 and f in (3) Art. 232 are Infinite, and therefore the centre of the section is at an infinite distance. Cyclic Sections. 242. Altliough we have already determined the positions of the planes whose intersections with conlcolds are circular, by treating such sections as particular cases of ellipses, it will be instructive to consider them from another point of view, since they have an interest peculiar to themselves in the solution of many problems both pure and physical. 243. To find the cyclic sections of a conicoid central or non- eentral. Let the equation of the conicoid be ax' + by' + cz' + 2dx + e = 0, this may be written in the form b [x^ + y' + 2-) + {a-b) x' - {b-c)z' + 2dx^ e= 0, hence, if the conicoid be cut by a plane whose equation is \/(a — h)x± sj{b — c)z =p \J[a — c), the coordinates of the points of the intersection satisfy the equation 6(a;' + y+2*) + 2? \/(«-c) W{a-b)x^■ sf[b-c)z]-{-2dx^-c = 0, which is the equation of a sphere. 166 CYCLIC SECTIONS. Tlicsc plane sections will be real, if a, J, c are in order of magnitude when all are positive, if a > & when c is negative, and if c> /», without regard to sign, when h and c are both negative. Also if a = 0, the sections are real when h and c are both positive, and oh. Hence, cyclic sections of central surfaces are parallel to the mean axis in the ellipsoid, to the greater transverse axis in the hyperboloid of one sheet, to the greater conjugate axis in the hyperboloid of two sheets. If a be the inclination to the principal section (a, J), cosa _ sina _ 1 \/{a — h) ±\I[b — c) ±V'(«-c)* Cyclic sections of the elliptic paraboloid are parallel to the tangent at the vertex of the principal section of greatest latus rectum ; and for these sections putting a = 0, cosa _ sina _ 1 ^Jb ± '^[c — h) ± V(c) * It is obvious that these are the only cyclic sections, since a plane not pai'allel to one of the axes, as that of y, being of the form Ix + my + nz =p^ could not reduce the expresssion [a- h) x^ - {h —c) z^ to a linear form, so as to ensure that the points of intersection with the conicoid should lie on a sphere. 244. Generation of a conicoid hy tlie motion of a variahh circle. From the last article it appears that the central conicoids and the elliptic paraboloid can be generated by the motion of a circle, the plane of which is parallel to cither of two fixed planes, and the diameter of which changes so that it is always a chord of the section which is perpendicular to the line of intersection of the two fixed planes. The centre of the circle of course describes in each case the diameter conjugate to the chords which it bisects. 245. Def. The point-circles in which the variable circle ter- minates are called wnhilics^ those are real only for the ellipsoid, the hyperboloid of two sheets and the elliptic paraboloid. CYCLIC SECTIONS. KIT I For the conlcoid ax' -{■ hi/'' -i cz'^ = 1 , the four unibllics arc a> h> c. For the ellijitlc paraboloid, %* + cr = 2.r, oh the two cz I umbihcs at a finite distance are civcu by -77 ,n = ±-77;^) *= -^ \/{c-h) \\h) 2x = y , and y = 0. 246. Ani/ two cyclic sections of opposite systems lie on one $2)1 ere. The equations of the planes of two cyclic sections of opposite systems are y{a-b)x- >^{h-c)z-k} y{a-h) X + ^/{b - c) z - Jc] = ; or, (a-h) a'- [b-c] z'- {h+ k') ^/{a-h) x- [h - h') ^J{h-c)z-f hlc = 0. Hence they intersect the surface in a sphere whose equation is j b {x'+ y'+ z') - 1 + (/.-+/.•') ^f{a -h)x^ {k- k') ^/{b -c)z- W = 0. 247. It Is an Instructive problem to deduce the positions of the cyclic sections directly from the equation (4) obtained I in Art. 230. This equation may be written aX' + hfjb' + cv' = (\- + fjb- + v") p, and since, for a cyclic section, the values of r, and therefore of p, are equal for all values of X, /i, v consistent with the equa- tion l\ + )nfx -f iii^ = 0, it follows that : ip-a) {miM + nvY + T [{p - h) p,' + {p - c) v'} = : is true for an infinite number of values of /i : v, the coefficients , of /i", /xi/, and v' are therefore each zero ; .'. [p — a) vin =■ 0. If p = a, cither ^ = 0, or /3 = Z» = c, in which latter case the surface is spherical, and the equation Is satisfied for any values of /, VI, ?<, i.e. for any direction of the plane. Also if m = 0, the coefficient of p.' = p- b = 0, and similarly, for ;( — 0,p = r. 1G8 . CYCLIC SECTIONS. Hence, If the surfiicc be not splieiical, we must have /, w, or n = 0. Suppose m = 0, then w„ = h^ and the coefficient of v" = {p- a) n' + (/3 - c) P = ; JL - 'L - JL ' ' b — a c — b c- a^ which give real values for I and n only under the same circum- stances as are already stated in Art. 243. The corresponding process for non-central surfaces can he followed out by the student. 248. Geometrical investigation of the direction of a cyclic section of an ellipsoid. In the ellipsoid let 0.4, OB^ 00 ha the semi-axes in order of magnitude, and if possible let a central circular section not pass through B^ but cut AB and BG in P, Q respectively, 0P= OQ, being the centre of the section; but OF is inter- mediate between OA and OB in magnitude, and OQ between OB and 06', which is absurd; hence, the central circular section must contain the mean axis. The inclination of the plane to GAB is the same as the angle BOA, OR being that radius vector of the section AG which = OB. It is easy to give a similar proof for the hyperboloids. 249. Thus a method of obtaining the direction of the circular sections of an ellipsoid is to find the inclination to 0^1 of a radius vector of the ellipse (a, c), whose length is i, the mean semi-axis of the ellipsoid ; if a be this inclination, ¥ cos" a b" sin" a -- ., 1 — = 1 = co3"a -f sura: a c PROBLEM!^. 1 ()9 XII. (1) All si)hcres wliich intersect a given conicoid in plane sections and pass through a fixed point on it pass through one of two fixed circles. (2) Find the equation of an ellipsoid referred to the jjlanes of its central circular sections and a central plane at right angles to them. When these are rectangular axes, prove that the squares of the axes are in harmonical progression, and that the equation takes the form (t i z? I y" ^ (x -z? -^ v' ^ 2 c* «* (3) Prove that tinough any point on an ellipsoid two planes of circular section can be drawn ; but that when the circles are equal, the points must lie on one of the principal planes passing through the mean axis. (4) If two circular sections of different systems be such that the sphere on which both lie is of constant radius mh, the locus of the centre of the sphere is the hyperbola — r; - n «= 1 - '«', V = 0; a, h, c, being in descending order of magnitude. (5) The sphere (x* 4 »/' 4 s' + a* - t' - c') := 2x meets X* v' 2' the ellipsoid - f -,- + -, = 1 only at umbilics. a' b^ r •' ((3) The locus of centres of all plane sections of a given conicoid drawn through a given point is a similar and similarly situated conicoid, of which the given point and the centre of the given surface are extremities of a diameter. (7) In a paraboloid of revolution, the eccentricity of any section is the cosine of the inclination of the plane to the axis of the surface, and the foci of the section are the points of contact with spheres inscribed in the surface. (8) A sphere is described, having for a great circle a plane section of a given conicoid; prove that the plane of the circle in which it again meets the conicoid intersects the plane of the former circle in a straight line which lies in one of two fixed planes. (9) A plane drawn through the origin perpendicular to any generating line of the cone x' (a* - rf') + t/' (i' - d") f s' (c' - d*) = 0, will intersect the ellipsoid — + — + -^ = 1 in a section of constant area, a' 6' c' x' y* - =' (10) In the hyperboloid -; + — ~ = 1 (a > c), the spheres, of which parallel circular sections arc great circles, will have a common radical plane. 170 PROr.LKMS. (11) On a central circular" section of the ellipsoid ax- + hif i c:» = 1 a right circular cylinder is constructed, shew that if h be an arithmetic mean between a and c, the cylinder will again intersect the ellipsoid in an ellipse, the plane of which will be given by ("ia - c) a; ± (3c - a) z = 0, and that the area of the ellipse will be -^- {2 (a* + c*) - Si-jl (12) Prove that the difference of the squares of the axes of a central section is proportional to the product of the sines of the angles which it makes with the planes of circular section. (13) Shew that if elliptic paraboloids have one of their cyclic sections coincident with a central cyclic section of az' ^by"^ + cz* = 1, a, b, c being in order of magnitude, the locus of their vertices will have the equations 2x z 11,^ — , and y = 0. V[(6 - a){c - b)} b - a b {e - a) z Also, that the equation of the plane of the other cyclic section common to the conicoid and one of the paraboloids will be ± - /( — ]z + a; + — = 0, where I is the latus rectum of any section parallel to the plane of xy. (14) If sections of an ellipsoid — 4- -^ + -^ = 1 be made by planes passing through the centre, and through another given point (x'y'z'), the sections of greatest and least area will be at right angles to each other, and ... , vabc Trabc , . , . o ■, the areas will be — ^ , - , r,, r^ being the semi-axes of the section made by the plane — 4 "-^ + — = 0. Shew that the product of the areas ^ a b c ^ will be constant if the point lie on the curve of intersection of the ellipsoid and a concentric sphere. (15) The locus of the axes of sections of the surface nz'^ + ly^ -r c:' = 1, ... . , ,. a; « z . , •which contain the line - = - = - , is the cone I in n [b - c) yz {mz - ny) + {c - a) zx {7ix - Iz) + (a - i) xy {ly - i»x) = 0. (16) Shew that the foci of all parabolic sections of the surface V* 2* — + r = ^> lie on the surface (^-^?)(^^^?g*^) (17) Prove that the foci of all the centric sections of the conicoid ax' 4 i^* + cz* = 1, lie on the surface (x' 4 I/' i 2'J (1 - fli* - h^ - cs») {a (c - by 7/z' ib{a- cf zV 4 c (6 - ayxY) = (ax' 4 by' i cz') {{c - by y'=' 4 (a - c)' s'x' 4 (J - fl)' xyj. CHAPTER XIII. TA^'GENTS. CONICAL AND CTLINDRICAL ENVELOPES. NORMALS. CONJUGATE DIAMETERS. 250. On many accounts it is desirable that the student should be early acquainted with the chief properties connected with tangent lines and tangent planes to conicoids, before he is led to consider more general surfaces. We shall therefore give in this chapter some of the principal propositions relating to tangency in the case of the conicoids as represented by their equations in the simplest form. We shall also explain the properties of conjugate diameters and diametral planes. Tangent Lines and Planes. 251. To find the condition that a straight line shall touch a given conicoid, at a given point. Let the equation of the conicoid be ax' -\- Inf -\-cz' = 1, the equations of a straight line drawn in a direction (X,, yu,, v) through the given point P, (,/,,'/, h)^ are A, /i V ' ^ ' the values of r at the points 1\ Q^ where it meets the conicoid, arc given by the equation « (/+ >^'-)' + ^ (,'7 + H'^Y + c (A + vrY = 1 ; or, since of + Inf + c//' = 1, 2r {afK. + hg^l -\- chv) + r' («V + hfji' + cv') = 0. If the direction be such that Q coincides with /*, the straight line will become a tangent, and in this case the two values of r will be zero ; therefore af\ + hgfM + chv = 0, (2) is the condition of contact at the point (/, ,7, h). 172 TANGENT LINKS AND PLANES. 252. To find the equation of a tangent plane at a given point of a conicoid. The locus of all tlic tangent lines which can be drawn through the point (/, //, /i), is found by eliminating X, /a, v between the equations (1) and (2) of the last article, giving ^/(•^ -/) + Ay Cy - ^) + c^^ i^ - ^ = 0, or ofx -f hgy + chz = 1. (3) The locus Is therefore a plane, and this plane is called the tangent plane to the surface. If^ be the perpendicular from the centre upon the tangent plane 1 = aY 4 hW + cVi' (Art. 70). Cor. 1. The generating lines of a hyperboloid of one sheet through the point (/, ^, h) being two of the tangent lines, the tangent plane contains these lines, which together form what we have called the line-hyperbola In Art. 231. Cor. 2. Since any generating line Is intersected at every point by some line of the opposite system, no two of which lie in the same plane, it follows that the tangent plane to the hyperboloid at any point in a generating line changes its position as the point moves along the line. 253. To find the equation of a tangent plane to a conicoid drawn in a given direction. Let (Z, 7n, n) be the given direction of the normal to the tangent plane, so that Its equation Is Ix + my + nz =p ; com- paring with the equation afx + hgy 4- cliz = 1 , 7 in n Vf^Vg^dr^'' and, since af^ •}- hg'' + cli' = 1, , /^ m' n' ^ a b c ^ and the equations of the two tangent planes In the given direction are determined. TANGENT LINEH AND PLANES, 17o 254. The equation of a tangent plane to the cone ax* + hi/' + cz^ = 0, is a\x + hjMy + cvz = 0, it' the line of contact be in direction (\, /i, v) ] and lx+nii/ + nz = if the tangent plane be in direction (/, ?/«, ??), subject to the condition r 711' n' - + ^ + - = 0. a c 255. To find the equations of an asymptote to a central conicoid. Let the equation of the conicoid be ax' + hi/ + cz^ =\^ and let (^, 77, t) ^^ f^'Wy point in the asymptote whose equations are ^ — = ^ = - — - = r. then the two values of r are infinite in the equation « (I + '^rY -}- Z* (7; + ixrY + c (^+ vr)' -1 = 0; .*. rt\' + Z'/A* + cv'' = 0, and a^X + Ji7//. + c^t/ = 0, and, if a^' + 5?;'' + c^""' — 1 be not finite, the straight line lies entirely in the conicoid. Hence, every straight line drawn in a tangent plane to the cone aa^ + hif + cz^ = 0, parallel to the line of contact, is an asymptote, including the generating lines in which it may intersect the conicoid. 2o0. To find the nature of the intersection of a central coni- coid with the tangent plane at a given i^oint. Let the equation of the conicoid be ax^ -\-h7f -\-cz' =^^ that of the tangent plane at (/, ^, h) is afx + hgy + chz = 1, we have also af'' + h(f + cK' = \. At the points of intersection {^r+hg\ax^ + hf)-[afx^-lgyy = {\-ch^){\-cz'^)-[\-chzy; .'. ah {gx -fyY + c (s - hy = 0. For the ellipsoid and hyperboloid of two sheets the only solution is - = - = ^ = I, since ah^ and c are of the same sign ; for the hyperboloid of one sheet the section is two lines, since ab^ and c are of contrary signs. 174- TANGENT LINES AND PLANE!:^. 257. To find the magnitude and direction of the axes of the section of a central conicoid made hy a given i^lane through the centre. Let the equations of the conicoid and plane be ax"^ + by" + cz'^ = 1 , and Ix + my + nz = 0. The equation of a sphere of radius r is x^ -\- y"^ + z^ = r"^ \ there- fore the cone [ar' - 1) x^ + [h^ -1)3/'+ {cr' - 1) 2' = is the locus of all diameters of the conicoid which are of equal length 2r; the cone, therefore, intersects the given plane in two lines which are the direction of equal diameters of the central section, and if r be chosen so that these directions coincide, the given plane will be a tangent plane to the cone, and the line of contact will be an axis of the section ; therefore, r m^ n^ by Art. 254, — 5 — - + , .^ , + — i — : = 0, which is the quad- •' ' ar — 1 or - 1 cr^ - 1 ' ^ ratic giving the lengths of the semi-axes. x\nd, by the same article, if (\, /z, v) be the direction of the axis 2;-, \ {ar' -\) ^^l [br^ - 1) _ v jcr' - 1) I m 11 258. To find the locus of tJie points of contact of all tangent planes which pass through a given point external to a given conicoid. Let (/, g^ h) be the given point, ax' -i by- + cz^ =1, the equation of the conicoid. The equation of a tangent plane at any point (^, 7;, ^) on the conicoid is a^x + brjy + c^^ = 1 , and if it pass through the given point af^ + bgrj -+ ch^= L The tangent planes at every point of the conicoid whose coordinates satisfy this equation pass through the given point, the locus required is therefore the section of the conicoid by the plane whose equation is afx + bgy + ch,z = 1. 259. Def. The plane containing the points of contact of all tangents from any point to a conicoid is the Polar Plane CONICAL ENVELOPES. 175 of the point, and the point is the Pole of the plane, with respect to the conicoiiL Tliis will be a definition whether the point be external or internal, if we consider that imaginary tangent planes have a real plane containing the imaginary curve of contact. Another definition will be given which does not involve the consideration of tangency. One of the most important propositions connecting the pole and polar plane is the following. j 260. If U he the polar 2ilcine of any jwint P with resjiect to f a conicoid, the polar plane of any po/«^ Q in the 2)Iane U u-ill pass through P. For, if ax'^ + hy"^ 4- cz" = I be the equation of the conicoid, and I (/,,[X7j + cvz - \)\ Normals. 26o. To find the equations of the normal to a conicoid at any point. Def. a normal at a point is the straight line drawn per- pendicular to the tangent plane at that point. i ^^ [fi Oi ^0 ^^ ^^^^ point, the equation of the tangent plane is j ^fa + ^jy + chz = 1 ; therefore the equations of the normal will bo af hg ch - P' if r be the distance between (.c, g, z) and (/, //, 7^), and p be i the perpendicular on the tangent plane from the centre. 2GG. To shew that six normals can he draion from a given I point to a central conicoid. Let the equation of the conicoid be ax'' + hg^ + cz^ = 1. A A 178 NORMALS. The equations of a normal at a point [jc, ?/, z) are ax bij cz ^^ ^ ' if this pass tlirougli a given point (/, q, Ji) f=x{pa + l), = y{ph+\\ ]i = z{pc+\); af~ h(f cli' _ •'• {pa + iy "^ pTTf ^ {pc + lY~ ' which gives generally six values of p determining the feet of six normals from the given point. I 267. To shew that the locus of a pointy from which three normals can he drawn to a central conicoid^ which have their ; feet in a given plane section of the conicoid, is a straight line^ ' and to find the condition to which the given plane section must he suhject.^ ■3? y"^ z' ' Let the equation of the couicoid be — 4- y^ + -; = Ij and that | of the given intersecting plane l-^m^+n- =1, (1) a h c ^ and let (^, ?;, ^) be a point from which if six normals be drawn the feet of three of them will lie on the given plane section, the i other three must then lie on some other plane section given by Z"-+m'f +n'-=(7. (2) a b c Hence the six feet lie on the surface Z - + w -f + w - - 1 U' - -f m 'j 4 w' - - fZ \ a c J \ a c J r^ ?/ ■^'^ a be And it is easily shewn by Art. 266 (1) that the six feet also lie on each of the three surfaces U= [U' - c') yz - Vriz + c'^^ = 0, F = (c« - a") zx - c'^x + a'^z = 0, W= [a' - h') xy - d'^y + h'rjx = 0, * Qxmrterly Journal, Tol. VIII,, p. 69. NORMALS. 1 70 and therefore on the surface \U+fiV+vW=0. (4) Now wc can make (3) and (4) Identical by writing for the equation ("2) 7- +-^+ — +1 =0, la mo nc and equating the remainder of the coefficients, so that .^. -,^_^_^-^ he' .(c'-..) = (^,^) 1 (5) ^("■'-^•) = (^7) 1 ah' vV^V-t^c^^ = [l - 1) 1 a' Xc^-va-'l = (».-!) 1 (6) ^. ^3 Vc ; are the direction-cosines of three straight lines at right angles to each other, and we know therefore (Art. 143) that they are equivalent to the systems ax; + ax; + ax; = by; + %,/ + by; = cz; + cz; -f cz; = 1 , 2/1^, + 1/-/-2 + y-A = ^.«^. + ^2^2 + ^3^3 = ^ J, + ^2^. + ^sVs = 0. (3) x^ ?/'■' 2^ Hence, in the ellipsoid -^ + 'rr, + -, = l, we shall have ' ' cr b c x;+x;+x;=a% y;+y;+y;=i^, z; + z;+z;==c% n) or the sum of the squares of the projections of three conjugate diameters on one of the axes is equal to the square of that axis. If (?, 7n, n) be the direction of any line, by (3) and (4) [ix^ + my^ + nz;f + (^ + my^ + nz;j' + [ix^ -I- my^ + nz;; = Pa + m''b^ + n-c"^ =])\ r; - [Ix^ + my^ + nzj +. . .= a' + F + c' -;/ ; but [Ix^ 4 '"?/, + nzy and >•,'■' — (?jj, + my^ + w^;,)' are respectively squares of the projections of r, upon a line and a plane whose directions ai-e given by (/, ?», «), hence it follows that The sum of the squairs of the projections of three conjuyatc diameters on any line or any plane is constant. 272. To find the relations which exist between the lengths of a system of conjugate diameters of a central conicoid and the angles betioeen them. CONJUGATE DIAMETERS. 183 Let the equation of the surface referred to Us principal axes be ax' + bi/'' + CZ' = lj and, referred to a system of conjugate diameters inclined at angles a, /3, 7, let it be a'x' -\- h'lf -\- cz^ = 1. The invariants derived from li [x^ + y'^ + z^) — ax'^ - hy' — cz\ and the transformed expression, see Art- 153, give the equations 111111 ,,, - + 7^ + - = - -f- 7 + - , (1) a b c a b c sin'a sin'^/S sin'7 -111 ,. be ca ab be ca ab - 1 — cos'a — cos"/3 — cos^7 + 2 cosa cos/3 C0S7 1 .„. and Tjr-, = -.- . (3) abc abc When the surface is an ellipsoid, all these lengtlrs are real, and we sec from (1) that the sum of the squares of three con- jugate radii is constant ; from (2) that the sum of the squares of the faces of a parallelepiped having three conjugate radii as conterminous edges is constant ; and from (3) that the volume of such a parallelepiped is constant. In the hyperboloid of one sheet, since ahc is negative, and 1 — cos'''a — cos'''/3 — cos^7 + 2 cosa cos/3 C0S7 is always positive, a'h'c must be negative, but «', Z>', and c cannot all be negative, hence one and only one is negative ; that is, in a hyperboloid of one sheet, two of a system of conjugate diameters meet the surface in real points, and the third does not. In the hyperboloid of two sheets, ahc^ and tiierefore a'b'c is pojitive, hence two, or none, of the three «', ?>', c are negative. If none be negative, writing -r, , — j^ , — 5 for «, 5, c respec- tively, we must have both a' — b'' - c^ and V'c^ - a^ {J/ + c') positive, which are easily shewn to be inconsistent. Hence two must be negative ; or, in the hyperboloid of two sheets, one and only one of a system of conjugate diameters meets the surface in real points. 273. The relations (I) and (3) may be obtained geometri- cally. We will give the proof in the case of the ellipsoid, and leave the other two cases as exercises for the student. Let Ox^ Oy^ Oz be the directions of the axes of the surface, 184 CONJUGATE DIAMETERS. Ox\ Of/\ Oz those of a system of conjugate diameters; yl, B^ C, A\ B\ C\ the extremities of the serai-diameters along those axes; a, J, c, a', h\ c their lengths. Also, let the sections AB^ A'B' intersect in ^,, let OB,^ be the semi-diamctcr con- jugate to OA^ in the section A'B\ and let CB,^ meet AB in B^ ; Then, the plane ^'i5' being conjugate to 06", 0^1,, 0^^, 00' will be a system of conjugate radii, or 0.1, will be con- jugate to the plane C'B.^ and since OA^ lies in a principal plane, C'OB.^ will be perpendicular to that plane, and will therefore contain OC; and OC, OB^ being the principal semi-axes of the section J5,0', 0^„ Oi?„ 00 will be a system of conjugate diameters, and 0^1,, OB^ will be conjugate in the section AB. Hence we have the equations and from these we obtain the relation (^^ + l''Jrc"' = d' + h'' + 6\ Also, since the parallelogram of which 0^1', OB' are con- terminous sides, is equal to that of which OA^^ OB^ are con- terminous sides, and similarly for the section B^C\ AB^ w^ have vol (a, ?/, c) = vol (n-,, h,^, c) = vol (r^„ Z*,, c) = vol («, J, c) ; denoting any parallelepiped by three conterminous edges. This is equivalent to relation (3) of the last article. CONJUGATE DIAMETERS. i. ^. •^ ' 185 V^ - :'/ ■<■ I / 274. To find the diametral playie bisecting a given system of. ^^ parallel chords^ in the case of non-central conicoids. ^V 'f. /'. Let the equation of the surface be "^ -i — = 2a;, aod fetj • (\, /i, v) be the direction of the chords, the equation of the,/-', diametral plane will be shewing that all the diametral planes arc parallel to the common axis of the principal parabolic sections ; a fact which might have been anticipated from the consideration that these surfaces have their centre on that axis at an Infinite distance. "We cannot in these surfaces, as In the central conlcoid, have a system of three conjugate planes at a finite distance, but we can find an Infinite number, such that, for two of them, each bisects the chords parallel to the other and to a third plane. By taking the origin where the intersection of these two meets the paraboloid, and referring to these three planes, the equation of the surface will assume the form b c Let the equations of the two diametral planes be »«,«/ + nj3 = l, (1) and let the direction of the third plane be [l^^m^n^. The direc- tion-cosines of the chords bisected by (1) are in the ratios I : hn^ : c»,, and If these be parallel to (2) and the third plane, we shall have hn^m,^ + cn^n^ = 0, 7„ + Im^m^ + cn^n^ = 0. Similarly, in order that (2) may be conjugate to the inter- section of the other two, we shall have bm^m^ + cn^n^ = 0, l^ + hm^m^ + cn^n^ = 0. One of these Is coincident with one of the former, and, there being thus only three relations necessary between the four quantities determining the planes, an infinite number of such systems can be determined. B li 'J 186 POLAR PLANE. Polar Plane. 275. We shall conclude this chapter by proving a theorem which gives rise to the definition of the polar plane of a point, alluded to in Art. 259. Def. The 2)olar flane of any fixed point, with respect to a given conicoid, is a plane, which, with the conicoid, divides harmonically all straight lines passing through the fixed point. 276. Through a fixed point a straight line is draicn meetinq a central conicoid^ and on this line a point is taken, such that its distance from the fixed point is a harmonic mean between the segments of the line made hy the conicoid ; to find the locus of the point. Let the equation of the conicoid be ax^ + 1)^^ + cz^ = 1, and let the equations of the straight line through the fixed point x-f ^y—l ^z_-h ^ ^^^ I m 11 The values of r at the points of intersection are given by the equation a (/+ IrY + b{g + mrf + c{h + nrf = 1 . 1 1 _ 2 {qfl + bgm + cJut) '*. '''2 «/ ' + ^f + <^^^' - 1 * If now r be taken for the distance from (/, g, h) of the point, whose locus is required, - = — h — ; therefore af + bg" -f cU' - 1 f {afl + bgm + chn) r = 0, and, since Ir = x - /, etc., the equation of the locus will be ofx + hgg -\- chz = 1. 277. The corresponding locus for an arithmetic mean is a conicoid similar to the given one, of which a diameter is the line joining the given point and the centre. For a geometric mean, the locus is a similar conicoid, which meets the given conicoid in the polar plane of the given point. PROBLEMS. 187 XIII. (1) Find the equation of the tangent plane upon the principle that no other plane can pass between it and the surface in the neighbourhood of the point through which it is drawn. (2) Prove that the three surfaces X* i/ _2z x^ !/' _ 2s x^ y' _ 2= will have a common tangent plane, if w. 2 I (3) Tangent planes are drawn to an ellipsoid, and are such that their intersections with the plane zx are parallel to the line ex V(6' - c') + az V(a* - i') = 0, shew that the points of contact all lie on a circular section. (4) The locus of the centres of sections of ax"- f bij'^ + c:' = 1 by planes which touch ax^ + fiy- + 7s' = I, is (5) Prove that the tangent planes of the cone a' {b' - c^) b' ^ir - c') e (a- + b^) X* y* s' cut the surface -; + — — ^ = 1 in rectangular hyperbolas. a b c (6) Find the locus of the feet of the perpendiculars let fall from a point (a, (i, 7) on the tangent planes to the cone ax^ + bi/ + «'* = 0; and proTe that if the locus be a plane curve it will be a circle, and tliat, if o > a and a 1' cz' e negative, the point must lie on one of the lines y = 0, , = " , . b - a c - b (7) The tangent planes to an ellipsoid at points lying on a plane section \mI1 intersect any fixed plane in straight lines which touch a conic section. (8) Two similar and similarly situated ellipsoids have tl.cir axes in the ratio of 1 : «, ;» > 1 ; from any point on the exterior as vertex a cone is drawn enveloping the interior, shew that the plane of the curve of intersec- tion with the exterior ellipsoid touches another similar ellipsoid whose axes are to those of the interior as n' - 2 : >i. (9) If the area of the central curve in which a cylinder touches an ellipsoid be equal to that of the section containing the greatest and least 'xes, ppove that tl;e axis of the cylinder will lie on one of two planes. lH^ PROBLEMS. (10) If an ellipsoid be placed on a horizontal plane with an axis 2c ver- tical, shew that the altitude of a star which will cast on the plane a circular shadow is tan"' — , , where d is the distance of the foci of the horiaontal a principal section. (11) The normal at any point P of an ellipsoid meets the principal planes in G^, G<, G^; prove that PG^. PQ^.PG^ varies as the cube of the area of the central section made by a plane conjugate to the diameter through P. (12) If r be measured inwards along the normals to an ellipsoid so that pr = »»' a constant, ^; being the perpendicular from the centre on the tangent plane, prove that the locus of the point thus obtained will be aV i'v' cV 4- -1 -If 1 (a* - m*)' (6' - vi'f (c- - nff What does the locus become when m is equal to one of the principal ase» of the ellipsoid ? x^ y* z' (13) The normals to the ellipsoid — ^ + r; + —^ = 1 at points on the planes — +J+ -=+lall intersect the straight line a b c ° ax (6« - c") = hy (c' - a«) = cz (a» - 6'). (14) The enveloping cones which have as vertices two points on the same diameter of a conicoid intersect in two parallel planes between Avhose distances from the centre that of the tangent plane at the end of the diameter is a mean proportional. (15) If (a?!, j/p Si), (xo, j/j, Cj), (Xj, 7/3, 23) be the extremities of three x^ V- 2* conjugate diameters of the ellipsoid -^ + r; + 1 = 1> the equation of the plane passing through these points will be ^, (xi + X. + 0:3) + ^^ (y, 4^2 + 5/3) + ^ (si + z, + Z3) = 1. If one of the ends [x^, y,, Zi) be fixed, shew that the perpendicular from the centre on this plane will describe the cone a»x' + b'^y^ + cV = 3 [xxi + yy^ + zr J». (IC) The locus of the centre of gravity of the triangle formed by joining . .r* V* s" 1 the extremities of three conjugate diameters is - + vs + -5 = - » anJ t'le locus a b c 3 of the intersection of tangent planes drawn through their extremities is a* i' c' PROBLEMS. 189 (17) Prove that the sum of the products of the perpendiculars from the two extremities of each of the three conjugate diameters of a conicoid upon any tangent plane is equal to twice tfhe square of the perpendicular from the centre. (18) Prove that the sum of the squares of the distances of any point of a given sphere from the six ends of any three conjugate diameters of a given concentric ellipsoid is invariable. (19) If three straight lines be conjugate diameters, and the planes per- pendicular to them conjugate planes, find their direction-cosines. (20) The locus of points, from which rectilinear asymptotes can be drawn to the conicoid ax* + hy^ + cs* = 1, at right angles to each other, is the cone a' (b + c) x' + 1/ (c f a) t/* + c" (a + b) z* = 0. (21) The locus of the intersection of two tangent planes to the cone x' V* s' — + -v + — = 0, which are at right angles, is the cone a c o D (6 ^^ c) x^ i {c i a) 2/* -f (« + h) z- = 0. (22) If two planes be drawn at right angles to each other touching the central conicoid ax' + hy' + cz* = 1 , and having their line of intersection in a given direction (/, m, n), shew that the locus of their line of intersection will be the right circular cylinder t ,1 ' ^, m^ ^ 71^ n" ¥ I- r + Mi" X* + t/* + z* = {Ix + mi/ + iizf + + — V — + . (23) If the non- central conicoid — 4 - = 2x, be taken in the last pro- a blem, the locus will be 21 {Ix + my + nz) - 2x = a («' + /*) + 6 [l' + ?«*). (24) The locus of the intersection of three tangent planes to the conicoid ax* + by* + cc' = 1, which are mutually at right angles, is " a b c and to the conicoid — + ^ = 2x, is x = - -^ . a b 2 (20) The locus of the intersection of three tangent lines to the ellipsoid b* + "rj + 'j = 1, mutually at right angles, is (t» + c') X* -i {c^ T «') y' + («' + b') z' = b'c' + cV + a'bK (26) Shew that the cone, whose vertex is (/, ff, h), which envelopes the X y z ellipsoid -, + h + "2= 1> will be cut by the plane of xy in a rectangulai 190 niOBLEMS. hyperbola, if the vertex lie on the spheroid-- — v-„ + - = 1. Also, if P be the centre of this section when V is the vertex, shew that the locus of P will be the inverse of the projection on the plane of ay of whatever curve V is made to describe upon the spheroid. (27) A cone whose vertex is any point of the hyperbola x = 0, 2* v" x^ V* s' •7- - "f-, = 1, envelopes the ellipsoid -i; + "r-, + -7= 1, whose least semi-axis isc; and h and k satisfy the relation h^ - c ■= — — ^^^ — - + — ^^ ; shew that the directrices of all the sections of the ellipsoid made by the planes of contact lie in one or other of two fixed planes. (28) The points on a conicoid, the normals at which intersect the normal at a given point, all lie on a cone of the second degree having its vertex at the given point. (29) The six normals drawn to the ellipsoid — + V; + -, = 1 from the rt- b' c- point (/, g, h) all lie on the cone 0/ _ c^) /- + (c- -a') ^ + («» - b^) '' ,- = 0. ' x-J ^ ' y -9 ^ ' z-h (30) The six normals drawn to the conicoid ax* + by^ + cz- = 1, from any point of the lines a {h - c) x = ±b {c - a) y = ± c [a - h) t, will lie on a cone of revolution. (31) A section of the conicoid ax* + hy""- 4 0:'' = 1 is made by a plane parallel to the axis of z, and the trace of the plane on xy is normal to the ellipse -, + ;, SI = — r- — TT. ; prove that the normals to the conicoid ' (a - cf {b - cy (c^ - aby ^ at points in this plane will all intersect the same straight line. (32) The normals to the paraboloid •j^- = 2x, at points on the plane j)x f (^y + rs = 1, will all meet one straight line if ;/* {h - c) f 2j, {,/b - ;-'c) = 2 ^'^J ^ '"'"'^ . - c (33) Prove that a tangent plane to the cone -; + •/ - 1 = will meet b - c b c the paraboloid y + - = 2x in points, the normals at which will all intersect in the same straight line, and the surface generated by the straight line will have for its equation 2 (6 - c) {X {by'- - cz') , be {y' - z')]' = {by'- - cz') {bf + c='/. PROBLEMS. 191 (34) Shew that any three equal conjugate diameters of an ellipsoid lie on the cone % (2«' -h"-- C-) + r" ('2i' - c^ - "') + - C-'f' - «' - h-) = 0, and that the planes containing two of three equal conjugate diameters touch the cone X* if ^ a' (2a* - i^ - c*) "^ 6« (2b'- c' - a") ^ r ('Jc^ - a' - h^) ~ ' c*, the equation of the surface may be put into the form x^ \ :^ - x^ = d^, *nd if a, li^ '-I be the angles between (t/z), {zx), (ry) in this case, £0S a (cos/5 cos-/ - COS «) = ^ 2 (a^ f 6' - c^ • CHAPTER XIV. CONTOCAL CONICOIDS. FOCAL CONICS. BIFOCAL CHORDS. CORRESPONDING POINTS. 278. In the preceding chapters we have considered the intersections of planes and straight lines with conicoids, in this chapter we shall discuss the mutual relations of conicoids grouped in a particular manner and called confocal conicoids, and prove certain theorems relating to their intersections, which will be useful hereafter when we treat of the curvature of surfaces and geodesic lines. A knowledge of the whole theory of confocal surfaces is essential for the solution of many important problems in Physics ; in fact, it was in the study of the attraction of ellip- soids that Maclaurin was first led to consider the properties of this class of surfaces. The theory may be said to have been completed by Chaslcs,* although many valuable propositions are due to M'Cullagh.t 279. Dee. Two conicoids are confocal, when the foci, real or imaginary, of their principal sections coincide ; or, when the directions of their principal axes coincide, and their squares diifer by a constant quantity. Another definition will be afterwards given, but for our present purpose this definition has the advantage of greater simplicity. If -? + 111 + -5 = 1, and ^ + 'ttt, + -75 = 1 be the equations a b c ^ a h c ^ of two confocal surfaces, * Briot and Bouquet, Geometric Anah/tique. Aper<;ti Uistoriqne, L'Acad. Bnix. 1837. t With reference to the relative claims of Chasles and M'Cullagh to priority in certain investigations, see Liouville's Journal, vol. XI., p, 120, and Proceedings of the Irish Academy, vol. II., p. 501. CUNFOCAL CONICOIDS. 193 or a'-V^ a' - h" and a' -c' = a" - c'\ These relations have given rise to two methods of stating the equation of a group of confocal conicoids, called, for the sake of brevity, confocaJs. In one method a Is called the ])riina7')/ semi-axis, and the equation of the group is written I •! I- = 1 a a — p a - y ' /3"' and 7" being constant quantities, the individuals of the group being determined by assigning particular values to the primary axis. In the other method the equation x^ y'^ z'^ of — k b' — k c^ — k represents all conicoids confocal with by assigning arbitrary constant values to k. I 280. To shew that three conicoids can be drawn through a I given pointy confocal with a given central conicoid^ and that [ these three will be an ellipsoid and the two h/perboloids.* ' Lot -5 + 7w + -^ = 1 be tlie given conicoid, in which a p 7 ° ' a' > ^'^ > y\ where yS'' or y'^ may be positive or negative, and I let (y, <7, //) be the given point. All confocals are given by x"^ f z' a-'-k^ ^'-k i'-k ' when any such passes through the point (/, y, /<), k Is determined by the equation Apei: Hint. (30), p. 392; Proc. Tr. Acad., vol. 11.. 196. 194 CONFOCAL CONICOIDS. or {k - a') [k - yS'O (/. - 7'1 +r i^ - n (^- - yl + / (^ - 7') (^'^ - a') + ^^' {J^ - a') L^^ - /3') = 0. If now we write for k in the left side of the equation, successively a'"*, ^\ k, + + +, I3'>k>y% + + -, a'>k>^% + - -, which proves the proposition. Cor. Two confocal ellipsoids or two confocal hyperbololds of the same kind cannot intersect. 281. If two parallel tangent planes he drawn to two confocals^ the difference of the squares of the p)erpendiculars from the common centre on these planes will he constant.^' For, if Z, 7H, n be the direction cosines of the planes and p, iJ be the two perpendiculars, f = ?V + m'h^ + «V, (Art. 253) p" = l\c"^m-'h"' + n'c"', .-. 2)' -p" = r [d' - a") + nr {P - h") + n' {6' ^ c"') = a- - d\ Cor. If an ellipsoid and hyperholoid he confocal^ all tangent planes to the ellipsoid drawn parallel to tangent planes to the conical asymptote of the hyperholoid will he at the same distance from the centre. For p=0] .'. p' = d' - a\ 282. The poles of a given plane^ taken with reference to each of a series of confocals^ lie on a straight line perpendicular to this plane.^ Let -. + •' „ ■ + -5 i = 1 be the equation of any one of * Aptr. Hut. (37), p. 393 ; Proc. Jr. Acad., vol. Ii., 491. t Ajycr. Jlist. (50), p. 397. ELLIPTIC COORDINATES. 195 ilio surfaces, /3 ami 7 being constant, \x -i- /j,i/ + vz = 1 that of the given plane ; and let (^, ?;, ^) be its pole with respect to the surface ; therefore the equation of the plane must be the >;une as d' ^ a' - IS' ^ d' - i' ' these arc the equations of the locus, which is evidently perpen- dicular to the given plane, and, since the point of contact of the particular confocal which touches the given plane is the polo with reference to that confocal, the locus is the normal at the point of contact to the confocal to which the given plane is a tangent. Let (/, (7, h) be the point of contact of the particular confocal which touches the given plane, 2a its primary axis, 2a that of any other confocal, and N the part of the normal intercepted between the plane and the polar plane of (/, g^ h)^ whose cqua- , fx qxi liz , . , , tion IS "-r- 4- ., ,.; + —, :. = 1, whicu may be written a' a — 13- a'-y' ' -^ a art' \rt a and x—f=~~^ Np^ &c., iV being measured inwards, and p be- ing the perpendicular from'the centre on the given plane ; ... Np = a'-a\ Elliptic Coordinatts. 283. The position of a point on an ellipsoid is deternwied^ when the octant on xchich it lies is known^ by the primary axes of the confocdl hyperholoids which pass through it. For if — +yij-f— = 1 ^c the equation of an ellipsoid and a - i' = ^', rt' — c' = 7'"', 7 > yS, the primary axes of the confocal 196 CON FOCAL CON ICO IDS. hjpcrboloitls, which pass through the polut (|, 77, ^ on the ellipsoid, will be given by the equations a a - /3 a — 7 a^ a — p a - 7 whence -,-^ + ^^ — i^im, — ^^ + r^- — .n 7^ t, = 0, or a^ + ^a^ + ^'fX = 0, (I) if, then, «', a" be the primary semi-axes of the two hyperboloids, the roots of the equation (1) are a'*, a''^ \ ^ f.'i a a a and similar expressions for t]^ and ^^ Def. The primary axes of the confocal hyperboloids, which pass through any point of an ellipsoid, are called the elliptic coordinates of that point. It follows that the equations, in elliptic coordinates, of the two curves of intersection with the ellipsoid are a or a" = constant, 284. When three confocals pass through a pointy each of the normals to the confocals at this point is perpendicular to the other two* This will be proved, if we shew that at every point in the curve of intersection of two confocals the normals are at right angles. Let the equations of two confocals be x' y' z' , - +75 + - = 1, a c . x^ y"* z'^ and , -;, + ,1,--,., + -^ — ,- a' - k^ y -k' c* - k Apcr. Ilkt., (30), p. 392. CONFOCAL CONICOIDS. 197 [f (I, 17, ^j be any point in the line of intersection, we find by subtracting J^+ ^" + ^' ^Q. therefore if /, m^ n and /', vi ^ n be the direction cosiues of the normals U + mm + nn = 0, (Art. 253) which proves the proposition. I 285. Three confocah pass through a point P, and a central ^section of one of them is made hy a ])lane parallel to the tangent plane to it at P ; to shew that the axes of this section are parallel to the normals at P to the other confocals. Also, if'^a-, 2«', 2a" he the lengths of the primary axes of the confocals, the sqiiares of the semi-axes of the section icill be d^ — a'^ and c^ - a"^.* T ^ x^ y'^ z' , x^ 11^ z' Let -1, + fi + -, = 1, — — j + ~-^ + -i 7 = 1 represent a c a— Kb— kc— k, the three confocals through P[f .7, h), by giving k the two vahies k\ k" derived from the quadratic a' [d' - k) ^ V [W -k)^ & {& -k) ^ ' If (/, m, 7i), (?', m\ n'), and {I", m", n") be the directions of the normals at P to the three confocals ^' - ^- = — and — -^ = .^ = ^ (2) aH ■ b'm c'n {a'-k) I' [F-k] m' {6'-k) n" ^ ^ and we shall obtain from (1) and (2) the equations d'r h'm' c'd' a — k — k c — k. d'-k l b'-k m' c'-k n and — . , = ,., . — = -. — . — . a I 6" m c' n By Art. 234 or 257, k is the square of one of the scmi-axe3 of the section by the plane, Ix + my 4- W2 = 0, parallel to the tangent plane at P, and [l', m\ n) is its direction ; also, since a' - k = a"\ d' - a!'^ is the square of the semi-axis which is * Prw, Ir. Acud., vol. 11.. p. iW. 198 CONFOCAL C'ONICOIDS, parallel to the normal to the confocal corresponding to Jc\ and similarly for k". If a', a" belong to the hjperboloid of one and two sheets respectively, a' > a", so that a' — a' is the square of the smaller semi-axis. Cor. When two confocals intersect, the normal to one of them at any point of the curve of intersection is 2ya^allci ^ an axis of a section made on the other hy a plane parallel to the tangent plane at the point. And diameters of each confocal, siqyposed central, drawn parallel to the normals to the other at every point of the curve of intersection are of constant length, being real for one and imaginary for the other.^ 286. To find the lengths of the perpendiculars from the centre npon the tangent planes at a given point to three confocals lohich pass through that point in terms of the py^imary axes. Since - =-^"4 +'t4 + -5 , p' a c the equation ., , — ,,, +...= reduces to the form a' [a — a;') 7 4 ATi a b C k* - Ak' + — ., =0; .: /k"k"' = ci^l/c" and / = j^, "^^ ^ , p" = &c. ^ ^ [a - a )[a - a -)^ -^ If, with the normals of the three confocals as axes, three new confocals be constructed, of which the semi-axes are re- spectively a, a, a" ; b, b', b" ; and c, c, c" ; the squares of the coordinates of their points of intersection will be , .--?vX— -.x=;^&c. (Art. 283); (rt - a ) (a — a ) ^ ' ^ ^ ' they will therefore pass through the centre of the first three confocaU, and the squares of the perpendiculars upon the tan- gent planes through that point will be ,— ^ — m7"s sv = I"? ^^c. ; Pruc. Ir, Acft'l, vol. ii., p. 4t»9. COXFOCAL CONICOIDS. 199 therefore the principal phones of the first three confocals will be tangent planes to the second three.* 287. If 'p he the ■perpendicular on the tangent plane to an ellipsoid at any point of its curve of intersection loith a confocal hi/j)erboloid and d be the central radius parallel to the tanrjent to the curve^ pd icill he constant. ^ , aW Hence, if 2a' be the primary axis of the hyperboloitl, since the tangent to the eurve of intersection is a normal to the other confocal hjpcrboloid, d' — d^ - a'"' (Art. 285) ; •'• pd=T-^ ^1 which is constant. [a -a)^ 288. To find the directions of the principal axes of a cone vJiich envelojjes a given central conicoid. x' it z^ Let "1 + j:, + — = 1 be the given conicoid and (/, ,7, h) the £•1 'i 72 vertex of the cone, then writins; u^ iox —, ^ '{:, -\- — — 1, the ^ ^ a' C equation of the enveloping cone, referred to the vertex as origin, is The centre (|, 77, ^) of a section made by a plane lx-i-7ng + nz = q, '• being written for"^ + 'fr + -$? is given, as in (3) Art. 229, by u^^ —fv u^ij—gv _ uX~ ^^ _ ri^v — {u^-\- \)v _ —V Let ^ + -^^ + i^ = 1 be the equation of a conicoid through a P 7 Aper. nifU, (30), p. 303. 200 CONFOCAL CONICOIDS. (/, ,7, /<), the tangent plane to it at that point being lx+7ny+nz=p referred to the original axes ; •'• 2'' — If'^ '".7 + *'^* ^^^^ fp = ^a"* . . . ; If a' — a^ = ^"^ — h'^ = y^ — c^ = /»■, or the conicoids be confocal, 1 = 1 = ?. / m n ' Hence, the centres of all sections parallel to the tangent plane to any confocal through the vertex lie in the normal to that confocal, which is therefore a principal axis of the cone, or The 2^^'i>icij)al axes of a cone enveloping a given conicoid are normals to the three confocals drawn through the vertex. 289. To find the equation of the enveloping cone referred to the normals to the confocals through the vertex as axes. Since these are principal axes the equation of the cone is of the form Ax'+Bf+Cz'^O, and, if p^, p,^^ p^ be the perpendiculars from the centre of the conicoid on the tangent planes to the three confocals, the equa- tion of the line joining the centre and vertex of the cone, referred to the same axes, will be — = — = , and since this 1\ Ih 1\ line passes through the centre of the curve of contact, the plane of contact will be parallel to the plane conjugate to this line, whose equation is Ap^x -}■ B2\y 4 Cp^z = (Art. 2G9), but the equation of the plane of contact will be X 71 z ^ + =/ + _ = ! \ fjb V ' if \, /u., V be the intercepts on the three normals ; .-. Ap;\ = Bp.^fj, = Cp^v^ benoe, if «,, a.,, "^ be the primary semi-axes of the tjircc con- CONFOf'AL CON'ICOIDS. 201 focals, since ^>,X = (?/"-<<''' v.K:c. (Art. 282), the equation required Avill be a, — a a„ - a a^ — ir 290. To find the cquntion of the enveloped conicoid referred to the three normals throxicjh the vertex. Since the equation of the ph\ne of contact is _;^■^• . Pdl , J\^ _ 1 ■i 'i ' 2 2 ' a •-' ' 5 (A| — a a,, - a a.^ — a tlic equation of the conicoid is of the form x' ir z^ — a' a^, — a a^ — a' \a^ — a a.' - a a^ —a J ' if, therefore, we transform the origin to the centre (-7',, -/>.,, — ;>3), by writing ^— /', for a*, &c., the coefficients of a-, ?/, z being i equated to zero, we shall have p + -. — . + -v^2 ■»- -f^. + 1 = 0, a^ — a a,^ —a ^3 — « j hence the equation required will be \a, - a^ a^ - a' a^' - a J \a^ — a' «/ - a' «/ - aV Vfl, -«' o,f-a a/ -a J 291. If from any point of a central conicoid a line be drawn I touchinfj two given confocalsy the j)ortion of this line intercepted letwfen the point and the plane through the centre^ parallel to the tangent plane at the pointy will he constant \ I If, with P the given point as vertex, two cones be described I enveloi)ing the two coiifocals, the line under consideration will be one of their common sides. ♦ Scott, Quarta-hj Jnurnnl, vol. vr., p. -.'js. f Ibid. t /V.H-. /(•. Acfui. vol. II., p. VM. 202 CUNFOCAL CONICOIDS. Let (^^'rlo.^ be the primary scml-axcs of the given conicoid and the two confocals drawn through the point P, a that of cither of the conicoids to which the line through P is a tangent. The equation of the cone enveloping the conicoid (a) referred to the normals to the confocals through P is a?" V' 2' a, - a and X =2\ '^^ ^'^^ equation of the central plane parallel to the tangent plane at P to (aj, therefore the square of the portion of the common tangent Intercepted between P and the central plane is 2\^ + ;/'' + ^^ where a - a o,^ — a <^3 — « in which the two values of a are «, a the primary semi-axe» of the two given confocals. If rtj'"' — a' = Z-, the equation may be written or [p: + / + z-^) Ic^ +... + p;^ [a: - a/) [a^ - a/) = ; and ^/ + 7/" + ;;'' = ^-V^^* .tV-V — ^ hence, the square of the intercepted portion is constant. 292. CoK. Tvo conicoids can he drawn confocal with a given conicoid and toucJiinq a given straight line. 293. 1/ a chord of a given central conicoid touch two other surfaces confocal with it^ the length of the chord will be pi'oportional to the square of the diameter of the first surface parallel to itr' Let a central section be taken containing the chord PP' ; draw CQ a radius of this section parallel to P'J\ and produce ^ ♦ Proc. Ir. Acnd.. vol. II.. p. d08. CONFOCAL CONICUlDS. 203 it to meet the tangent at P In T) let CN blseet PP\ and /'J/, parallel to NC^ meet CQ in J/, then CM.CT=CQ^ ] licnee, since CT Is constant by the last article, PP' = 2 CM x CQ\ 204, When two confocnls are viewed by an eye in any position^ tin iv aj)pa7-ent boundaries cut one another at right angles wherever they appear to intersect.'^ The boundaries will appear to intersect in any line drawn from the eye so as to touch both surfaces. Let the points of contact of such a line be P, (/, //, h) and /*', (/", y', //'), and lot the equations of the two confocals be - + -^r^, + :, = I and — + -r/—-:, + —, , = 1, a a — p a - 7 a a — l:i a — 7 since PP' is a tangent line, the points P' and P are respectively in the tangent planes at. Pand P'; . ff\ Oil . M _i a a — p a — y and -'4 + --/-^ p. + -75 5 = 1 ; a a — p a — 7 therefore, subtracting and dividing by d^ — a"^, ff 99 ^'^*' which shews that the tangent planes at P and P' are at right angles, and proves the proposition. There may be four, two, or no apparent points of intersec- tion, and, when they exist, they will be in the direction of the common generating lines of the two enveloping cones of which the eye is the common vertex. 295, The following method of dealing with tangents to confocal surfaces is due to Gilbert ;t it enabled him to solve with great facility many of the problems in this subjcit. • Ai>er. flist., (33), p. 392. Proc. fr. Acad., vol, H., p. 501. t .V.;/(i-. Annrilcs, vol. vi.. p. 529. 204 CONFOCAL CONICOIDS. He slicws that, if wc have pouits P, Q on two confocals wliose equations are and ^,+ Q2_^; + -(98_yi-lj and, if (^, X) denote the angle between the normals at P and (), measured outwards, (S, X) the angle between P(? (= S) and the normal at P, and p\XH be the perpendiculars from the centre on the tangent planes at Pand Q respectively, then will cos(^,X)=^j— ^ {i>eCOs(S, X)-;;xCOs(S, 6)]. Let a:, ;/, z and x^ y\ z\ be the coordinates of P and (), h ,r, ^ s ^^ ini zz icos(S,^) = ^4+J^'.= +.-^.-., Similarly ^ cos (S, «) = "^ + ^•'f ^, + ^, - 1 r5 ^ ••• - cos(a,X)-- cos(S, ^) and co3(0,X) = ^-? . ^'^^+... = ^,-^^ {j)^ cos(8,X)-y>,^ costs, 0)j. Cor. If FQ be a tangent at Q to the surface (0) cos (8, ^) = 0; .-. cosfO, X) ^t/^'I, cos(S, X). {/ — X 296. // tivo covfocals toncJi the same sfranjht line, tlie taiujeut planes at the points of contact will he at ricflit angles. For, if cos (5, X) = and cos (8, ^) = 0, then co3(^, X) = 0. 297. J/\, /J., V he the primarT/ semi-axes of the confocals 2^oss- irifj through P, that of a confocal enveloped hy a cone of which FOCAL CONICS. 205 r is the vertex^ ?, ?», n the direction cosines of any generating line of the cone icith reference to the normals to the three confocalsj P on' n' For co9(S, ^)=0, or cos(S, \) co3(^, X) + cos(S, fi) cos{6, fi) + cos(S, v) cos(^, v) = ; • by Cor Art -05 ^^^'-^ + ^^^^^ + ^^-^i^ - which proves the proposition, and shews tliat the equation of the cone referred to the three normals is x' f -' . " 208. If any jjoint P he taken in a fixed plane Z7, and on the normals to the three confocals passing through P lengths equal to the primary semi-axes he set off^ the sum of the squares of the projections of these lengths on a normal to the plane U loill he constant for all positions of P in that plane^ viz. the square of the primary semi-axis of the confocal touching UJ^ Let 6 be the primary semi-axis of the confocal touching Z7, X, /i, V those of the three confocals, then cos (S, X) cos (^, X) -f . . .= ; .-. (6'-'-X0 cos^(^, X) 4- {ff'-ix'') co8'(/^, ^l)■Y[&'-v') coi'[e, v) = 0, or X"' cos'(^, X) + ti"" cos' ((9, fi) + v' cos"''(^, v) = d\ Focal Conies. 299. Among the surfaces of the system of confocals obtained by giving all values to h in the equation x' //« z' there are two which have a particular interest. If «' > Ir > c" be nil positive, suppose h' to increase gradually from zero, the surfaces will change from ellipsoids to hypcrboloids of one * Com. Revd., vol. XXII.. p. 07. 206 FOCAL CONICS. sheet, as h passes through c^^ ami from hypcrboloids of one to those of two sheets as it passes through h'\ When /.: = c'*, z^ = 0, and the eonfocal may be considered as two phanes coincident with that of xy^ it being the limit of a very flat elhpsoid or hyperboloid of one sheet, as h is a little less or greater than c", the boundary of both being the ellipse a —c b -c ' In the same manner the hyperbola is the boundary of the two flat hypcrboloids of one and two- sheets for which k is a little less and greater than l>\ Th<3se conies are called the focal ellipse and hijixrhola of any of the eonfocals ; they pass through the foci of the two prin- cipal sections containing respectively the least and the mean axes of the ellipsoids of the system. The focal hyperbola also passes through the umbilici of the ellipsoids, for which x" ^' 1 {d' - k) [a:' - b') (c' - /.•) {If - c') ci' - 6' ' But there arc other properties which make the term focal conies peculiarly appropriate, and which we shall discuss in the next chapter. 300. To find the eonfocal hyiJerholoids which pass through a jJoint in the principal plane section of an elhpsoid which contains ^/«e j'jrimary and least axes. Let the equation of the ellipsoid be x' y' z-" a a - p a —y and let (/, 0, h) be the point in the plane of zx. W a bo the primary semi-axis of a eonfocal hyperboloid /' 0_ h' therefore either a' = /9, or d'='^-., . a 0; rocAL CONKS, 207 'J'lio latter solution gives tlie liyperboloid ;/" "•/ ■^- -P 1 ; 7 " a \ (I J ■\vlii(.-li is a livperboloid of one or two sheets aceorcling as < — , /.('. as the point is one side or the other of the 7 focal hyperbola. The other solution gives the focal hyperbola, which must be considered as a flat hyperboloid of two sheets or one, according to the position of the point. oOl. "We may observe here that as these focal conies belong- to the group of confoeals, many of the propositions given above can be applied to them. For example, a cone on a focal conic as base corresponds to an enveloping cone, since the focal conic IS in this case the curve of contact of a flat ellipsoid enveloped by the cone ; and the normals to the confoeals through the vertex are axes of the cone. e302. To find the locus of the vertices of all right cones irhich nivelope a given conicoid. Since the positions of the principal axes of such cones, which are perpendicular to their axes of revolution, are indeterminate^ we must consider three confoeals through the vertex of some enveloping cone for which the directions of the normals to two of them will be indeterminate. It is evident that if we draw normals to a conicoid of which one of the axes Is infinitely small, these normals will be parallel to that axis, unless the points at which they are drawn are indefinitely near the edge, and in passing round this edge from one side to the other the nwmals will assume every direction in a plane perpeudiculai- to the tangent to the bounding focal conic, and this tangent being the normal to the third confocal will be the axis of a right cone. Hence, the vcj-tices of right cones must lie in one of tlie focal conies. 303. CoK. The locus of the vertices of right cones on an elliptic base is an hyperhola in a j^lane perpendicular to its plane and vice versa. 208 BIFOCAL CHORDS. x' 9/ For any ellipse — + "t^ = 1 may be looked upon as the focal . x^ z' ellipse of a conicoid of which the focal hyperbola is — , „= 1, if (j^ — d^ = 1/ = '-f\ and the vertex must therefore lie in the focal hyperbola ; hence the equation of the locus of the vertices . x' z' , a —hh Bifocal Chords. 304, Def. a hijoccd cliord of a conicoid Is a chord which intersects two focal conies of the conicoid. These conies being the limits of confocals of the conicoid, the properties proved in Arts. 291 and 293 are true for these two particular confocals, whence, if a bifocal chord be drawn through a point P on the conicoid, the portion inter- cepted between it and the diametral plane parallel to the tangent plane at P is the primary semi-axis of the conicoid, obtained by writing for c^ and a'""* in the formula a'^ — c^ and a^ - h'^ ; and the whole length of the chord is proportional to the square of the diameter parallel to it. But in order to obtain a simple geometrical construction for the position of the four bifocal chords passing through a point, we have been obliged to deal with the problem in a direct manner, and wc shall therefore give an independent solution of the problem concerning the intercepted lengths of the chords as a preliminary step. 305. If P he one extremity of a bifocal chord of a conicoid, the portion of the chord intercepted between P and a plane through the centre, parallel to the tangent plane at P, ivill be equal to the primary semi-axes of the conicoid. Let /, m, n be the direction cosines of a bifocal chord drawn -+-, = 1, whose through a L point P (/, g, /') ot a conicoid a real focal conies are X' / = 1 1 ^' and , ^, a - b a^-c'^ l/-e BIFUt'AL CHORDS. 209 then ^4 ^ + li r^ = » (Art. 172, Cor. 1 , 1) a—c h —c ^ ' multiply (1) by -, :, , and (2) by ^, - -, , and add ; then C CI U it a c' a"^ 6/-c' \c- a J \b a J ' (P q' Ji\ /f m' n\ (If mq . nJi\' «"■' i'' 6'"" «'* hence, if P6r the normal at P, and P^ the bifocal chord, meet the central plane perpendicular to PG in P, E respectively, a' cos' I:PF=FF% and PF=PEcosEPF- .: PE=a. 306. If a tangent plane he drawn perpendicular to the bifocal chord^ the distance from the centre to the point where the chord meets this plane will he equal to the primary semi-axis. This can be shewn by midtiplying (1) by a' - c\ and (2) by j a* — 6*, and adding ; whence ! p + fj^ + U^ _ (Z/+ mg + nh )' = d' - / V - niT - w V. 307. To shew that the four hifocal chords through any point P of an ellipsoid lie in two planes passing through the normal at P and intersecting the primary axis of the ellipsoid in the feet of the normals at the umhilics. The bifocal chords of an ellipsoid through any point P(/, <7, h) are the generating lines common to the conical surfaces, whose common vertex is at P, and whose guiding curves arc the focal ellipse and hyperbola 210 BIFOCAL CIIOnDS. Now the equations of tlie two cones referred to parallel axes through P, are (fz-l,xY (cjz-hjY _,. „^„_„ „. d'-c^ ^ b'-c' -^. »'"-". l') ancl(^PffI-L&^^):=/,o.-. = 0; (2) and a common axis is the normal at P, whose equations are a^x V^y c'z The equation of two planes containing the bifocal lines and the normal is — = — , where w,, v„ are the values of ?/, v ob- talned by writing ~ , ^ , — for x^ y^ z] and it is easily proved Multiplying (1) by jr, , and (2) by ., , and subtracting, we obtain c' {a' - b') a' ib' - c') b' (c' - «') _ 1 /c'.2' _ 2yz jy\ and \d' {¥ - c') J.+ b' (c^ - a') ^+c' {a' - b') jV = («-c)(«-M^(x-7J' llencc, the four bifocal lines lie In two planes whose equa- tions referred to the axes of the ellipsoid are a' (?/ - c') J. + F (c^ - a'O '^ + 0^ [a' - V') j^ which pass through the points \±- \/{a'-b^) \/(a'' - c"), 0, oL that is, through the feet of the normals at the urabillcs. CORUEsroXDlX({ I'UfNTS. 211 Corresjyonding Points. 308. Di:f. If (.r, //, z), {x\ /, z) be two points F and P' situated respectively on the ellipsoids a^ f z' , J x' r ^' . - + 7^ + -. = 1, and — + -fr, 4 -, = 1, a c a b c P and P' are said to be corresponding points if - = — , -j- ='jj ^ '- = ~- . Ivory first made use of points so connected in order to ( stablish a relation between the attractions of an ellipsoid on an external and on an internal point, proving the following pro- position : 309. //* P, Q he two jyoints on an ellipsoid j and P\ Q tlie corresponding iwints on a confocal ellipsoid, PQ' = P'Q. Let {x, ?/, z), (^, 77, ^) be the points P, Q on the ellipsoid ~ + 't^j -)■ T' ^ ^5 ''^"^^ ^^^ V'*') ?/: ~')j (1} v\ ^ ) he corresponding ijoints P', Q' on the confocal -Tr+ ;-4 4 -7;; = 1. a' (j' c y -^ *' 1 ^ f 1 bniee - = — and - — - , , we have a a a a and similarly for the other coordinates; or PQ'=-.P'Q. 310. Since x'' — x"'= [n' — a') '—. . if be the centre of the 'a ellipsoid, then OP' - OP" = cC - a". 311. If an?/ concentric ellipsoid he drawn tliroufjh the vertex of a cone envelopinrj a given ellipsoid, the tangent plane at the point 'corresponding to the vertex loill meet the second ellipsoid in an ellipse, every point of tohich. icill corresjwnd to a point in the plane of contact; and, if the ellipsoids be confocal, the lengths oj , the tangents from the vertex will he equal to the corresponding radii of that ellipse. 212 CORRESPONDING POINTS. %^i + %=hO], '-i^d let ~ + ■^pr+ %^^ =1, (2), be CI u C Ct O i- Let (/, g^ h) be the vertex of the cone enveloping the ipsoid -12 -f T5 + -^= 1, (0, and I a b c an ellipsoid on which the vertex lies. Tire pLinc of contact of the cone is — + '7^ +— ^ = 1 in which ' a h c ^et (^, 97, t,) be any point, then, if (f, 1?', ^') be the correspond- ing point on (2), and (/', q\ h') correspond to the vertex, we havc/^=/r; therefore (^', 77', ^') is on the plane which touclies (1) at (./",,9'', /')• Also, if the ellipsoids be confocal, the latter part of the proposition is obvious by Ivory^s theorem. 312. If three points on an ellipsoid he the extremities of three conjugate diameters^ the three corresponding points on any other ellipsoid loill he also at the extremities of conjugate diameters. For the corresponding points on a concentric sphere are the same for both ellipsoids, and these are obviously at the ex- tremities of three perpendicular radii. / 313. Confocal ellipsoids are cut hy a fixed confocal hyper- , holoid ; to skeia that if any point he taken on the curve of '' intersection of one of the ellipsoids^ the corresponding point on any I other will lie on its curve of intersection. 'Z V! 2 If f.r, ?/, z) be a point on the Intei-sectlon o^ '-, + 'jr. + —. = I \ 1J7 J 1 a b c x^ ?/^ zf- •with the hyperbolold — + '-^ + — = 1 confocal with it, if (.c, ?/', z) be the point corresponding to (.r, ?/, z) on the con- •111 I X- ?/ Z X X- focal ellipsoid — , + ;,r, + "—. = \. writing , for - , &c., a b' c- ° a *nn' -t p5 = 0. (12) Shew that three non-central conicoids can be drawn through a given point, confocal with a given non-central, and that these will be a hyperbolic and two elliptic paraboloids. Shew also that the three normals to the confocals at the point are mutually at right angles. (13) Three confocal paraboloids intersect in S; a cone having its vertex at S envelopes a fourth confocal paraboloid ; find the equation of this cone referred to the normals to the confocals at S as axes. (14) Prove that the polar of the foot of a normal to an ellipsoid with respect to the focal ellii)se is the polar of the foot of the ordinate with respect to the principal section of the ellipsoid; also that the line joining the two feet is a normal to an ellipse similar to the principal section. (15) "When the two confocal hyperboloids through a point degenerate into flat surfaces bounded by the focal hyperbola, explain the perpendicu- larity of the three normals at the point. (16) Find the three confocals of an ellipsoid through a point m one of the focal conies. ^■2 J.2 2* If (/, 0, h) be a point on the focal hyperbola of the ellipsoid — + ts + -,.= 1. shew that the ellipsoid which passes through it is **_ (a* - c") v* 2« _ a*- c* (> - c')J ' ^ (6' - c»)V* + («* - b'f A' "^ (a' - b')h* " (a* - 6«) (6« - c') ' (17) a', h', c' and a", b", c" are the semi-axes of the confocal hyperbo- . ^- V* s' loids which pass through a point P in tlie ellipsoid ^ + V^ + -5 = ^ > 2^)2^') P' are the perpendiculars from the centre upon the tangent planes at P ; shew that the equation of the plane of the focal ellipse referred to the three normals at P is — + ■^+-,. + 1=0, and that the bifocal lines lie in the two c» c" c"* , a*x* o'*v* planes = — ^ . (18) Shew that the equation of the circular cone containing the four bifocal lines through any point of an ellipsoid is (— - 1) z* = y' + z', re- ferred to the three normals to the confocals through the point, ;; being the perpendicular on the tangent plane to the ellipsoid. 216 (19) If X be the length of a bifocal chord of the paraboloid — + - = a-, which makes angles /3 and 7 with the axes of y and z respectively, 1 _ cos*/3 cos' 7 (20) Find the locus of the point corresponding to a given point of an ellipsoid, on a system of confocal ellipsoids. (21) Corresponding points on an ellipsoid of semi-axes o, h, c, and a sphere of radius r, being defined by the relations f=^ y^J/^ z^£ a r ' b r ' c r ' (x, y, s) being on the ellipsoid and {x, y', z') on the sphere, prove the following theorems;: (i) The points on a system of confocal ellipsoids corresponding to a fixed point on the sphere are on the intersection of two confocal hyper- boloids. (ii) The curve on the sphea-e corresponding to the curve of intersection of the ellipsoid and a confocal hyperboloid lies on the asymptotic cone of the hyperboloid. (iii) If four curves on an ellipsoid of the kind described in ii, form a small rectangle of sides ds, ds', there will be a corresponding rectangle on the sphere whose sides rfo-, rfo-' are oonnecrted with ds, ds' by the relations rds = X^da-, rds' = X^da', X,*, X/ being the differences between the squares of the semi-axes of the ellipsoid and the two hyperboloids which intersect it in two adjacent sides of the small rectangle. CHAPTER XV. MODULAR AXD UMBILICAL GENERATION OF CONICOIDS. PROPERTIES OF CONES AND, SPHERO-CONICS. 317. The modular and umbilical methods of generating conicoids, invented by MacCuUagh and Salmon respectively, may be stated as follows : For the modular method, " The locus of a point whoso distance from a fixed point is in a constant ratio to its distance from a fixed straight line, measured ixircdkl to a Jijced p/a/^e, is a surface of the second degree." The fixed point is called a modular focus, the fixed line a directrix, the constant ratio the viodulus, and the plane the directing plane. 318. Since this locus contains ten disposable constants, viz. three dependent on the position of the fixed point, four on that of the fixed straight line, and two on the direction of the fixed plane, and one more, namely the constant ratio, the locus may, in general, be made to coincide with any sui'face which can be represented by an equation of the second degree in an infinite number of ways, since there will be only nine equations con- necting the ten disposable constants. If all but the three coordinates of the focus be eliiiiiiiatcd, there will result two final equations determining a curve locus of such points; such curves are called /wa/ conies, being the same as the limits of the confocals discussed in the last chaj)tcr. Again, if all but the four constants wiiich determine the position of the directrix be eliminated, there will be three final equations which, with the equations of the straight line, will determine a ruled surface, called a dirigent cylinder, the trace of which on the plane of the focal conic is called a dirigent conic. FF 218 MODULAR AND UMBILICAL 319. For tlie ximlnUcaJ method, " The locus of a point, the square of whose distance from a fixed point bears a constant ratio to the rectangle under its perpendicular distances from two directing planes, is a surface of the second degree." This locus contains ten disposable constants ; three dependent on the position of the fixed point, three on the position of each of the directing planes, and one more, namely the constant ratio. The fixed point, therefore, will not generally be unique, but may be on any point of a curve locus. The fixed point is called an umbilical focus ^ the intersection of the planes a directrix^ the constant ratio the umbilical modulus, and the locus of the focus the lunbilical focal conic, the trace of the dirigent cylinder on the principal plane being the umbilical diriyent conic. 320. To find the locus of a point whose distance from a focus is in a constant ratio to its distance from a directrix^ measured parallel to a (jiven directing idane. Let S the focus be taken for origin, &, Sy parallel to the directrix DX and the directing plane respectively, and let a, /3 be the coordinates of D in xy, e the modulus, and m the angle of inclination of the directing plane to the plane of xy. >s: z I ^ p ^ r / y/ Let PiV be drawn from the ])oint (x, y, z] to the directrix parallel to the directing plane, NM parallel to ^y, and PM nKNURATIOX OF C'OXITOIDS. 219 perpendicular to XM; then PJAVwlU be parallel to tlie ilirecting plane. Hence we shall have JAV=?/-/9, and r21={x — a) secco; and, P being a point in the locus, SP=e.Py'j .-. a;' + 7/ + z^ = e'{{x- af sec"^ « + (3^ - ^Y] ; this is the equation of the locus required, which is a surface of the second order. Since z = x\an(o + h Is the equation of any plane parallel to the directing- plane, wc have, at the points of intersection with the surface, x' sec' Qj + f = x'' + if + (2 - li)\ which, combined with the equation of the locus, shews that the curve of intersection lies on a sphere, except when e=l, in which case it lies on another plane ; hence, all sections parallel to the directing plane are circles, or when e = 1 straight lines. 321. That the section by a plane throuf^h S parallel to the directing plane is a circle is obvious geometrically, for, if this plane cut the directrix in //, the section is the locus of a point whose distances from a9 and H are in a constant ratio, and is therefore a circle, unless ^=1, in which case it is a straight line, and the surface is a hyperbolic paraboloid. 322. To find the locus of a pointy the square of whose dis- tance frmn a focus IS in a constant ratio to the rectauQle under its distances from two fixed directing planes. Let the focus S be taken for the origin, the planes bisecting the angles between the directing planes being parallel to the planes of rr?/, yz. Let also &> be the Inclination of the directing planes to the plane of a-?/, a, 7 the coordinates in the plane of zx of any point in the directrix, and e the constant ratio. From any point /*, let PQ^ PR be drawn perpendicular to the directing planes ; .-. SP'=ePQ.PP; the equations of the directing planes will be {x — ot.) sin w i '- - y cos w = ; 220 M(JI>ULAK AND UMBILICAL therefore, if j:, ?/, z be the coordinates of P, ^ -^y' -^ z' = e [[x - ay sln*6j — [z - or -^ + - .^'- = 1 ; whence It is the polar of (|', 97'), the foot of the corresponding directrix, with respect to the section In xy. Also, since d^ (^' - |) = c"|', and Z^-' (7?' -rj) = c'v'7 the equation of the tangent may be written a^(f-|)+y(V-'7) = c'; it is therefore pcrpendlcuhar to the line joining (^, 7;) and (^', 7^'), whence the second part of the proposition. 326. If a section of a comcoid he made hy a plane perpen- dicular to that of a focal conic^ so that it contains a directrix, to shew that the distance of any j^oint of the section from the directrix will have a constant ratio to the distance from the cor- resjwndinq focus. For if (I, 7;, 0) be the focus corresponding to the directrix (f, 7^'), the equation of the conicoid may be put into the form {-^ - ^7 ^{y- vY + z' = \[x- ^y + ^ (^ - -nj, and, If the equation of the plane be — ,-- = ~ = ^j then for any point P of the section [x - ^)* +[y- vf + 2' = (XZ"' + fini') r' ; therefore SPa: I'Q, if PQ be perpendicular on the directrix. 327. Cor. If the plane containing a directrix be perpendi- cular to the focal conic, the corresponding focus S will be a point In the plane (Art. 325), and will therefore be a focus of the section ; hence, Every point of a focal conic of a conicoid is a focus of the section made by a plarie perpendicular to the focal conic at tliat p)oint. 328. To find cohere a coJiicoid is intersected hy its focal conies. If the directrix be parallel to Oz for the conicoid x' f z" , the equations of the focal conic will be v_ -t^ -1- ,-^"^ =1, 2 = 0, GENEKATION OF CONICOIDS. 223 aud it will intersect the conicoid if -^rr'-i. ?> + -TT7Tir~~-r\ = ^ a [a - c) {Ir - c ) give X' : y^ positive. Hence, if tlie focal conic and the corresponding principal sec- tion be both ellipses or both hyperbolas they do not intersect ; but, if they be not of the same kind, they will intersect in the conic. Now, referring to the equations of Arts. 320 and 322, we see that in the modular method of generation the focus cannot lie on the conicoid, but may do so in the umbilical method, thus the umbilical focal conies correspond to the umbilical method 329. To find the focal and dirigent conies for non-central surfaces. For the paraboloids, comparing the equation (^-ir+(i/-'7r+(~-?r=^(^-rr+/^(z/-'?T with '^+- = 2^:, we have X = l, ?> (1 -;a) = c = ^ - f , V = m\ ?= 0, and |' -f rj' = Xp + ixr)"" ; .-. /.= l-^-,andc(|-fr)=^^(^^4-e-l The focal conic is therefore a parabola, which has its vertex at the focus of the parabolic section parallel to the directrix, and is confocal with the section in its plane, since the abscissa of its focus is ^c + ^ (i — c) = ^b. Also, the equation of the dirigent conic is 7/'^ = " (|' + :t ) , which is the reciprocal of the focal parabola with respect to the section 1/ = 2hx. It will be found that the focal conic of an elliptic or hyper- bolic cylinder is the two straight lines containing the foci of the principal sections ; and that that of a parabolic cylinder is two straight lines, one of which is at an infinite distance and the other contains the foci of the principal sections. 224 MODULAR AND r.MBILICAL 330. In order to apply tlic modular method of generation to the investigation of properties of conicoids, the modulus and directing plane must be real, aS well as the focal and dirigent conies, and, referring to Arts. 320 and 324, we obtain the lollowing conditions: j^ ij'^ ^ I. For the ellipsoid — + W + "^^ = Ij « > ^ > c, the only focal conic which is applicable being ^' - + -^ = 1 CI — 6^ h'' — I For an bblate spheroid a — b and co = 0. The prolate spheroid, for which J = c, cannot be generated by the modular method. x^ y'^ z^ II. For the hypcrboloid of one sheet — -fy^— ^ = 1, h> a. ^^ a' b' c' ' ' the directing plane is parallel to 0^, and both the focal conies — 7 + ^~ — ., = 1 and zr:r — •> ^ 5 = 1 are applicable, the corresponding moduli e, e being given by {e^—])b'^ = c^ and ( 1 - e'^) h' = rt\ where cos'"' ft) sin" ft) so that the hyperboloid of one sheet can be generated by means of foci lying in a focal ellipse or a focal hyperbola, the greater modulus corresponding to the ellipse. x' if z" III. For the hyiK'rboloid of two sheets —,—\., r, = 1, i > r, •' ' a b c ' the directing plane is parallel to (9y, and the focal conic is - 2 2 - 7/ J = 1) tlic modulus being given by (I — c'^) b" — c\ it ~T C — C The hyperboloid of revolution of two sheets, where b = c^ cannot be constructed by the modular method. if z^ IV. For the elliptic paraboloid y H — = 2.c, b > c, f = cos(y, /> (1 — r'^) = c = Z' slu'w, the focal conic is / = 2 [b - c) [x- ^c). GEN'ERATIOX OF COXICOIDS. 225 V. For the livpcrbolic paraboloid *-t = 2^-, h c ^ (' = 1 , l)[\ — sec' (w ) = - r, or h tan" co = c ; the focal parabolas arc y- = (Z> + c) (2j; + c) and 2;' = - (i + c) (2x - J), each of which satisfies the modular method. 331. To trace the changes of the surfaces and real focal conies corresponding to changes of the modulus from to cc . If we transfer the origin used In the equation of Art. 320 to the centre, and have regard to the sign of the constant terra, we shall obtain the following results : f cosa) and <], Surface at first an hyperboloid of two sheets, passing through a cone, to an hyperboloid of one sheet, conjugate axis perpendicular to the directrix. Focal conic at first an hyperbola, transverse axis perpendicular to the directive axis, passing through the asymptotic limit, viz. two straight lines, to an hyperbola, transverse axis parallel to the directive axis. ' = 1, Surface an hyperbolic paraboloid. Focal conies two parabolas. ' > 1, Surface an hyperboloid of one sheet, conjugate axis ])arallel to the directrix, including an hyperboloid of revolution. Focal conic an ellipse, transverse axis parallel to the directive axis. The hyperboloid of revolution of two sheets is lost between c= 1 and e = cosa). 332. The directrix In the umbilical method of generation being parallel to the intersections of the two scries of circular c, (J 226 MODULAR AND UMBILICAL sections, the plane of the focal conic is known, and by Art. 322 the umbilical modulus can be found in the same manner as in the modular method, and it will be seen that all surfaces can be generated, except the hyperboloid of one sheet, the hyper- bolic paraboloid, and the oblate spheroid. Properties of Conicoids deduced hy the modular and unibUical methods. 333. Every plane section of a conicoid^ wliich is normal to a focal conic at any pointy has that point for a focus. For, if S be the point through which the plai>c section passes, the corresponding directrix also will be in the plane ; let PQ be perpendicular to this directrix from a point P on the section. If the focal conic be modular, let PR parallel to a directing plane meet the directrix in i?, then the ratios SP '. PR and PR : PQ will be constant for every point in the section ; and, therefore, SP : PQ will be a constant ratio. If the focal conic be umbilical, let PM, PN be perpendiculars on the planes through the directrix parallel to cyclic sections ; then SP'ccPM.PN, and for all points of the section PiM: PQ and PN : PQ will be constant ratios, therefore ySPx PQ. 334. If a section of a conicoid he made hy a plane perpen- dicular to the plane of a focal conic, it will contain tioo directrices ; to shew that the sum or difference of the distances of any point of the section from the two corresponding foci loill he constant. Let QD, Q'U be the two directrices, and S, S' the corre- sponding foci, which in this case will not be necessarily in the plane of the section ; draw through any point P of the section QPQ' perpendicular to the directrices. If the focal conic be modular, draw RPR' parallel to a directing plane meeting the directrices In R and R'. JSinca the modulus is the same for both foci, SP: PR:: S'P:PR'; .: SP: S'P:: PR : PR' :: PQ : PQ', and SP i S'P : PQ J PQ' :: SP : PQ. GENERATION OF CONICOIDS. 227 Now PQ + PQ or PQ - PQ' is constant, according as P is or is not between the directrices, and SP : PQ is constant, since PR : PQ is so ; therctbrc SP ^ S'P is constant. If the focal curve be umbilical, draw PJ/, PN perpendicular to the planes through QD parallel to the cyclic sections, and let P}[\ PN' be corresponding perpendiculars for Q'D' ; then PM: PQ :: P^P : PQ' and P.V : PQ :: PN' : PQ', also SP' = c,PM.PN, S'P'' = e.PM'.QN'', .-. SP: S'P::PQ:PQ', and the argument proceeds as before. 3;35. i/a chord of a conicoid meet a directrix, the line Joining the 2)oint in the directrix with the corresponding focus loill bisect the angle between the focal distances of the extremities of the chord or its supplement. Let the chord PP' meet a directrix in Q, and let S be the corresponding focus, then SP '. PQ :: SP' '. P'Q\ .-. SP: SP' ::PQ: P'Q, which proves the proposition. 336. Cor. If PQ be a tangent to a conicoid at P, meeting a directrix in Q, and S be the corresponding focus, the angle PSQ will be a right angle. 337. A straight line touching a conicoid makes equal angles with the lines drawn from the jwint of contact to the foci which correspond to the directrices lohich the line intersects. For, if P be the point of contact, Q, Q' the points in which the tangent meets the directrices, S, S' the foci, since the modu- lus is the same for botli foci, we shall have SP : PQ :: S'P : PQ'. Also the angles PSQ, PS' Q' are right angles, therefore the triangles are similar, and the angles QPS, Q'PS' are equal. 338. If a cone, having its vertex in any directrix, envelope a conicoid, the plane of contact will pass through the corresponding focus, and he jyerpendicular to the line Joining the focus with the crtcx. 228 MODULAR AND UMBILICAL GENERATION. If V be the vertex and S the focus, and VF be any side of the cone touchuig the surface in P, PSV will be a right angle. Hence the hicus of P, which will be the curve of con- tact, will be in a plane through S perpendicular to VS. 339. If the vertex of a cone be any point in a focal curve of a conicoidj and the base be any plane section of the conicoid, the line joining the vertex with the point in which the corresponding directrix meets the p)lane of section loill be an axis of the cone. Let S be the vertex, and let the plane section cut the direc- trix in E, and EP, EP' be tangents to the section at P, P', thea /5P, SP' will be perpendicular to SE^ the intersection of two tangent planes to the cone through SP^ SP' ; therefore SE will be an axis. Cor. 1. The second plane of section of the cone and coni- coid will intersect the corresponding directrix in the same point as the first plane. Cor. 2. If the fii'st plane of section pass through the direc- trix, the second will do so also, and in this case, since there will be an infinite number of axes of the cone, it will be one of revolution. 340. If the vertex of an envelop)ing cone of a conicoid be a point on a focal conic of the conicoid^ the cone icill be one of revolution, and its internal axis loill be the tangent to the faced curve at the vertex. Let V be the vertex of the cone, VP, VP' the tangents to the trace of the conicoid on the plane of the focal curve, then PP' will be a tangent to the dirigent conic at the foot of the corresponding directrix (Art. 325) ; and since the plane of con- tact is perpendicular to the plane of the focal curve, it will contain the corresponding directrix, the cone therefore will be one of revolution (Art. 339, Cor. 2). Also, since the tangent at V to the focal conic is perpen- dicular to the directrix, and to the line joining V and the foot of the directrix (Art. 325), it will be perpendicular to the plane of circular section, and will be the internal axis of the cone. CONES AND Sni ERG-CON ICS. 229 Cones and Spliero- Conies. The properties of cones of tlie second degree, and of their intersections with a sphere whose centre is at the vertex, called aphero-conks^ have been discussed in an elaborate manner in two memoirs by Chaslcs.* In these investigations he has made use of certain reciprocal properties of the cyclic sections and focal lines, by which any theorem relating to cyclic sections involves a corresponding theorem concerning focal lines. "We can only make a selection of some of the innumerable propositions given by Chaslcs, in the proof of which we shall generally employ the properties of focal lines, in place of the reciprocal properties of the cyclic sections, employed with so much skill in those memoirs, for which we refer the student to a valuable translation by Graves. 341. Focal conies of conical surfaces. Since a cone may be considered as the limit of cither of the hyperboloids when the axes are made indetinitely small, if a, Z>, c be finite quantities proportional to the principal semi- axes of an hyperboloid, supposed indefinitely diminished, we obtc-un the equations of the cone -r, + 4^ ^ — 0, a > i, and the ^ a 0' c ^ ' corresponding focal conies, viz. a' + c' ^ l;' + c' ' _ -^ - J =0 1 ^^ ^'^ and -r, — n — i-, ; = 0. a' -y b' + c' The same consideration shews that the line joining any focus with the foot of the corresjiondinQ directrix is perpendicular to the focal line containing the focus. The student should obtain these results by a direct com- parison of the equation of the cone witii such an equation as * Suuc, Man, de PAcad. Roy. de Brtueelkf, vol. vi. 2;'»0 CONES AND SriIERO-CONICS. which will give the focal and dirigcnt lines a'-b' h' + c- ' a ^ c' If he compare with the equation (•^ - ^f + (y - vY + ^' = V {-c - IT ^f-'y- v')\ he will obtain the equation ^^ + ,."-.=0; hence, when the directrix is in the axis, the vertex is a modular focus, \ and V are the squares of the moduli in the two cases, and are equal to 1 r, and 1 + -^ . The cone has therefore tlie property that all the three focal conies are real, having a common point in the vertex, two of them being ellipses evanescent in the transition between real and imaginary existence, and the third the limit of an hyper- bola consisting of two right Unas intersecting in the vertex. The vertex is therefore not only modular, but doubly mo- dular, since it is a point in two modular focal curves, and it is also an umbilical focus, as we see from the fact that the cone is the limit of two hyperboloids, for both of which the real focal hyperbola is modular, and for one the real focal ellipse is modular, while for the other it is umbilical. Of the two moduli in the modular generation of the cone, the less modulus belongs to the focal lines, and is called by MacCullagh the linear vwdulus^ while the other, to which only a single focus corresponds, is called the siiujidar modidus. 342. Cydic sections of a cone. x' ?/* z"^ . ■ The equation of a cone '— + "/r, = -ij may be written a' b' c x^ -f y' ■+ 2^ = ( 1 + 7, ) '~'' - [jr. - 1 ) y' \ therefore, any plane sec- tion which is parallel to one of the planes ( 1 + '^ J ^^ = ( 7^ — ^ ) y'j lies on a sphere a^id is circular. ai CONJUGATE DIAMETERS OF A CONE. 2.'51 The planes through the vertex, to which circular sections c parallel, are called cyclic planes. 343. A §2)]ierc^ ivhich passes t/iroutjh the vertex of a cone and any circular section^ toUclies the cyclic plane of the 02)posite system. A sphere can be described through any two circular sec- tions parallel respectively to the two cyclic planes ; let the plane of one of the circles approach indefinitely near to the vertex, in which case the circle degenerates into a point-circle lying on a cyclic plane, which is therefore a tangent plane to the sphere. Conjugate Diameter's of a Cone. 344. Take any line VA through the vertex of a cone, let VBC be its polar plane, and VAC any plane through VA inter- secting VBC in VCj then VB the polar line of VAC will lie in VBC', also VC will be the polar line of the plane through VA, VB. Thus, if any plane cut VA, VB, VC in A, B, and C, the triangle ABC will be self-conjugate with respect to the section by the plane. If then a section be made by a plane parallel to VBC, the polar of the point in which this plane will cut VA will be at infinity, and the point will be the centre of the section. Vxi is therefore the locus of the centres of all sections by planes parallel to VBC; and VB, FC have the same relation to VAC, VAB respectively. VA, VB, VC, therefore, form a system of conjugate diameters of the cone. COK. If a plane cut a system of conjugate diametral planes of a cone, the triangle formed by the lines of intei'section is self- conjugate with respect to the section of the cone by the plane. Reciprocal Cones. 345. If a cone he constructed whose sides are jJcrjyendicular to the tangent planes of any given cone, the tangent planes to it will he pfrp> ndieular to the sides of the given cone. Let any two tangent planes be drawn to a cone ^1, then two corresponding sides of the other cone B, perpendicular to those tangent planes, will be perpendicular to their line of intersection ; 232 RECIPROCAL CONLS. the line of Intersection of tlic tangent planes to A is, therefore, perpendlcuhar to the plane containing the corresponding sides ofi?. Proceeding to the limit, the line of intersection becomes ulti- mately a side of the cone A^ and the plane containing the sides of ^ a tangent plane to B; whence the truth of the proposition. From this reciprocal property the cones are called reciprocal cones. x^ ir z^ It "2 + 7^ ;^ = be the equation of a cone, aV''+Z'"y- cV = will be that of the reciprocal cone. 346. Any plane through the common vertex, having re- lations to one of the cones, has perpendicular to it a line which has reciprocal relations to the other cone, and the plane and line are said to correspond. If two lines cori'espond respectively to two planes,- they will each be perpendicular to the line of intersection of the planes, and the plane containing the two lines will correspond to the line of Intersection of the two planes ; also the angle between the planes will be equal to the angle between the corresponding lines. 347. The student will have no difficulty In establishing the following theorems: To a line through the vertex of a cone and its ^wlar 2'>lfin3 with reference to the cone^ correspond a p)lane and its p)olar line ■with reference to the reciprocal cone. To three conjugate axes of a cone correspond three conjugate diametral planes of the j-ecijn'ocal cone. 348. The cyclic planes of a cone correspond tn the focal lines of the reciprocal cone. The equation of the cyclic planes of the cone ci^x^ + Vy* = c^z'^ is [a' - b'') x' = [1/ + c") z\ and that of the focal lines of the , x^ ?/ z'^ . x' z'^ . ^ , ,. i-eciprocal cone -r; + yr, = -. is — — — = ,., — ;, ; these focal hues ' a h c a - // b -f c ' are therefore perpendicular to the cyclic planes of the reciprocal cone. rjJOPEUTIES OF CONES OF THE SECOND DEa^fe^ 'i^S ' / The relation between the focal lines of one cotic £m'(^ the /■ cyclic planes of the reciprocal cone is deduced geometrically thus : ' / To a cyclic plane corresponds a line VS perpendicular to it^ ' i^ any two conjugate axes in the cyclic plane are at right angles ; v'-* therefore any two conjugate diametral planes of the reciprocal cone through VS are at right angles. Let a plane be drawn perpendicular to VS through any point >S', this plane will meet the two diametral planes in two perpendicular lines, and, by Art. 344, Cor., the pole of one of these lines with respect to the section of the cone will lie on the other line ; therefore this pole is on the directrix, and 8 is the focus of the conic section ; VS is therefore a focal line, having the focal property proved for any conicoid in Art. 333. The locus of these directrices is called a dirigent plane by McCullagh and a director plane by Chasles, and this plane, with the perpendicular planes through the focal line, form a system of conjugate planes; it corresponds, therefore, with the third axis, conjugate to two perpendicular lines in a cyclic section, which contains the centres of circular sections parallel to that cyclic section. 349. The method of dealing with propositions connected with focal lines, by the modular and umbilical methods, may be seen by the following, in which we shall state the reciprocal theorems. Properties of Cones of the Second Degree. 350. The sines of the angles^ lohich any side of a cone maJces ii-ith a focal line and the corresponding dirigent jilatie^ are in a ''>ns(ant ratio. Let a plane pass through any directrix UQ and the corre- sponding focus >S', and let P be any point in the section of the cone made by this plane ; V the vertex of the cone. Draw PIlj FQ perpendicular to the dirigent plane and directrix. II II 234 ritorERTiES of cones of the second degree. Then SP : PQ and PR : PQ, and therefore SP : PR arc constant ratios; and BS being perpendicular to VS^ VS is perpendicular to the plane of section, and PSV is a right angle. Hence, the ratio of the sines proposed is -jjTr' pi/j and is therefore constant. Reciprocal theorem. The ratio of the sines of the angles made hy tangent planes with a cyclic plane and with the jjolur line of this cyclic plane is constant. 351. The prodact of the sines of the angles which any side of a cone makes loith the directing or cyclic jtldnes is constant. If V be the vertex of a cone, P any point on the cone, PZ, PL' perpendicular on the directing planes through F, then hy the umbilical generation of the cone (Art. 322) PF" is propor- PL PL' tional to PL. PL'] or -^^ . -pp- is constant, which is the pro- perty enunciated. Recij>rocal Theorem. The product of the sines of the anghs ivhich each tangent plane to a cone makes with the two focal lines is constant. 352. A tangent ^;?a?ie to a cone maJces equal angles icith the planes through the side of contact and each of the focal lines. For, let the tangent QPQ perpendicular to the side VP meet the dirigent planes in the points Q^ Q', and take S, S' the foci corresponding to the directrices through Q, Q' ; then SQ is per- pendicular to VS, and also to PS, and therefore to the plane VPS; also VP is perpendicular to SQ and PQ, and therefore to SP; hence SPQ is the inclination of the planes VPS, VPQ, and being equal to S'PQ' (Art. 337), the proposition is proved. Reciprocal Theorem. A tangent plane to a cone intersects the two cyclic jylanes in two straight lines, ichich make equal angles tvith the side of the cone along which it is touched by the tangent plane. 353. The last theorems arc particular cases of tlie two following ; rROPKRTIES OF CONES OF THE SECOND DEGREE. 2C5 The planes ijassing tliroiigh the two focal lines of a cone^ and throuf/h the intersection of two tangent jihuies to the cone^ make equal angles toith these tangent p/«??c5. And the reciprocal theorem : A plane containing tico sides of a cone intersects the cyclic planes in two straight lines^ which respectively make equal angles with the two sides. "We give Chasles' proof of the reciprocal theorem as a gooil example of the geometrical treatment of problems connected with cyclic planes. Take two circular sections of opposite systems of the cone, the plane of tlse two sides cuts the planes of the two circles in two chords, which, with the portions of the sides of the cone intercepted, form a quadrilateral inscribed in the circle in which the sphere containing the circular sections Is cut by the plane of the two sides ; two opposite angles of this quadrilateral are supplementary, hence the chords make equal angles with the sides of the cone, and, since they are parallel to the sections by the cyclic planes, the theorem is proved. ^554. Simple propositions for the circle can be transformed into others relating to the cone with the same facility as In plane geometry properties of conies are obtained. This is effected by considering the lines and points In the circle as the intersections of planes and straight Hues, jiasslng through the vertex of a cone, with the phine which cuts the cone in this circle. It will be sufficient to give two examples of this trans- formation. 3.j5. Two tangents to a circle make equal angles with the chord which joins the two points of contact, hence Two tangent planes to a cone and the plane of the two side.t of contact intersect a cyclic plane in three straight lincs^ the third of ichich bisects the angle between the other two, Tlie reciprocal theorem is, If planes be drawn through a focal line of a cone, and two sides of the cone^ ctnd through the line of intersection of two 236 SPIIERO-CONICS. jyJancs touch inu the cone along these sides ^ the third plane will bisect the angle hetioeen the first two. 356. Two tangents to a circle make equal angles with the line joining their point of intersection with the centre of the circle, hence Two tangent planes to a cone, and the j^la^ic passing through their line of intersection^ and through the conjugate of a cyclic 2>lane, meet that cyclic plane in three linesj one of which hisccts the angle between the other two. The reciprocal theorem is, The planes pxissing through a focal line of a cone and tiro sides of the cone make equal angles ivith the plane p)assing through the same focal line and the straight line in lohich the plane - containing the two sides intersects the dirigent plane. Sphero-conics. 357. If a cone of the second degree be cut by a sphere whose centre Is at the vertex of the cone, the complete curve of intersection will be two closed curves, which will be plane curves if the cone be one of revolution. Chaslcs observes that we obtain three distinct curves if we consider the portions of the complete curve of intersection con- tained on the three hemispheres cut off by the three principal planes of the cone. First, consider the hemisphere whose base is perpendicular to the interior or principal axis of the cone, the figure Is then a closed curve, and may be called a spherical ellipse^ the foci of ■which are the points where the focal lines cut the hemisplicrc, having, It will be seen, properties In all respects corresponding to the foci of a plane ellipse. Secondly, consider the hemisphere whose base is the other principal plane perpendicular to that containing the focal linCs, the figure Is then composed of two halves of spherical ellipses, ■which may together be called a spherical hyperbola whose foci . lie within the concave portions, and it will be seen that sections of the sphere by tlic cyclic planes have properties similar to those of asymptotes. Thirdly, consider the hemisphere whose base is the plane sniERo-coNics. 237 containing the focal lines, the figure is then formed by two halves of spherical ellipses and has four foci and a centre where the minor axis of the cone meets the hemisphere. We shall consider a sphero-conic to be one of the first two of these curves, viz. the spherical ellipse or li^pcrbola. The curves in which a sphere cuts two reciprocal cones, of which its centre is the common vertex arc called reciprocal sjiltcro-comcs. The principal reciprocal property connecting the two may be stated thus : Every point of a sphero-comc is tlie pole of a great circle which touches the reciprocal spihero-conic. 358. The intersection of a central conicoid icith a concentric spJierc is a s])hero-conic. 1 and x^ ■{■ if ->r z^ = r^ be their cqua- ion lies on the cone [ar' -\)x'-\- {hr' - 1 ) / -f [cr - 1) ^^ = ; this cone is evidently concyclic with the conicoid. 359. We give below two or three of the numerous properties of sphero-conics, which arc the counterparts of properties of plane conies, each of which has its duplicate obtained by forming the reciprocal proposition. The proofs of these can be gathered from the previous articles ; but as exercises in spherical trigo- nometry the student may take almost any ordinary property in plane conies relating to foci and directrices, and to asymptotes of hyperbolas, which correspond to the cyclic arcs, and find analogues to them in sphero-conics ; he may also find equations corresponding to the jiolar equation of a conic or of a tangent to a conic, or of the auxiliary circle, or of the locus of the intersection of perpendicular tangents. A tangent to a sphero-conic makes An arc of a great circle which equal angles with the radii vectores touches a sphero-conic and is cut off drawn from the foci to the point by the cyclic arcs is bisected at the of contact (Art 3j'_'). point of contact. The sum or difference of two radii The sum or difference of the angles drawn from the foci to any point of which a tangent to a sphero-conic a sphero-conic is constant. makes with the cyclic arcs is constant. 238 SPHERO-CONICS. The first theorem is proved by limits as in plane conies. The product of the sines of arcs The product of the sines of arcs drawn perpendicular to the cyclic drawn from the foci at right angles arcs from any point of a sphero-conic to a tangent to a sphero-conic is is constant (Art. '6ol). constant. We give tlie following as an example of the mode of applying spherical trigonometry. 8G0. The locus of the intersection of i^erpendicular tangents to a sphero-conic is another sphero-conic for tchich the j^t'oduct of the cosines of the distances from the foci of the first sphero-conic is constant. Let tangents at P, P' intersect at right angles in (), 2a the major axis, 27 the distance of the foci S^ S\ /S'P=r, SP' = r\ SQ = p, S' Q = p\ L SQP= L S' QF = ir^\htxy cos27 = cosp cosp' + sinp sinp' sin2-»/rj and if ^;, ji be perpendiculars on PQ from S^ S' sin2J = s\n'\lr sinp, sin^/ = cos\/r sinp', and sinp sin//, being constant, is equal to sin (a — 7) sin (a -f 7); .*. cosp cosp' = cos27 - 2 sin (a — 7) sin(a -f 7) = cos2ot, from which the equation of the cone determining the sphero- conic may be easily deduced, viz. (,;« _ //) ^.2 + (,/ _. d') f _ (a' + h') z' = 0. This equation may also be obtained as follows : The e({uation of two tangent planes to a cone ax^+hf'+cz'^O, drawn through a point (^, 77, t) is (af 4 hv' + c^ (^^' + h" + ^'1 - i"^--^'- + h!/ + c^^T = 0, or [Ix + my + nz) [J'x + ///// + n'z) = 0, //' mm' _ nn' •'• ^^-'TW) ^ M^rN- af ) ^ c (./r+ hrf] ' and, when the tangent planes are at right angles, a {Inf + cD + h (rr + ^D + c (.rf + h-n') = ; rKOBLEMS. XV. (1) Prove tliat the equation of two circular sections of an ellipsoid in elliptic coordinates is a ' + a'" - ■^ w'a" = (a* - c*) .-^ , where r is the radius of each circle, d the distance of its centre from that of the ellipsoid, and 7; + a'r+r = 0, (I) b'^ + aif) + <•? + c" = 0, 244 UEXEKAL E(^UATIUN OF THE SECOND DEGREE. Considering |, 77, ^ as current coordinates, these three equa- tions represent three pLanes in each of which the centre lies. The three planes generally intersect in one point (I), but they may have one line common to them all (II), or they may all three coincide (III). I. In the first case, there will be one centre which may be at a finite (i) or an infinite distance (ii). i. If the centre be at a finite distance, its coordinates will be given by y ^ I «", c', h' X f + 5", Z>, a 0, and two similar equations. il. The centre will be at an infinite distance if any of its coordinates be infinite; thus \i ^ be infinite, A = ale - act'- - hh"' - cc'^ + 2a'b'c' = 0, and a"A + l"C' + c"B' must be finite; and we may notice that 7) cannot at the same time be finite, unless C : ^1=0, (Art. 3G4). II. In the second case there will be a line of centres, which may be at a finite (i), or an infinite distance (ii). i. The coordinates of the centre must be indeterminate, for which we have the conditions that A = and that the three expressions a" A + h" C ' + c'B\ a" C ' + h"B + c"A\ and a" B' -^ h" A' -\- c' C vanish, or a", «, c', h' h'\ c', A, a c", h\ rt', c I = 0. If A\ B', C be finite, Tp + TJ' + ^r = ^• ii. The line of centres will be at an infinite distance, (1) If the three ])lancs be parallel, and not more than two of them coincident ; the conditions for this are a c h' , c' h a -, = - = - and - =- = - c o a a c GLNKKAL IXilTATION OF TIIi: SEt'uND DEOKEE. '24') and that a'a'\ ////', c'c" shall not be all equal, hence in this case A\ B' and 6" all vani?"!). {'■?.) If one plane be at an Intiulte distance, and the other two be parallel oi' coincident ; in this case, if the first plane be that at an infinite distance, a, c', h' must all vanish and a" be finite, also -, = — and these must not be equal to „ if the two ' a c planes be parallel, but will be equal to -7, if they be coincident. (3) If one be indeterminate and the others parallel but not coincident ; suppose the first to be that which is indeterminate, (f, c', ?/, and a" must all vanish, and a'c'j cb" must be unequal. (4) If two be at an infinite distance, or if one be at an infinite distance and a second be indeterminate ; in this case all the quantities «, b, c, a, b\ c vanish except one of the first three ; if c be finite, a" and b" will be either one or both finite. Hence, for every case of (ii), A\ B\ and C all vanish. III. In the third case there will be a plane of centres, which may be at an infinite distance. In order that the three planes may coincide, we must have a c y a" , c b a b" -r = r = - = 777 and -=-=-=-; c a u a c c therefore all the minors vanish, and a'u" = b'b" = c'c". If the ])lane be at an infinite distance, all the coefllicients of u.^ must vanish, while one at least of «", ?/', c" is finite. 8G6. We have shewn that when A or //(«„) is finite, the terms included in ^{^ may be removed by transformation, without altering the directions of the axes, but that for every departure from the general case, in which there is a single centre at a finite distance, one of the conditions is that A shall vanish, and this condition is independent of all coefficients of u except those of terms of the second degree ; it is also the condition that the part ?/, containing the terms of the second degree shall be tiic product of two factors, real or imaginary, see Art. 88. So that^ in every case except where there is a single centre at a finite distance, by choosing coordinate planes, two of which bisect the 24G. GENERAL EQUATION OF THE SECOND DEGREE. angles between the planes it,, = 0, the general equation can be reduced to the form ^f + jz' -\- 2a"x + 2/3"?/ + 2y"z + 8 = 0. This is further reducible to ^/ + 72''' + 2a"a;+S = 0, by moving the origin in the plane of yz; and if a" be not zero, this finally reduces to /?/ + 72' +2a"u; = 0, or to /9?/ +73' + 5 = if a" = 0. If the two factors of u,^ be equal, the equation is reducible to 72;'' + 2a"x = or 7r + 8 = 0. 367. The loci of equations of the second degree may there- fore be classified according to the nature of their centres. I. Single Centre. i. x\t a finite distance. Ellipsoid. Hyperboloids of one and two sheets. Cone, real. Cone, imaginary (or point-ellipsoid), ii. At an infinite distance. Paraboloids, elliptic and hyperbolic. II. Line of Centres. i. At a finite distance. Cylinders, elliptic and hyperbolic. Line cylinder (limit of ellip. cylinder). Two planes, intersecting (limit of hyperb. cylinder), ii. At an infinite distance. Cylinder, parabolic. III. Plane of Centres, i. At a finite distance. Two planes, parallel (limit of parab. cylinder), ii. At an infinite distance. Two planes, one at an infinite distance. Two planes, both at an infinite distance. - 368. AVe have now to shew that it is always possible to choose such directions of the axes, that the transformed equa- tion shall contain no terms involving i/z^ zx, and .ry, the axes being in both cases supposed to be rectangular. CEXEUAL EgL'ATIUX OF TIIK SHCONlJ DECKKi:. 247 3Gt). Since our objects in this chapter arc, either to deter- mine what kinds of surfaces can be the loci of the general equation ; or, given a particular equation, to identify the sur- face which is its locus, we may avoid complications by con- sidering that if only one of the rectangles, say xy.^ appear in the equation, wo can by rotation of the axes of x and y make this term disappear, so that the equation will be reduced to the foi-m ax' + ^y' + r^£' + 2a' X + 2^"y + 2i'z + S = 0, and the nature and position of the locus will be at once de- termined. In dealing with the general ease we shall not therefore always examine the particular modification of the formulse which would be required if two of the three quantities a', h' and c were to vanish. 37(X To sluw tliat u^ can ahcays be reduced to the form ax^ + Py' + 'yz' by transformation of coordinates, a, ^ and 7 beiny real quantities. The quadric h^x? + y^ + z') — v.^ will become the product of two linear factors, real or imaginary, if /« satisfy the equation (/, _ a) [h - b) {h - c) - a" [h -a)- b"' [h -b)- c" [h - c) - 'ia'b'c! = (Art. 88). Since the equation is a cubic, one of the values of h must be real, and for this value h [x^ -\- y' + z') — u.^ = Qt is the equation of two planes which, whether real or imaginary, have a real line of intersection. Let this line be the axis of z in a new system of coordi- nates, so that Ii [x' + y^ + z') — u^ becomes Ax' + 2Bxy + Cy'^ on transformation, and the term xy may be made to disappear by simple rotation of the axes of x and y* Hence, referred to these new axes, zi,^ would be reduced to ax' 4 /By'' ■+ 73*, in which a, /3, 7 are real, although any one or two may vanish, the corresponding cubic being (/,_a)(/,_/3)(A-7)=0. ♦ This method was adopted by Archibald Smith in his Notes on tl»e " Undulatory TlicoiY of Us;ht:'—C'im'>. Maih. .hmr.. vol. r.. ji. .1. 248 GENERAL EQUATION OF THE SECOND DEGREE. 371. Tlic cubic given in the last article is called the dis- criminating cuhic^ the coefficients of which, as -vve have seen ia Art. 152, are invariants. Since the last term is — A, it follows that whenever all the roots of the discriminating cubic are different from zero, the locus of tlie general equation is a central surface. 372. To separate the roots of the discriminating cubic. The discriminating cubic is (70 - [h - a) [h - h) [h -c)- cr- {h - a) -...= 0, and — , c respectively; hence, since rt'A= B' C — AA\ we obtain by this substitution a' 4) (//) -^ [ah + A) [{h - h) \h -c)- a'] - [h'h + B') [c'h -f C) ; , ., .^ , , A' B' C and, it we write X, //, v tor r , rr , r ? ' a h c 4> [h) = {h - X) {{h - h] [h - c) - a'-^} - '' Jf ' [h -f^){h-y)- (1 ) therefore

[h] = [h - a) [h - h) [h -c)- h-' [h - h) - c" [h - c) ; tlicrefore (co ), (i), ^ (c), ^ (— cc ) are -f — | — supposing J > c, and the roots will be separated by h and r. Cauchy's method of separating the roots is given in Todhunter's Theory of Equations^ in the chapter on Cubics. 373. To find the conditions that the discriminating cubic ma>/ have equal roots. In the case of two equal roots, suppose yS = 7, thou n, can be derived by transformation from a./- + /S (/ -}- .r) or [a - /9) x' f /3 {.>■' + / + .:-, ; UENEIJAL E(iUATION OF THE SECOND DEGUEE. 24U .-. f/._. - (a - /S) [Ix + my + nz)' + /3 (u;' + 1/ + z') ; .-. a = (a - ;S) /"^ + /9, a' = (a - -S) »»*, /> = (a-/3)m-^ + ^, U ^[a.-iB)nJ, a ., //o' , cV/' a'/)' .'. p = a r = , r = c r • a u c These arc obviously tlie coijJitioua that the coijlcolJ may be one of revolution. If all three roots be e(iual, u.^ must have been a{x^ + y^ + z'^) before transformation ; therefore a = b = c and a =b' = g' = 0. o74. i\nothcj' form of the eonditions for two equal roots may be obtained ; for {jS - a) {^~c) = b"' and (/3 - a) {^-b)= c"' ; ... ^/3-a){b-c)=b'-'-c''', J/-' _ r" , c"- a'- . a" - b' a .-. ^ = a + =6+ =c+ , , a — b and we may observe that, if a' = 0, b' or c' = 0, ai)d if a, d be the two whieh vanish, /3 = //. 375. To find the equations of the coordinate axes lohicli inahe the terms in u,^ invoicing t/z, zx^ xy disappear. When u,^ has been reduced by transformation to ax^+^y'^ + yz^ one of the new axes is the intersection of the two plaijes whoso equation is, referred to the original axes, u^ - h [x^ + y'^ + z'^) = 0, where /< = a, /S or 7 ; therefore, by Art. 89, the equations of the axes are found by writing a, /i, 7 successively for h in x [b'c — [a - h) a] =y [c'a - [b - h] b'] = z [nU — (c — h) c\. These equations do not give the position of the axes directly if two of the three quantities a', b\ c vanish, but, if a', b' be the two which vanish, it is obvious, from the original equation, that the axis of ~ will be in the direction of one of the axes. Iv K 2oO GliNERAL EQUATION OF THE SECOND DEGREE. .376. The direction-cosines of the axes can be symmetrically expressed in terms of the roots of the cubic (/> [h) = 0. For [a'a + A'] {{a-h){a- c)-a"'] = {b'a + B'] (c'a+ C) (Art. 372) ; therefore [a'a + A'f [(a - l) [a — c) — a"^] is a symmetrical function of the coefficients, hence, if /, ?«, n be the direction-cosines of [p. - 1>) [a - c) - a"' {aL-c)[oi-a)-b"' [a - a) {a. - h) - c"' 1 1 (j>\a) (a-/9)(a-7)' AVe give also below the method of determining the directions of the axes by means of the definition of a principal plane. ,377. Tojind the equation of the locus of middle jwints of a si/stem of parallel chords of a conicoid determined by the general equation. Let the equation of the conicoid be f{-r^ ?/, z) = 0, and let (X,, /u., v) be the direction of the chords to be bisected, (^, ?;, f) the middle point of any chord. Then the equation /(| + Xr, t? +yur, ^-\-vr) = must have its roots equal and of opposite signs. This gives the condition , df df df , d^ drj dX or (r/f + c't) + b'K) ^ + [c^ + I'V + «T) y^ -f {l>'^ + (ii) + cX) V = 0, ■which is the equation of the diametral plane. .378. To determine the j^rineipal planes of any conicoid. A principal ])lane being perpendicular to the chords wliicli it bisects, we shall have the direction-cosines given by the thi'ce equations (i\ + (•' jX + b'v = .S'X, r'A. + A/i + «V = .s/i , (1) b'\ + «'/x 4- cv = sv^ where .s is a constant given by the cubic (.• - a) {s - b) (,v - c)-V'= (>, J GKNF.RAL Kca'ATION OF THK SF.COND DKC 1;F,1:. 251 tlic discriminating cubic wliicli has been already discussed. Since to each of tlie three vahies of s there corresponds one system of values of X : /i : v, there are, in general, three and only three principal planes. If, as in the case of a surface of revolution, there are an infinite number of values of \ : /x : v, we obtain from equations (1) a- s c h' , c' h — s a - , = , = - , and Y'= . = — ; e — s a b a c — s h'c J c'n ah' .-. s = a 7=0— J , = c - , , as in Art. 3 1 3. a b c 379. To shew tltat the three lyrincipal 2y^(tnes of any conicoid are mutiiaJhj at rujht angles. Let ., -}- Z>V, = .'?|X^, c'X, + hfi^ -f aV, = s,/i,, (1) Z^'Xj + (/'/X| 4- CV| = s^v^. ^Multiplying by X^, /x.^, v^ and adding, we obtain («X,^ -I- c>.^ + t'vj X, + {c'\ + hfi^ + aVJ /x, + {h'\ + «>.^ + cv,) v^ = 5, (X^X,^ + ^,^^+v,vJ; .-. s\^ . X, + s,^^Jb^ . fi^ + s,,v,^ . V, = .?, (XjX^ + ^l^f^^ + v,vj, whence (./^ + yz^ -{■ 8 = 0, in which a, /3, 7 are the roots of the discriminating cubic, for, on ti-ansformation, the coefficient of x" will be aX,' + hfi^' + cv^' + 2a'fj,^v^ + -^'»',^, + -t"'^,/*, and similarly, /3 = s , y^s^. 252 GENERAL EQUATION OF THE SECOND DEGREE. 380. To distinguish the surfaces represented hy an equation for which the roots of the discriminating cubic are finite. In this case there is a centre at a finite distance, to which if the origin be transferred, the direction of a new system of axes can be clioscn (Art. 3tO), such that u ^ ax"" + hf -t- cz" + dw' + 2a' yz + 21' zx + 2c xy -|- 2a" xw + 2l" yio + 2c"zio = 0, will become by transformation ax^ -\- ^y^ -\- 72;"''+ 8m?''' = 0, lo being written for the unit. The transformation will be effected by substituting Ix + my + nz + ^w for x^ and similar expressions for y and 2, w being unchanged ; the discriminants, being invariants, are therefore equal,* since the modulus of transformation L m, n, I n\ n\ 0, 1 .'.^[^c) = h\ a" a', h" c, c" rt", h", = a/37S a, 0, 0, 0, ^, 0, 0, 0, 7, 0, 0, 0, 8 a^7 * Hence, we have the following table for the case in which 0/37 or A is finite, and a > /3 > 7, by which it may be seen how the loci are distinguished : ax" + ^y' + 72' = a ^ 7 //(«) + + + + + + Ellipsoid Hyperboloid, one slieet + + ~ - - Hyperboloid, two sheets | - Cone, real i + + + Cone, imaginary, or jwint | + + + + Imaginary locus | ♦ Salmon's Higher Alf/cbra, Arts. 118 and 23, GENEKAL EQUATION OF THE SECOND DEGKEE. 253 In order that o£, /3, .iiul 7 may be all positive, a + h + c and A must be positive. If the locus be a point, or rather an in- definitely small ellipsoid, the section of i(^ = by each coordinate plane must be a point-ellipse ; therefore each of the quantities hc^a'^j ca — />'•, and oh — c"^ must be positive. The conditions for surfaces of revolution are obtained in Art. 373. ' 381. To distinguish the surfaces represented hy an equation^ for lohich one of the roots of the cubic vanishes, and the centre is single and at an infinite distance. The conditions that the centre may be at an infinite distance arc that A = 0, and that one or more of the three quantities bch)\v shall be finite, a" A +Z'"C"4 c"B'^ a"C'+h"B -t-c"^', a"B' + h"A' + c"C. The surfaces will be the elliptic or hyperbolic paraboloid, according as the roots of Ji^ - {a-\- b + c) h+ A + B+ C= 0, have the same or opposite signs, i.e. iis A -\- B -]■ C is + or - ; but, by Art. 363, A, 7?, C have the same sign, hence A, B, and C + gives an elliptic paraboloid, A, B, and C— „ hyperbolic paraboloid. 382. To distinguish the surfaces represented by the general equation when there is a line of centres at a finite distance. The conditions that there may be a line of centres are A = and II{u] = - [a" ^f[A) + b" ^/[B) H- c" V(C)}' = 0; or, if A\ B\ C a" b" c be finite -p + 77- + Tt =^- The equations of the line of centres arc A'^ — a a" = B'-ij — b'b" = C'^— c'c" = p suppose ; therefore, if \ic transfer the origin to any point in the line of centres de- fined by some value ofp, the equation of the surface will become ti., + a"^ + b"r) + c"^+ fZ= 0, an"' h'b"' c\r , ^ since the cocfiicicnt of p vanishes. 254 GENERAL EQUATION OF THE SECOND DEGREE. ir a', h',c' he all finite, A', B', C for instance, vanish, the other two being finite, yl' will be finite, and, by Art. 363, if B' = 0, then A = 0, and C' = 0, and h and c vanish ; hence, recnrring to the original equations for determining 1 -1 1 • 1 • 2a" b" ab"' , ^ the centre, wc easily obtain the equation u^ , H ^ +«= 0, and the condition b'b" = c'c". If two roots of the discriminating cubic be finite, since ii^ is reducible to the form ^7/ + jz\ the surfaces represented by the equation will be in the general case in which a'a"'^ b'b"'^ c'c"'^ 7 • ,- • — .r -f -w + ,v + " '^ tiuitey A, B, and (7-f , an elliptic cylinder, Aj Bj and (7-, a hyperbolic cylinder; when ., +...= 0, A, B, and (7 + , a line cylinder, A^ B^ and C— , two intersecting planes. If only one root be finite, 11,^ is reducible to yz'^j but in tJiI« case, since A + B + C=^0, A, 7?, C, being all of the same sign, must vanish separately, from which it follows that A\ B\ 6" also vanish, and there cannot be a line of centi-es at a finite distance. 383. To di'sti'nfjuish flic surfaces wJien there is a line of centres at an infinite distance. In this case A' ^B'= C = ; therefore A = B=C=0\ two of the roots of the discriminating cubic must therefore vanish ; also a'a!\ b'b", c'c" must not be all equal. Since aa = b'cj &c., u^ can be put into the form y z " + ;? + ? and the only surface represented Is a parabolic cylinder. 384. To distinijnish (Tie surfaces for icJtich there is a plane of centres. GENERAL Ec/L'ATION OF THE SECOND DEGl^EE. 25.3 In this case, as in tlie last, the minors all vanish, and wc have in addition a a" = h'b" = c'c" ; the equation Jnay therefore be written „, , fX 7/ Z\'^ , ,, fx 11 Z\ ahc[-+ 4 + - +2a'«" - + •/-, + -) + (Z=0. \rt be) \a c J The surface repi-escnted consists of two parallel planes unless aoca = a'a ", or d= - = , = -~ . \n whieii case they are ' a b c ^ ^ coincident. One of the planes will be at an infinite distance if a, b^ r, a', b\ c all vanish while one at least of the other quantities remains finite. 385. The results In the general case may be tabulated as -\- (L and v for ' "2 T 't'li r u -r 1 • ^ a a bo follows, if V be written for — yr + —frr + A B (a'a" - b'b"f + [b'b" - c'c")'\ where / denotes ' finite' and a, /9, 7 are the roots of -the discriminating cubic, a > /3 > 7. y A a /3 y i/M^^^'i ^ "■1 + + + + + - i 1 Ellipsoid + + + + ' Hj-perboloid, one sheet - - - 1 Hyperboloid, two sheets + + + ^1: 1 Cone, real + + + + 1 Cone, imaginary, or point-ellipsoid + If-] f 4- 1 Paraboloid, elHptic + ' - Paraboloid, hyi>erb<^)Hc 1 + + 1 + ' / Cylinder, elliptic 1 + + ' 1 + 1 Cylinder, line + - ! - f 1 Cylinder, hj-perbolic + — i - Planes, intersecting + i ■ / i Cylinder, parabolic « + ; 0, , Planes, parallel a"' //'* c"' Fur coincident i)lanes d= = , a b 256 GENEllAL EQUATION OF THE SECOND DEGUEE. For two planes, one at an infinite distance, a, J, c, a', h\ c' = 0, one at least of a", h'\ c" finite. For two planes at an infinite distance, d alone finite. 38G. Pi'OCf'Sses for Jindinej the locus of any given equation. When a particular equation of the second degree is presented to us, in order to discover what species of surface it represents, we would recommend the student first to form the discriminating cubic, and it will then be seen whether the last term A vanishes or not. I. If A be different from zero, we must find the centre, transfer the origin to it, and by changing the directions of the axes reduce the equation to the form ax^ + ^y'' + 72"^ + 6 = 0, where a, /S, 7 are the roots of the discriminating cubic, which can always be found approximately, at all events their signs can be determined by Des Cartes' rule ; and 5 has been shewn to be — -^- , or in particular cases may be found more easily without the use of the determinants. II. If A = 0, and A-\- B-\- C be not zero, In which case the two roots /?, 7 will be finite, it will be best to determine the directions of the axes which correspond to the three roots 0, /9, 7, And to suppose the origin so chosen that the equation becomes leithqr ^/ + 7.t;-^+2a"./; = 0, (1) or ;3/ + 7-"'+S = 0. (2) If we do not require the position of the vertex, the value of a" in (1) can be found by equating the discriminants, by which we obtain ^ja"^ = — j (a"A +h"C' + c"B')^. and if we find that a" vanishes, we take case (2). ]>ut if in case (1) the coordinates |, ?;, ^ of tlic vertex be required, we think that the best method is to transfer to this })oint, and observe that the result must be GliNKKAL LQUATION UF Till: SiaoND DKiiUKI-:. L'-x if (/, 7?/, «) be tlic direction of the new axis of x ; so that Avc obtain the equations for detenninln<^ ^, ?;, ^ and a", ''^ +077 + h'i^ + a" = la.", c'^ + hi] + rt'^ + h" = ))ioi", h'^ + at] + c^ + o" = »a", «"| + J"7; + c"^+ c/ + (/| + m-n + w^) a" = 0. From the first three equations a"A + h"C + c"B' = (LI + mC" + 7«S') a"., which gives a" without ambiguity, and the fourth equation with two of the former determin<3s f, ?;, and f. If a" = 0, we shall have in case (2) three equations, equivalent to two Independent equations, which will determine the axis of the surface, and we can obtain 8 from a fourth equation com- bined with two of the former; thus eliminating ^ and t] from the last three, h [a'c — hh') = l>" {b'b" — a' a") + c" [ba" - c'b") + d [a'c - bb'), since the coefficient of ^= a" A + b"C' -f c"B' = 0, when a" = 0. Tims the position of the locus Is determined completely in both cases. III. If zl=0 and A-\-B+ (7=0, the equation is reducllle to one of the forms yz'^ + 2a"x = (1), 2 + 18 = 0. The discriminating cubic is (A -32; (A- 1/-9;A-32)-128(^-1)-6.G4 = U, the last terra of which is and the roots 0, 36, — 2. The equations of the axes are known from [64 + 3 (A - 32;; X = [- 24 - 8 (/* - 1)) 3/ = [- 24 - 8 (A - 1;| z, which give 1x = y = z\ x = — iy = - -iz ; x = 0, y = z\ and fhe direction-cosines arc U, ?, \U (§V2, -Jv2, -iv'2), and 0, i , 2, - J n^2). OlINERAL EQUATION OF THE SECOND DECSREE. 2')d The equation being reduced to 36/ - 2z' + 2a".r = 0, by the equation of invariants -i(3.84G.lG + G.lG)' = -36.2a"'; .'. a" = ± 0. If (^, v^ ^) be the vertex referred to the original axes, when we transfer to this point the equation must reduce to ».^ + 2a"(i.c + iy+^^)=0; ... ?,'2^-Hr)-S^-3 = ^0L" - y|+ 7; + 3r-G = §a" - 8| + 3»7+ ^"-G = ?a"; .-. - 3-2.6-2.6 = 3a"; .-. ct" = - 9, and since the constant terra vanishes K(l + 2'7 + 2?)-3|-G;;-GC+lS = 0, or | + 2»; + 2^=3; 389. To find the surface ichose equation is x' - 2if + 2z^ -\-3zx-xij-2x-\-ly-bz-Z = {). The discriminating cubic is (A-l)(A^-4)-f(/. + 2)-i(/.-2) = 0, roots 0, '^-''^^ , H [u) = _ i [Aa + C'b" + B'cy = 0, and the equation is reducible to 3V(3)-H ,_ iV(3V-_l ,, , g^^, 2 ^ 2 " The equations determining any point (|, tj, ^) of the axis of the surface are - ^-l'? +7 = 0, (1) 3^ +1^-5 = 0, which give the straight line ^=-lr; + 7 = i(-ir+.>,; (2) 2Gi) GENERAL EQUATION OF THE SECOND DEGKEE. tlierelbrc multiplying the equations (1) by |, ■»?, t, and adding Tlio equation therefore represents two intcrseeting planes whose line of intersection is given by (2). 390. To find the axes of the conical envelope of a central couicoid. The equation of tlie cone referred to its vertex as origin is o- [ax^ + hjf + cz') = [afx + hgjj + chz)\ a being written for af'^ + hif + cJi^—lj the discriminating cubic in this case is [s - ca f af'] [s - ah + h\f) [s - ac + c'^h^) - b'^/h'' (s - <7a + df) - c'aVif {s - ah + hy) - a'hfY {s-ac + cT) + 2a'hVfyh'' = 0, or writing s^, .9^, s,. for s — aa^ s — ah, s — ac a'r h' direction-cosines of axes in the ratios {1, ± V('i) - 1. 1). (1. 0. - J)- 4. Hyperbolic cylinder. 5. Hyperboloid of one sheet, centre (J, ", - "/). 6. Cone, direction-cosines of axes {0, i v'(-). - ^ V(2);, {+ i \/(2), i, h]. 7. Ellipsoid, point or impossible according as t/ < = > 55, 8. Hyperbolic cylinder. (2) The equation Ix* + 6y* + 4s' - 7yz - llsj - 7x^ = a* represents an hyperboloid of one sheet whose greater real axis makes with the axis of s an angle tan"' ^,'('2). (3) The equation ax' 4 4^' 4 Uz^ t 12/yr i- 6zx 4 Ix// f 2'j".r ^ 2h'y 4 2c"z t = < 1. What surfaces will it re- present in the following cases : i. 36" = 2c", a> = y) sin 0, with similar clianges for y and s. Hence, defining an axis of a conicoid as a diameter, such that by revo- lution about it through two right angles every point of the surface returns to the surface again, deduce the ordinary cubic equation for the determina- tion of the axes. CIIAPTP]R XVII. .DEGREES AND CLASSES OF SURFACES. DEGREES OF CURVES AND TORSES. COMPLETE AND PARTUL INTERSECTIONS OF SURFACES. 391. Having already fully investigated the nature of the surfaces represented by the general equation of the second degree, we will proceed to the loci of equations of higher degrees, which we may consider as equations either in three-plane or four-plane coordinates : in the latter case we may suppose the equations homogeneous, without loss of generality. 392. Surfaces which are represented by rational and integral algebraical equations are arranged according to the degrees of these equations when plane coordinates are used, and according to classes when tangential or point coordinates are used. A surface is of the ?«'" degree when the equation of which it is the locus is of the n^ degree in the coordinates of any point of the locus; the geometrical equivalent being that a surface is of the w" degree when an arbitrary straight line intersects it in 7i points, real or imaginary. A surface is of the n"^ class when n tangent planes, real or imaginary, can be drawn to it through an arbitrary straight line. If p\ (?', /, s and ^/', q\ ?•", s" be the point coordinates of two planes, the coordinates of any plane passing through their line of intersection will be Ijj -\-m2)", Iq -\- mq"... [Art. 128], I : m being an arbiti'ary ratio, and the particular planes which touch a surface whose tangential equation is ^(p, i?, »*, s) =0, supposed a homogeneous algebraical equation of the ri"> degree, will be determined by the values of I : m which satisfy the. equation i^(/// + 7/y/', ...) = 0; the number of values of the ratio will be 72, and this will therefore be the class of the surface. M M 2GG COMPLETE AND PARTIAL 393. Curves and Torses arc arranged according to their degrees. A curve of the n^^ degree is one wlilch intersects an arbitrary plane in n points, real or imaginary. A torse of the n^^ degree is one to which 7i tangent planes, real or imaginary, can be drawn through an arbitrary point. Other classifications of curves and torses will be explained hereafter. 394. Among the various methods of treating of curves which have been proposed, one is to consider them as the inter- section of surfaces whose equations are given. In this method the difficulty arises, to which allusion has been made (Art. 13), viz. that extraneous curves may be introduced which are not the subjects of investigation. If any curve be supposed to be given in space, it is impossi- ble generally to determine two surfaces which shall contain no other points but points which lie on the proposed curve ; but among all the surfaces which may be drawn through a curve, it is desirable to obtain the simplest forms of surfaces of which the curve shall be the partial intersection. 395. The number of points in which three surfaces inter- sect, which are of the m^\ n*'>, and p^^ degrees respectively, is vin2)j unless they intersect in a common curve, in which case it is infinite. For the proof of this proposition, the student is referred to Salmon's Treatise, on Higher Algebra^ Lesson VI ir., on the number of solutions of three equations in three unknown quantities. The student may be able to satisfy himself of the truth of tlic proposition, by considering that the number of points in which tlie surfaces intersect will, by the law of continuity, be unaltered, if we substitute particular instead of the general forms of the surfaces. If the surfaces respectively consist of 7», ??, ^) arbi- trary planes, it is obvious that the number of their common points of intersection will be ?»??;>, each point being the inter- section of three planes, taken one from each system. iNTKKsix'TioNs OF si:ia\u'ic><. 2G7 390. Tlte comphte intersection of two surfaces of l/ie 7/i^'' and n^^ degrees respect i veil/, is a curve of the tnn^^ degree. Let a plane intersect the surfaces, the number of points of intersection of the pLine with the surfaces is mn, this is there- fore the number of points in which the phuie cuts the curve, and the curve is of the mn^^ degree. 397. To find the number of conditions ivhich a surface of the n^^ degree may he made to satisfy. The number of constants in the general equation of the »^'' degree is evidently the number of homogeneous products of four things of n dimensions, and is therefore ^ 4.5...(4 + n-l) ^ (/i + l)(n + 2)(» + 3) ^ 1.2. ..n ~ 1.2.3 ' but in estimating the number of constants with reference to the number of conditions which the locus can be made to satisfy, we must diminish this number by one, since the generality of the eciuation is unaltered if we divide by any one of the constants. The number of disposable constants, so obtained, is ( ;^ + l)(n + 2)(n + 3) _ (n^ + 6n +ll) ^ , 1.2.3 ~ ' 6 -^^ '' Thus <^(2)= 9, <^(3) = 19, (5) = bb^ (f> (G) = 83, and so on. Since, when a point is given, we may substitute its coordi- nates in the general equation of a given degree, and thus obtain a linear equation of condition between the constants, a surface of the third degree may be made to pass through 19 arbitrarily chosen points, and one of the fourth through 31, &c., and cj) [n] arbitrarily chosen points will completely determine the position and dimensions of a surface of the 7i^^ degree. A surface of the n^^ degree is also determined by (f) [n] inde- pendent linear equations of any kind between its coefficients. 398. All surfaces of the n^^ degree lohich pass through [n] — 1 given p>oints have a common curve of intersection. If ?f = 0, V = be the equation of two surfaces passing througli the given points, \u-\- fiv = Q will be the equation of another 268 COMPLETE AND PARTIAL surface of the ?i''' degree which passes through the ^(z^ — 1 given points I and since, by giving proper values to the ratio A,: yu, this surface may be made to pass through any additional point which is not common to the two surfaces u = 0, u = 0, this equa- tion will be the general equation of all surfaces which contain the ^ («) — 1 given points. But this equation is also satisfied by the coordinates of all points which lie on the curve of in- faces containing the [n] — 1 points be given, it will be possible to eliminate from the general equation of the surface of the n*^ degree all the constants but one, which will enter iflto the resulting equation in the first power only. This equation will then be of the form xi + X?; = 0, where m, v are of the ?i'" degree, and A, an undetermined constant. All surfaces represented by this equation will pass through the curve given by the equations which curve is therefore completely determined. For example, eight points determine a curve which is the complete intersection of two conicoids. In the case of complete intersections of surfaces the nature of INTERSECTIONS OF SURFACES. 269 the curve is not given when the degree is given, except in the case of prime numbei's, when it must be a phinc curve. For example, a curve of the twelftli degree might be the com- plete intei-sectiou of pairs of surfaces of the degrees (1, 12), (2, 6), (3, 4), and these different species, belonging to the same degree, would require a different number of given points to determine completely the surfaces. The following proposition serves to obtain the number of given points sufficient to determine a surface of the «'" degree which, bj its complete intersection with a surface of a lower degree, gives a curve of the yj/^'" degree : this is given by Pllicker, but may also be proved directly by a theorem given by Cayley.* 402. All surfaces of the n^ chgree lohich pass tlxroufjh given points of a surface of the [n)-cf>{p)-h lie on a surface of the rj^^ degree, where n =p -f q^ whose equation is My = 0, then u^u^ = will be one of the surfaces which contain the ^ (h) — 1 points, and may be obtained by giving a certain value to the ratio X : fi'm the equation \u + /it' = 0, so that \u + fivE. ? {n — q) — 1 points be taken on any fixed surface u^ = 0, all surfaces of the 7i"* degree, which pass through these points, will intersect the surface of the q^^^ degree in the same curve. Thus, if«^ = l, the proposition is reduced to the following:. All surfaces of the n^^ degree which pass through !« (« + 3) ♦ Xouvclles Annaks, xii., p. 3DC. 270 CoMPLKTt: AM) PARTIAL given points in a plane determine a curve of the 7i^'' degree in that plane. If 2' = 2, the proposition becomes : All surfaces of the ?«'^ degree which pass through n [n + 2) points on a conicoid intersect the conicoid in the same curve. 403. When it is said that a curve or surface is determined by a certain number of points, tliese points must be supposed arbitrarily taken, for it is possible so to select the points that this number would not be sufficient. Thus, a plane cubic is generally determined by 9 points, but, if those be the nine points of intersection of two of such curves, an infinite number may be drawn through them. A curve of the fourth degree of one species can be determined completely by 8 arbitrary points, but If these given points be the intersections of three conicoids which have not a common curve of intersection, taking these surfaces two and two, we may obtain three curves of that species passing through the same eight points. 404. If two surfaces of the n'^* degree pass through a curve of the nr^^ degree situated on a surface of the r^^ degree, they will also intersect in a curve of the n [n — rY^ degree, situated on a surface of the (n — r)^^ degree, because one of the surfaces which passes through the intersection of the two w*'*^ surfaces will be the complex surface formed of two of the degrees r and n — r respectively. Thus, If a conicoid intersect a cubic surface In three conies, the planes of each of these will Intersect the cubic In a straight line, making the complete Intersection ; and since the three plaucs form a cubic surface, part of whose curve of intersection with the original cubic surface lies on a conicoid, the three straight lines will He in one plane. 405. The theory of imrtial intersections of surfaces was first discussed by Salmon.* AVithout an examination of such partial Intersections it Is not possible to analyze different species of curves of the same degree. If we considered only complete * Qiuirtcrli/ Journal, vol. v. INTERSECTIONS OF SURFACES. 271 intersections of siu'ftices, curves of the third dcgTce couhl only be considered as phanc curves, whereas it will be seen that thej may also be imrtial intersections of conicoids. 406. Tojiml the surfaces oj xcliicli a given curve is the partial intersection. In order to find the surfaces which may contain a curve of the m^^ degree, it is observed that through ijc) points a surface of the U^ degree can be made to pass. Now, the total number of points which are common to a proper curve of the m^"^ degree and such a surface, supposing the curve not to lie entirely on the surface, is mlc^ since this is the number of points in which h planes intersect the curve ; and the law of continuity makes the statement general. If [h] = ink + 1 , one such surface can be drawn containing the curve ; if mk+ 1, two surfaces of the k^^ degree can be drawn, and therefore an infinite number. Thus, for a curve of the tliird degree, if Z- = 2, ^ (1-) = 9 > 3.2 -f 1, hence an infinite number of conicoids may be drawn containing any curve of the third degree. When ^ (Jc) —7nJc + 1^ one surface of the A;'^ degree contains the curve, and another of the k-\- If must also contain it, for (f)[k+l)- <}) [Jc]^ = -I [k -i 2) {k + 3), therefore ^•, all the rest can be found. 407. The number of arbitrary points through which a curve of the m^^ degree can be drawn cannot exceed a certain superior limit which is easily determined, for suppose k arbitrary points to be given, and a cone to be constructed containing the curve, and having its vertex in one of the assumed points, the degree of this cone will be wi — 1, since any plane through the vertex must contain m— 1 points of the curve besides the vertex, and therefore 7)1 — 1 generating lines of the cone, and the number of its gene- rating lines sufficient for its complete determination is the same as that of the number of points necessary to determine a plane c xi 71th 1 . wi (m + 1 ) . curve or the m—l\ degree, viz. — ^— — 1. The greatest value of h for which such a cone can be con- structed is — ^-- — ; this is therefore a superior limit, although lower limits to the number h may be obtained in general from other considerations. Thus, a curve of the third degree cannot be made to pass through more than six arbitrarily chosen points. 408. Jf {n) - 2 given planes, they will touch n'' - (f> («) + 2 additional fixed planes. Similarly for other theorems. In illustration of the points which have been considered In this chapter, relating to the intersection of surfaces, we give here some elementary properties of cubic and quartic curves. Cubic Curves. 413. If two conic.oids have a common generating line, any plane which does not contain this generating line will intersect the two conicoids in two conies which have four points in common, one of which will be in the generating line ; hence the curve which with the generating line forms the complete inter- section of the conicoids, being met by an arbitrary plane in three points, is a curve of the third degree; such a curve is called a cubic curve. Conversely, if we take any seven points upon a given cubic curve and an eighth on any chord of the curve, we can make an infinite number of conicoids pass through these eight points, which will have for their common curve of intersection the cubic curve and the chord, for each conicoid meets the curve in seven points and the chord in three, and therefore contains both entirely. 414. A cubic curve^ ichieh is the intersection of ttco conicoids having a common generating line^ intersects all the generating lines of the same system as the common line in tu-o iioints^ and those of the opposite system in one point only. Call the two conicoids A and B^ and the common line L. Any generating line of A intersects B in two points, neither of which will lie on Z, if it be of the same system as Z, but one will lie on Z, if it be of the system opposite to that of L] but the points which do not lie on L must lie on the cubic curve, which proves the proposition. 276 QUARTIC CUKVES. 415. The common generatinfj line of two conicoiJs ivJtich determine a cubic curve is twice crossed by the curve. A plane which contains the common generating line intersects each of the conicoids in a generating line of the opposite system, and these two lines intersect in one point only ; but the plane contains three points of the curve ; hence two of the three points must lie on the common generating line. 416. When tico cubic curves lie on a given conicoid^ to fad the number of points in which they intersect. Each of the cubic curves is the partial intersection of the given conicoid with another which has a common generating line with it. Call the three conicoids ^1, Z?, and B\ and the curves C, C\ and let the complete intersection of A and B be the curve G and the line Z, and let that of A and B' be the curve C" and the line U . The eight points which are common to A^ i?, and B' must be the intersections of the complex curves CL and C'L ; and two distinct cases arise according as Z, U are of the same or of opposite systems. If they be of the same system, L will meet B' in two points both of which will be on C" ; U and G will intersect in two points, therefore G and G' will Intersect in four points. If they be of opposite systems the two points in which h intersects B' will lie one on JJ and the other on G' ; hence Z, Z' ; Z, C" ; and Z', G will intersect in three points, and therefore G and C" in the five remaining points. Quartic Gui'ves. All. The intersection of two conicoids is a quartic curvcj since a plane must meet the two conicoids in two conies which intersect in four points ; but this is a particular kind of quartic curve. An arbltraiy quartic curve will intersect an arbitrary conicoid in eight points, and only one conicoid can be con- structed which will contain nine points of the curve, and therefore th» entire curve. QUARTIC ClIUVKS. 277 The general (juartic curve may tlicrefore be considered as the partial intersection of a conicoid and a cubic surface drawn througli thirteen points of tlie curve, and the remaining portion of the complete intersection must be either (i) two straight lines which do not intersect, or (ii) a conic which may be two inter- secting straight lines. i. In the first case a generating line of the conicoid which is of the same system as the two straight lines common to the two surfaces, meets the cubic surface in three points which must be on the quartic curve, while one of the opposite system meets the cubic surface in one point only, besides the points in which it cuts the two common lines, and therefore intersects the quartic curve once. ii. In the second case every generator of the conicoid meets the common conic in one point, therefore two of the three points in which it intersects the cubic surface lie on the quartic curve. If i<2 = be the equation of a conicoid containing the quartic curve, If, = that of the plane of the common conic, the equation of the cubic surface must be of the form i\u.^-\- u^v^ = 0, and the quartic curve, in this case, must be of the particular kind which is the base of a cluster of conicoids, viz. that determined by the equation \i(.^ + fi\\ = for all arbitrary values of \ and fi. 418. To find the number of jyoints of intersection of two quartic curves which both lie on the same cubic surface. Let the surface be denoted by S^ and the conicoids which contain the two curves by S,^ and S.\ and suppose the remain- ing parts of the complete intersections to be two non-inter- secting lines, so that the complete intersection of S,^ and >S, is C„ L, J/, and that of S; and S^ is (7/, L\ M'. The three surfaces S^^ S.^, and ^S^, intersect in 12 points, and since L intersects S^ in 2 points, and similarly for the other lines, 8 of the 12 points lie on the extraneous lines, and the two curves (7,, C'/ intersect in four points. This supposes that the lines X, L' do not intersect, but if they intersect, the modification is easily made ; for example, if the four lines form a skew quadrilateral, the numltcr of points. 278 PKUBLEMS. belonging to these lines will be reduced to four, and C'^, C/ ■will intersect in eight points. By similar reasoning, if the remainder of the intersection of S,^ and >§.,, or of S,^ and S^ be a conic, it can be shewn that 6'^ and C^ will generally intersect in four points ; and if the three surfaces all contain the same conic, by Art. 411 this \\'ill count as eight points of intersection, and therefore C,, C/ will intersect in the four remaining points. XVII. (1) Every cone containing a curve of the third degree, in which the vertex lies, is of the second degree. (2) Prove that an infinite number of curves of tlie third degree can be drawn through five points arbitrarily chosen in space, but that six deter- mine the curve : what limitations are necessary that such a curve shall pass through the points? (3) Through a curve of the third degree, and a straight line meeting the curve in one point only, a conicoid can be drawn, of which the generating lines, which do not inteisect the given line, meet tiie curve each in two i)()ints. (4) Tlirough any point in space a straight line can be drawn which meets a curve of the third degree, not a plane curve, in two points. (5) If P, Q be two points on a cubic curve, all the conicoids which contain the curve and the chord PQ have common tangent planes at P and Q. (6) A conicoid can be drawn tlirough a given chord of a cubic curve containing the curve and touching a given plane through the chord at a given point of the chord. (7) The projection upon any plane of a curve of the third degree, not plane, by straight lines drawn from a given point, is a curve of the third degree having a double point. (8) No straight line can cut a curve of the h'^ degree, not plane, in- more than M - I points. (9) The locus of the centres of a cluster of conicoids is a cubic curve. (10) 'i'hree conicoids which have a common generating line meet only in four points besides the generating line. (11) Through five points of a conicoid, wc can draw two curves of the third drgrce lying entirely in the conicoid. PROBLEMS. 27y (12) A quartic curve is the intersection of two conicoids, prove that a cubic surface can be construcled which contains the curve and two given conies, one on each conicoid, if these conies do not lie in the same plane. (IJ) Shew that, if normals be drawn to a conicoid from every point of a straight line, their feet will lie on a quartic curve. (H) Among the conicoids forming a cluster there are four cones, real or imaginary ; each of these cones has four of its sides tangents to the curve which is the base of the cluster, and the four points of contact are in one plane. (15) Through a chord AB of a quartic curve, which is the base of a cluster of conicoids, a plane is drawn determining a second chord ab; shew that, as the plane turns round AB, the chord ab generates a conicoid; shew also that a plane which passes through ah and a fixed point £ of the curve also passes through a fixed chord EF. (16) The projection of the base of a cluster of conicoids on a plane is a curve of the fourth degree having two double points, real or imaginary. (17) Two quartic curves which lie on the same hyperboloid, each in- tersect one system of generating lines in three points, j»rove that the curves intersect in six points if the generating lines be of the same system for both curves, and in ten points if the systems be opposite. (18) Through a given straight line planes are drawn touching the sections of a conicoid made by a plane passing through a second straight line; shew that the locus of the points of contact is a quartic curve passing through the four points in which the two straight lines intersect tlie conicoid, and four of whose tangents intersect both of these straight lines. (19) If three straight lines be the complete intersection of a culic surface with a plane, three planes through these lines will intersect the surface in three conies ; prove that one conicoid can be drawn containing the three conies. (20) Find the number of points in which two curves of the fifth degree on the same surface of the third degree, and on two conicoids, intersect. (21) The eight points given by the equations lx* = mif = nz* = rid', are so related that any conicoid passing through seven of them will pass through the eighth. (22) Three cones of the same degree have their vertices on a straight line, and two of their three curvcjs ot intersection are plane curves; shew that the third curve is also plane, and that the planes of the three curves intersect in a straight line. pr CHAPTER XVIII. TANGENT LINES AND PLANES. NORMALS. SINGULAR POINTS. SINGULAR TANGENT PLANES. POLAR EQUATION OF TANGENT PLANE. ASYMPTOTES. 419. In this chapter we shall, for reasons given in the eface, confine ourselves to the consideration of surfaces whose equations are given in Cartesian coordinates, and in discussing singularities of contact we shall only consider those of a simpler kind, reserving for a later portion of the work those which are interesting merely as subjects of pure geometry. 420. It will be convenient to state here, that we shall often employ the following notation : when the function F[x, y, 0), for which we shall write F, is used, U, F, W will be written for dF dF dF , , , ,, d'F d'F d'F d'F dzd^' d^d^' ^^"^ "^^'^^ ^=/(-^' y)^lh 1^ '•> h < will be written „ dz dz d'^z d'^z d'^z dx^ dy'' dx^ ' dxdy ' dij^ ' 421. To find the relation between the direction-cosmes of a tangent to a surface at a cjiven ordinary point of a surface. Let the equation of the surface be F= F{^, rj, ^) = 0, f , t), {" being the current coordinates of a point, and let [x, y, z) be the • point Pat which the line is a tangent. The equations of a line through P, whose direction-cosines are X, /x, v, arc L-'^ = n^=i_-_^=^ (1) At the points where this liiu* moots the surface, the values of r tan(;i:nt links and ri.AM-.H. /^ . /■ 281^/-' ' '/^ '^Z' '/^ arc given by the equatloij i'(.c + Xr, y -\- iir^ ^ + ^n'y^\ ^^^^\y v^ since F[x^ y, ^) = 0, this equation may be written ' '^ /' '■(^ciia/ji another point (} will become coincident with 7', and the line will be a tangent line. • dF dF dF- ^ „ . , , At an ordmary pomt -r- , -7-, , do not all vanish, but •' ' dx^ dy ^ dz ' there may exist points on a smface for which this docs happen ; such points are called singular points : we shall presently considiu* this peculiarity. 422. To find the equations of the tantjent plane and the. normal to a surface at an ordinary point. The equation of the locus of all the tangent lines which can be drawn through an ordinaiy point is found by eliminoting \, /i, V between the equations (1) and (3\ which gives ,^ ,dF , ,dF ,^ ,dF ^ shewing that the tangent lines all lie in a plane, whic Ii is called the tanrjent plane. The normal is perpendicular to this plane, and its equations arc fz "^ ^ - .V _ ^- ^ r dF dF dF /{fdFv 7dF7' TTTT^^ dF /\l''J\ (dF\' /dFy dz \/\[dx) "^ \dy) "^ [dzj dx dy dz \/ [\dxj \dyj \dz J ) the equation (3) represents that the normal is perpendicular to every tangent line. 00 282 NUI.'.MALS AND TANCiKNT PLANKS. 42iJ. To Jiiul the niunher of normals which can he drawn from a given point to a surface of the n'" degree. Let F[^^ 7]^ ?)=0 be the surface. The number of normals Avill be the same from whatever point they be drawn, the number may therefore be found by Investigating the number of normals which can be drawn from a point at an infinite distance, which we may assume in Ox produced. The number will therefore be equal to the number of nor- mals parallel to 6>.r, together with the number of normals to the section by a plane at an infinite distance. Jf (.f , _?/, z) be the foot of a normal parallel to Ox, F' [y) = 0, F'[z) = 0, which combined with the equation F {x, ?/, ^) = give n [n - \Y solutions. Again, any plane section of the surface will be of the n^^ degree, and the number of normals drawn to any curvey (a-, ?/) = of the n^"^ degree is, in like manner, the number of normals parallel to Ox, together with the normals which can be drawn to points at an infinite distance, the mimber of the latter is n, and the number of normals parallel to Ox is given by the num- ber of solutions of /' [y) = 0, and f'[x, y) = 0, which Is n[n — 1) ; hence, the number of normals to the plane section at an infinite distance is 7i^ Therefore, the number of normals which can be drawn to the surface from any point = n {n - ])'' + n^ = ii^ - n' -f n. 424. To obtain the form of the equation of the tangent iilane irhen F{^, t], ^) is represented as the sum of a series of homo' (jcneous functiu/is. Let F{x, y, z)^F^-\- F„_^ +...+ F^ + c where /'', denotes a homogeneous function of the .s'" degree in .r, y/, z ; then, by a known property of homogeneous functions, / d d d\ ^ „ therefore the equation of the tangent plane may be written ^dF dF ^dF „ , , -, TANGi::vr lim:> and I'Lani;s. 283 01', since l'\ 4 7^,,_, +...+ <; = 0, ^'1 + "',/y + f',/f -^ ^■■- + =^- +•■■+ (" - » -'•'. + '" = "■ 425. To find the equation of a tanrjent plane to a surface^ when the direction ofthej^lane is given. Let {Ijtn^ n) be the given direction, and 1^A-mT]->rnt=p tlie equation of a tangent plane to the surface jF'(^, ?/, ^ = *^ 5 *''<^" if (a-, ;/, z) be the point of contact, since this erpjation must be identical -with ^dF dF ^dF dF dF dF dx dij dz dx ^ du dz ' •vve have I dx m dij. 11 dz ^; V dx "^ dy dz ) \ and these equations, with that of the surface,, give the coordl- ivates of the points of contact of any tangent plane in the given direction, and also determine a relation between /, ?», n and ^7, sueh as was found in Art. 253 in the case of a conicoid ; this relation is the tangential equation with the JBoothian coordinates I m n P' P' P' 426. To find the locus of the jyoiats of contact of tiufjent planes drawn to a giccn surface from a given point. Let F=F{^, 7], ^) = be the equation of tlie given surface of the n^^ degree, and let (/, //, h) be the given point. ]f (x, 1/j z) be one of the points of contact, the tangent plane to the surface at (a:, ?/, z) must pass through (/, y, //). This gives the condition •which, combined with the equation of the surface, determines the required locus, or the curve of contact. It has been shewn, (Art. 124), that .r , +V , + .- , mav dx ^ (III dz by means of the equation of the surface be reduced to an ex- 2^4 SINGULAR POINTS. presslon of the [n- 1)*^ degree in u", ?/, z ; the equation (4) so reduced gives a surface of the (« — l)'** degree, called the frst polar^ Avhose intersection with the given surface is the curve of contact. The curve of contact for any conicoid is therefore a conic, the first polar being in this case a plane. The general theory of polars will be considered hereafter. /Satgalar Points. 427. To find the relation between the direction-cosines of a tangent line at a singular point. oince at a singular point i , -- , - and -r- separately vanish, the coefficient of r in equation (2) vanishes for all values of \, yti, v, which shews that the line (1) meets the surface in two coincident points, in whatever direction it be drawn through P; in this case we find the direction of any tangent line by taking a point Q near the double point P, and moving it up to P until a third value of r vanishes, the direction of PQ will then be that of a tangent line, and the relation •between its direction-cosines will be ^ d d d ax dg dz or. written in full, «X' -^ vix" + wv" 4 2u'fiv + 2v'v\ + 2w'Xfj, = 0. (5) If all the partial differential coefficients as far as those of the {»— 1)'" order vanish, the equation (3) will have 5 roots equal to zero, and the point will be a multiple point of the 5*" degree ; it is easily seen that the directlon-coslncs of any tangent line win satisfy the equation ^ d d dy ,, which shct\-s that the tangent lines all He In a cone whose tertcx Is the point I*. This cone is called a tangent cone. SINGULAR I'uINT.S. 285 428. To find the equation of the tangent cone at a laidtiph 2)0 Int. If tlic multiple point be of the s^ degree, the direction- cosines will satisfy the equation and the equation of the tangent cone is found by eliminating X, /i, V between this equation and the equations (1), we thus obtain {(i-)^+(.-i/),:J,+(r-^);^.fi'-=o, where it must be remembered that in the performance of the operation indicated ^-cr, ii—y and ^—z must be treated as constant, in other words, the symbol of operation must be ex- panded before the differentiations are performed. 429. To find the equation of the normal cone at a douhle point. The equation of the tangent cone at a double point Is, by (5), and that of the tangent plane at any point of a generating line of this cone whose coordinates are x + Xr, y + /^r, z + vr Is [u\ + tc'iM 4- v'v) (I - a-) + [io'\ + i"/i + u'v) [v-y) -f- (i-'X4 u'fi^icv)[i;-z) = (\] lience, If , , = ' , = ~ , be the equation of the normal ' X /i V to this plane at (a-, y, 2), «X + 10 fi + v'v xo'\ 4- VIM + \iv v'\ A- u'fJL + in X' /a' v .'. mX -f ic'fi + vV — X'p = 0, io'\ + Vfi + u'v — fi'p = 0, r'\ -f n'fi + irv - v'p = 0, X'\ -|- yti'/i 4 v'v — ; = P 283 srxciUi-Ai: points. M, v.- , V \' i v' (^) X', fi' and tlic equation of the locus of the norraals to all the tangent planes to the tangent cone is where p = vw — u"\ ^/ = v'w — ini\ &c. 430. The condition that the tangent cone shall degenerate into two tangent planes is u\ 10 ')) becomes 0, Art. 88, and in this case, the equation - {pX' + r'fju' + q'v") 0, Art. 3G4 so that the normal cone degenerates into two coincident planes ; this may be accounted for geometrically in the following manner: the generating lines of the normal cone are each perpendicular to the plane containing two of the generating lines of the tangent cone taken indefinitely near to one another ; if then the tangent cone become two planes, we can take the two generating lines on one plane, which gives a normal to that plane ; or we may take one on each close to the line of intersection of the planes, which will give a normal in any direction we please in the plane perpendicular to the line of intersection, and a double plane will be formed, because these two generators may be on either side of the double point. The equations of the line of intersection of the two tangent planes will, by Art. 89, be [v'lc'- uii) (I — x) =[ic'u'— vv') (77 - 1/) = [n'v- wio') (^- z)^ or ]>'{^-x) = q{rj-ij)==r'{^-z), unsymmi:ti:ical equation. 2S7 ami tliat of one of the coincident planes In which the normals lie p{^-^) + r'{v-?/) + q'{^~^) = o, and this plane is perpendicular to the line of intersection, since J>p' = •' (Art. 3(33). h' T=0, X=0 be the respective conditions that the tangent and normal cones may become two planes, and I\ (J^ li, i", fj\ R be tke minors of A', X= Pp + R'r -^-Q'q, but 2' = qr-2)"' = uT, Art. 363, F ' = q'r' — pp' — u 2] &c. ; .-. N=T{up-i-lc'r' + v'q')=T\ 431. To find the equation of tlie tangent plane and normal at arty jioint of (he surf tee (jlven hi/ the equation ^=f[^-, v)' Let a Hue be drawn through .r, ?/, sr, whose equations arc ^-.t'=m{^-z), rj-ii^n{^-z), the points In which this line meets the surface are those for which ^ is given by the equation r - z =f[x + m (^- z), y + n (?- z)\ -f[x, y) = [mp + nq) (^- z) + | [nn^ + Ismn + tn') (^- z)' +. ... , and If the line be a tangent line two values of ^- ^ are zero ; therefore 1 = mp + nq^ and eliminating m and n by means of the equations of the line, we obtain the locus of the tangent lines ^- z =p) (^ — yl(), PAq at the multiple point. If the pLane move past V to the position ir, the curve of intersection Avill gradually assume an oval form, which will degenerate into a conjugate point at a. It is clear also that a plane may meet the ring in the circle GIIKL, in which case it is a tangent plane at every point of^ the curve in which it meets the surface ; this curve is composed of two coincident circles, as may be seen by moving the plane inwards parallel to itself. It will be shewn also, that a tangent plane, drawn through a line COD perpendicular to Oz^ intersects the ring in two circles. OF SURFACE AND TANGENT PLANE. 291 436. To find the equations of the tangent line at oni/ puint of the curve of intersection of a surface loith its tangent plane. Let the equation of the surface be F[^^ 77, ^)=0; that of the tang-ent ph\ne at [x^ y, z) will be ,^ , clF , , dF ,^ , dF ^ or i^-cc)F'ix) + [v-!/)F'{^) + {^-z)F'{z) = 0. (1) Let the equation, of the tangent line at any point {x\ ?/', z') of the curve of intersection be L^' -V-!/^_ ?: z since this line lies in (1), \F'{x)^tiF'{y) + vF'[z)=i), (2) and since it meets the surface in two coincident points at (-^'j y'l ^'), \F' {x') + fiF' if) + vF' [z') = ; (3) these two equations determine X : /j, : v when (a;', y', z') is an ordinary point on the curve and the surface. 437. To find the singular points of the curve of intersection xoith the tangent plane at any point. If the point be a singular point on the curve of intersection, any line drawn through this point will have two points coin- cident at the point considered ; hence, the two equations obtained in the preceding article will be satisfied by an infinite number of values of X. : /* : v; this will happen in any of the following three cases : (i) When F' [x].^ F' {y) and F' [z) vanish simultaneously, which occurs when there is a singular point at (.r, y^ z). in which case there arc an infinite number of tangent planes. (ii) When F' {x'\ F' [y) and F' [z] vanish simultaneously, in which case [x\ y\ z) is a singular point on the surface. H ^vi,on^;-g)=£:g;)=^;;j,i„ „.,..,. ...e .1,0 tangent plane at (.r, y, z) is a tangent plane at (.<•', //', z') also. 292 KULED SURFACEi?, In case (1) one of the tangent planes to the tangent cone touching it along a generating line (V, /u,', v') must be the l)lane considered, and the equation (2) must be replaced by [uX + iv'/jk' + v'p') \ + {w'X' + Vfi' + u'v) IX + {v'X + u'[x' + icv) V = 0, (Art. 4t^9), thus the ratio \ '. [x : v will be determined, except in cases where {x\ y\ z) is a singular point on the surface, or where the tangent plane considered is also a tangent plane to the surface at(a;',2/', s').^^ In case (ii) a third point at least must be coincident with (a;', y\ z\ and the equation (3) must be replaced by where s is 2, 3, ... according to the degree of multiplicity of the singular point {x\ y\ z'). In ease (iii), if neither [x, y, z) nor (a:', y'^ z) be singular points of the surface, the equations which determine \\ [iw will be Xi^' {x) + iiF' [y] + vF' [z] = 0, whether (cc', y'j z.) be coincident with (a;, ?/, z) or not. This ease includes the singular tangent plane, a portion of whose curve of intersection consists of two coincident curve lines, which will be considered immediately. riuUd Surfaces. 438. The student is already familiar with certain surfaces which are capable of being generated by straight lines, or through eveiy point of which some straight line may be drawn which will coincide, throughout its length, with the surface. For example, — a plane, a cone, a cylinder, an hy[)crboloid of one sheet, an hyperbolic paraboloid. He is aware that any portion of two of these, the cone and the cylinder, may, if supposed perfectly flexible, be developed into a plane without tearing or rumpling. DEVELOPABLE SURFACES. 293 We shall now give sonic account of the general character of surfaces which have this property, distinguishing them from those which, although capable of being generated by the motion of a straight line, arc incapable of development into a plane. 439. Def. a liuJed Sioface Is a surfiice which can be generated by the motion of a straight line; or a surface through every point of which a straight line can be drawn, which will lie entirely In the surface. A ruled surface, on which each generating line intersects that which Is next consecutive, Is called a Devehimhle Surface^ or Torse. A ruled surface, on which consecutive generating lines do not Intersect, is called a Skew Surface, or Scroll. Developable Surfaces. 440. Explanation of the developunent of developable surfaces into a plane. Let Aa, Bb, Cc, ... be a series of straight lines taken in order, according to any proposed law, so as to satisfy the condition that each Intersects the preceding, viz. In the points a, b, c, ... . Since Aa, Bb Intersect In a, they He in the same plane, similarly the successive pairs of lines Bh and Cr, Cc and Dd^ &c. lie In one plane ; thus, a polygonal surface Is formed by the successive plane elements AaB, BbC, &c. This surface may be developed Into one plane by turning the face AaB about Bb, until It forms a continuation of the plane BbC, and again turning the two, now forming one face, about Cc until the three AaB, BbC, CcD are in one plane, and so on ; the whole surface may, therefore, be developed into one plane without tearing or rumpling; the same being true, however near the lines Aa, Bh, ... are taken, will be true in the limit, when the surfixce will become what we have dcfnied as a developable surface, this name being derived from the property just proved. 294 DEVELOPABLE SURFACES. Edge of Begresslon. 441. The polygon ahcd^ ... whose sides arc in the direction of the lines Bh^ Cc^ ... becomes in the limit Ji curve, generally of double curvature, which is called the Eihje of Rcgressioii, from the fact that the surface bends back at this curve so as to be of a cuspidal form. Every generating line of the system is a tangent to the edge of regression, which is therefore the envelope of all the generating lines. DEVELOrAHLE SURFACES. 295 In the case of a cyliuder, the edge of regression is at an Infinite- distance. For a practical construction of a developable surface having a given edge of regression, see Thompson and Tait, Nat. FltiL^ Art. 149. 442. To find the general nature of the intersection of a tan- gent plane to a developable stafiace with the surface. The plane containing the element DdE of the surface repre- sented by the figure evidently becomes in the limit a tangent plane to the developable surface at any point Z> in the genera- ting line Dd, since it contains two tangent lines, viz. Dd and the limiting position of a line joining such points as D and E, which ultimately coincide ; and again, supposing I)dE in the plane of the paper, Ff meets this plane in e, Ggf meets it in some point /', Ilhg in g'j &c., and similarly for Cc, Bb, . . . on the other side. The complete intersection of the surface and tangent plane is therefore the double line formed by the coincidence of l)d^ Ee^ and the limit of the polygon ah' clef g ... which is a curve touching the double line Dd at the edge of regression. Cor. To find the nature of the contact of the edge of re- gression and the tangent pilane . The plane containing the generating lines A/, Ee contains the three angular points c, <:/, e of the polygon in the limit, therefore the tangent plane contains two consecutive elements of the edge of regression, and is, as will be seen later on, what is called the osculating plane at that point. 443. The shortest line which Joins tico j^oints on a develop- able surface is the curve., the osculating plane at every point of which contains the normal to the surface at that point. If the surface be developed into a plane, the shortest line must be developed into the straight line joining the two points. If on the polygonal surface in the figure on page 294, ADCD...K be the polygon which in the limit becomes the shortest line joining A and iv", since on development this becomes a straight line, two consecutive sides EF^ EG must be iuclincd at equal 206 SKEW SURFACES. angles to line Ff. Ilcnce a straight line, drawn through F perpendicular to the line Ff in the plane bisecting the angle between the planes EFf\ GFf^ will evidently lie in the plane EFG^ and bisect the angle EFG. This line will be in the limit the normal to the surface, and the plane EFG will be the osculating plane of the curve ABCD ... at the point F. Therefore the shortest line is the curve, the osculating plane at every point of which contains the normal to the surface at that point. Such a line Is called a geodesic line of the surface, and it will be hereafter shewn, that the property enunciated for de- velopable surfaces is true for geodesic lines on all surfaces. If the geodesic line, joining two given points, be drawn on a right circular cone, the equation of the projection upon the base can be shewn to be - sin (7 sin a) = j sin [0 sin a) + - sin [(7 — ^) sin a}, a, h being the distances of the given points from the axis, 7 the angle between these distances, and a the semi-vertical angle of the cone. Skew Surfaces and Curves of greatest densitij. 444. Let AA\ BB\ CC\ DD\ &e. be straight lines drawn according to some fixed law, such that none intersects the next consecutive; let aa\ hh\ cc\ dd\ ... be the shortest distances. Suppose now that we take two of the generating lines as CG\ 1)D\ and imagine DD' twisted about c so as to be parallel to CC", and united with It by means of an uniform clastic membrane : if now DD' be returned to its original position, the portion of the membrane near cc being unstretched, will be denser than any other portion. If the same process be adopted for every line, the series of membranes will gene- rate a surface which will ultimately, as the lines approach nearer to one another, become a skew or twisted surface. The curve which is the limit of the polygon formed by joining «, h^ c, fZ, ... at which the Imagined membranes would have the greatest density, is called the curve of greatest density ; It is also called the lino of strict ion. SKEW SURFACES. 297 It may be observed, that the shortest distances between the conscciitive generating lines of a scroll are not generally elements of the line of striction. 445. To explain the nature of the contact of a tangent plane to a shew surface at any point. Let P be any point of a skew surface, AA' the generating- line passing through P, suppose a plane to be drawn tlirough 1* containing BD' the next consecutive position of the generating line, this plane will intersect the third line CC in some point 7/, QQ 298 SINGULAR TANGENT PLANE. and, if PR be joined, it will meet BB' in Q ; PR will therefore be a tangent line at P having a contact of the second order at least, so that, if the surface were of the second order, it would lie entirely in the surface. The tangent plane at P is the plane containing A A' and PR\ R will change its position for any change of position of P, thus the tangent plane at any point in AA' will always contain AA'^ but it Avill move about AA through all positions, as the point of contact moves along AA'. The tangent plane, therefore, at any point of a skew surface contains the generating line and some other curve which must be a straight line in the ease of a surface of the second degree. 446. To shew that the equation of the tangent plane to a developahle surface contains only one 'parameter. Since the general equations of a straight line involve four arbitrary constants, we must, in order to generate any ruled surface, have three relations connecting the constants, so that it may be possible, between these equations and the two equa- tions of the generating line, to eliminate the four constants, and thus obtain the equation of the surface which is the locus of all the straight lines. In developable surfaces the generating straight lines are such that any two consecutive ones inter- sect, and the plane containing them is ultimately a tangent plane to the surface. The equation of this plane will then involve the four parameters, and by means of the three relations we may eliminate three, so that the general equation of the tangent plane to a developable surface will involve only one parameter, and we may write it in the form a being the parameter, and 0(a), "^[o) functions of that para- meter, given b any particular case. Singular Tangent Plane. 447. Def. a singular tangent j)la)ie is a plane which, instead of touching a surface in any finite number of points, touches along the whole of a curve line. SINGULAR TANGENT PLANE. 299 If the curve of intersection of any plane with the surface be composed, in part at least, of two or more coincident lines, the other part being made up of simple curves, either the plane will be a tangent plane to the surface at every point of such a mul- tiple curve, or it will contain a multiple line of the surface, such as would be generated by the rotation of a cross round any fixed Hue not passing through the angle of the cross. Conversely, if a tangent plane touch along a curve line on the surface, this curve line will be a multiple line on the tangent plane. Thus, in the case of the anchor ring (Art. 435), the plane which touches the ring along a curve has for its curve of inter- section the two circles coincident in LKII] also the taiigent plane to a cone contains two generating lines which ultimately coincide, and is therefore a tangent plane at every point of the generating line which it contains; any more general develo])- able surface is an example of the case of a tangent plane which contains a double line, at every point of which it is a tangent, combined, as shewn in Art. 442, with another simple curve. A surface of the fourth degree admits of the case of a double conic, as in the example of the anchor ring, or of a quadru])le straight line, as when it is made up of two cones touching along a generating line. A surface of the fifth degree might be composed of one of the third degree and one of the second, in which case a tangent plane might meet the former in a triple and the latter in a double straight line. 448. To find tlie condition that a tangent plane may he sin- ffidar. Since a line, drawn in any direction in the tangent plane through any point of the double curve in which the tangent plane touches the surface, will contain two coincident points, but if it be drawn in the direction of the two coincident tangents to the curve of contact it will contain four coincident points, we have to express that at every point of the double curve there are two coincident tangents, and that a line in their direction contains four coincident points ; and we may observe that 300 SINGULAR TANGENT PLANE, = 0, these tangents arc what have been called inflexional tangents (Art. 432). Since the two inflexional tangents coincide, their direction- is given by the equations u\ + to'fi + v'v _ w'\ + Vyw + tt'v _ v'X + u'fi + wv Ur - V ~ W '' and \U+fxV+vW=0. Tlic condition that these equations shall hold i» w, w', v\ U w\ Vj u'j V v', m', w, W U, V, IF, and the condition that a fourth point may become coincident is that for the values of \ : /^ : v given by the above equations the further conditions wlien the curve has a higher degree of of multiplicity may be easily obtained. 449. The conditions of the" existence of a singular tangent plane may also be found, by considering that the point of con- tact, determined by the equations of Art. 425, may be any point of a curve line, and its coordinates are therefore indeterminate. 4.")0. For a surface given by the unsymmetrical equation ^ =/(!■, 7j), the equation of a tangent plane at any point (a:, ?/, z) is a tangent line whose equations are meets the surface in points for which p is given by vp = (p\ + ^At) p + y {rX"" + 2sX/M + tfjC') ANCnOK RINO. 301 If the tangent plane be singular, for all the points in wliicli it meets the surface, v=pA, + 2/A, and for all the points of the double curve four values of p are zero, and the two inflexional tangents coincide ; /. 7-V + 2s\^l + tiji' = and (x ^ + fi -^ J z = 0, and the former has equal roots, therefore rt = 5^, and either ?-X + *f/x = or s\+ f/j. = 0. 451. Every tangent plane to a developable surface is a sin- gular tangent plane, since it contains two consecutive generating lines, hence its curve of intersection with the surface consists of two coincident straight lines, and, as shewn in Art. 442, a single curve line. The analytical conditions of singularity are satisfied, since, if \, yu., v be the direction-cosines of the double line, which lies entirely in the surface, the coefficients of all the powers of p will vanish, and rt = s^ in consequence of the coincidence of the two lines. At any point of the single curve, the values of p, q being y, q, the direction cosines X., /i, v of the tangent are given by v = \p-\- fiq and v = \p' + mq', which are independent equations, since //, p and q, q are generally unequal. 452. We have selected the following illustrations of the points which have been considered in this chapter, and we call attention especially to those relating to cubic surfaces and the wave surface as of intrinsic importance. 453. Tangent plane to an anchor ring. Let the plane containing the centres of the generating circles be taken for the plane of a-?/, and the axis of rotation for the axis of z ; and let r be the distance of any point (a?, y, z) from the axis, c that of the centre of the generating circle, a its radius; then r^ = x^+f/^ and z^ -^^ {r - cf = a'' ; the equation of the anchor ring is ie + V' + r + c' - ci'f - ic' (r 4- rf) = ; 302 IIELICOID. that of tlic tangent plane at a point (a?, y, s) is x{r-c)[^-x)Ary [r - c) [rj -7j) + zr{^- z) = 0, or (r - c) {x^ + 1/7]) + rz^= ?•' (r - c) + rz' = r {a" + c{r-c]]. To find the curve of intersection of the surface with a tangent plane which j^asses through the centre. Suppose that it passes through axis of ?/, and is inclined at angle a to that of a^, so that a — c sin a; and at any point of the curve of intersection r = c— a cos^, e = a sin ^, x = z cota ; .'. y^ = ?•'■' — x'^ — c^ — 2ac cos^ + a^ cos""*^ — a^ cot"''a sln^fl = c'"' — 2ac cos 6 + a" cos^ 6 — (c' - a^) sin' = (c cos^-a)"; •'• {y ± cf = c^ cos'"' 6 J and x' + z'' = c^ sin"' ^ ; .-. x' + [ij± af + z' = c' ; hence the curve Is two circles which intersect in the points of contact, forming two double points. To find the form of the curve EAF in the figure of the ring. The equation of the tangent plane is ^ = c — a^ and the form of the curve of intersection is given by the equation [ri' + r + 2c (c - a)Y = Ac' {v' +{c- a)'], or iv" + D' - 4«C77'^ + 4c (c - a) ^' = 0. When c = 2a, the curve is the lemniscate of Bernoulli. 454. Tangent plane and normal to a Helicoid. Def. The Helicoid is a scroll generated by the motion of a straight line which intersects at right angles a fixed axis, about which it twists with an angular velocity which varies as the velocity of the point of intersection with the axis. If the axis be taken for the axis of z^ and that of x be one position of the generating line, the equation of the surface generated will be r=rtan->|, and the tangent plane at a point (.r, ?/, z) will bo EXAMPLE OF SINGULARITIES. 303 or (a;" + y^) (^- s;) = c {xrj - yf ) ; at the point (a;, 0, 0), the equation becomes x^^ct], hence the tangent of the angle which the tangent plane at any point makes with the axis varies as the distance of the point from the axis. The equations of the normal at (a?, ?/, z) are ^- X ^ v-y ^ c (g"- z) ^ y -X x'^if ' and for the normal at (.r, 0, 0), ^ = a;, xr]-\ c^= 0, hence the locus of the normals at points taken along a generating line is an hyperbolic paraboloid ; which is true for any scroll. 455. Tojind the singularities of the surface lohose equation is {z' + 2x' + 2/)' - [■^' + f) [x' + y'+\Y = 0. Wc consider this surface as represented by the given equation, in order to illustrate the general methods given in Arts. -4:27 and 448, for discussing singular points and planes; but the student will see clearly the results to which we shall be led, if he first trace the plane curve whose equation is ?/^= a? (1 — a;)^,* and then imagine the form of the surface which would bo generated by its revolution round the axis of ?/, which it is easily seen is the surface proposed. To find a singular point we have, writing r"^ for x'' + y\ U =2x [Iz' -iJ^^r"- 3/) = 0, V =2y [Az' _ 1 + 4r' - 3/) = 0, TF= Az [z' + 2r') = 0. The systems of values of x, ?/, z which simultaneously satisfy these equations, and that of the surface are ^ = 0, and either (i) cc = 0, ?/ = 0, or (iij r'' = 1 ; (i) shews that the origin is a singular point — it will be found that the tangent cone of Art. 428 becomes an infinitely slender cylinder or cone, given by V +/*'' = ; (ii) gives a circle of singular points — the conical-tangent at any point (a?, y^ 0) of this circle becomes the two tangent planes [x^ + yTj—}y=z ^\ * Frost's Curve Tracing, Plate II., Fig. 8. 304 WAVE SURFACE. To find a singular tangent plane vvc have, by Art. 448, the equations 2 [Xx + fxy) [Az" - 1 + 4r'' - 3/) + 4s {z' + 2r') v = 0, and 2 [X' + /x*) (4s' - 1 + 4r'' - 3r*) + 4.v^ [Zz' + 2r') + 32s [\x + /x?/) v + ^[\x + firjY (2 - 3r') = ; there will be two coincident tangents if j/ = and 4s''-l+4r'-3r* = 0, also by the equation of the surface [z^ + 2ry — r'^ (1 + r")' = 0, the only solutions of these equations are s"'' = 0, r''=l, and z^ = -^j^ r^^^f the first solution gives no tangent plane, but two cones intersectingN^ a circle, any generating line of either of which is a tangent line; the second solution gives two tangent planes s = + § \/3, each of which is a singular tangent plane touching along a circle ic'* + ?/^ = ^, the direction of the tangent to which is given by \x-\- fiy = and v = 0, the re- maining part of the curve of intersection is a single circle of radius |. In this case the condition of four points being coincident is X-f -^ H- -r] F=Oj which becomes [3 V dx "^ d7j) 2,0 (^xx + fxy) {\' + /.^) - 8 [\x + li^yY = 0, it is therefore satisfied by Xx + fiy = 0. Wave Surface. 456. The equation of the Wave Surface may be written in either of the forms y' z' + -/^, + -i 5 = 1, r' — a^ r'^ — ¥ r^ — 6^ aV V'f d'z' ^ or -s i + -r,-^, + -5 Q = 0, r -a r — b r ~c where r' = rc^ + ^^ + z* ; and we shall suppose a>h> c. The existence of singular tangent planes to this surface is of great importance in explaining a peculiarity in the transmission of light through a biaxial crystal. In order to shew that such planes exist, we shall employ the method of Art. 449. WAVE SURFACE. Wi) 457. To jind the point of cojitact of a tangent plane whose equation is Ix -f my -{ nz =pj and the relation between ?, ?>?, n and p. Tlic equations for cleterminlng the point of contact arc V V W xU-\-yV+sW -y = - = — = ' = -2cr suppose, I m n p 117 Op where U =-^i^^-..-2jl\ r - a' r' — c p 0^ v" ^^ .-. X Z7+ yV-\-z W= 2 - 2/-'P= - 2 - c Also by squaring and adding, and observing (2), ^=^^f^^'-f^-fT 1(7^. ^ (^73^3 + ^TZTri+Zi (3) The equations (1) and (3) give the values of x, y, z at the point of contact, and (2) is the required relation between the constants. R It 306 WAVE SURFACE. If a, y9, 7 be the Bootbian coordinates of tbc tangent pUino, VIZ. — , — , — , and a. o . c be written lor - , 7- , - , tbe tan- gcntial equation of tbe surface will be aV h'^^-^ eV p — a p —0 p —c wberc p*"* = a'^ -f ^'^ + 7^^, an equation of tbe same form as tbe Cartesian. 458. To find a singular tangent idane of the xoave surface. Tbe point of contact in tbis case being any point in a curve, tbc coordinates must be indeterminate ; now y Avill be of tbe form — if ^; = b and m = 0, and tbese values also make r' indeterminate, and tberefore x and z. And since, by (2) and (3), r^-f-P \[f-aj [f-6y\' T li' 1 we baVC -^ jr, = y^j -„ = -^ :, . a —b b —c a —c Tbe curve of intersection is a circle given by tbc })lanc lx + nz= p, and eitber of tbc spberes , d'-b"" . b'-c' or r' - — — z = c\ nb ' p = a, l — O and p = c^ n = give imaginary planes, bonce tbcrc are four real singular tangent planes. 459. To find tlie singular 2^oints and the corresponding normal Tbc singular points may be founxl by investigating for wliat definite points of contact /, ?/?, n can be indeterminate, wo sball tbus obtain ax c z a c ,j^O,r = !,, ana ^.~^, = ,.,-, = -^. , (4) wbicb determine four singular points. CUBIC SURFACES. 307 By equations (1) of Art 157, a" — b' , d^ 1 h* — c' c' — l = p and ?? = w ; X P ^ V a^_J-^ y^-c"- a'-c' .-. 1+ n = -; X z p .'. [a^xl + c^zn) [Ix + nz) = aV, by (4), or (iVf" + c'zhf + {a' + c') xzln = d^c\ ... (,r - h') r + {h' - c'-') n' + ^^^^ V(«' - 1^') \/{h' - c') In which gives the eqiuation of the normal cone at the singular point. Cubic Surfaces. 460. On every surface of the third degree there are 27 straight lines and 45 triple tangent jylanes^ real or imaginary. This theorem was iirst discovered by Cayley.* An arbitrary straight line intersects a cubic surface in three points, given by an equation of the form u^ Du.r-^\D-u. r' + 1 Dhi . r' = 0. Now the four constants in the equations of a line may be chosen so us to satisfy the equations m = 0, Du = 0, D''u = Oj l)'u = 0, and, since the above equation will then be satisfied by all values of r, all straight lines having such constants will lie entirely in the surface ; and the number of such straiglit lines will clearly be limited, speaking generally, although in particuKir cases, as in that of a cylindrical surface, it may be infinite. If a plane be drawn in any direction through such a straight line, its line of intersection with the surface will be composed of that straight line and a conic forming a group of the third degree; and the two double points in which the straight line ♦ Camhriil'je and Dnhlin Matkernnticnl Journal, vol. iv. 308 CUBIC SURFACES. intersects tlic conic arc two points of the surface at which the plane is a tangent plane to the cubic (Art. 434). JSow, there will be five positions of the plane for which the conic will become two straight lines. For, if the axis of a? be a line which lies entirely in the surface, the equation of the surface will be of the form where m.^, v^ are quadric functions ; and if the surface be cut by a plane whose equation is ^ = - = r, the conic, which is part of the line of intersection, will have for its equation /u,u'„ + vv\ = 0, where u\^ v',^ have each the form a/' -f h^x' + c^ + 2clpc + 2(?/ + 2/ra = 0, in which a^, e^, f^ are homogeneous functions of /i and v of the degrees denoted by the suffixes. Hence the equation of thd conic will be cuf + /9,a;'' + 7, + '2h^x + 2 a./ + 2^/a! = ; it will therefore become two straight lines if ^A% + 2a,£,r, - o^K' - ^.<'iX: = 0, which gives five values of the ratio A, : /i. In each of the five particular positions of the plane the com- plete intersection is three straight lines, which give three double points, and the plane is a triple tangent plane touching the surface at each of these double points. Through each of the three straight lines In a triple tangent plane four other triple' tangent planes besides the one considered can be drawn, giving rise to 12 new triple tangent planes and 24 new straight lines, making in all 27 ; and the surface cannot contain any but these 27 lines, for the point in which any line on the surface meets a triple tangent plane ABC must lie on one of the three lines AB, BC^ CA, which form the complete intersection of ABC with the surface, and the plane which passes tlu-ough the new line and AB, supposing this to be the line which it cuts, must contain a third line, and, therefore, must be one of the five triple tangent planes drawn through AB; the line considered must therefore be one of the 27 lines. LINE OF STRICTION. 309 Five triple tangent planes can be drawn through each of the 27 Ihies, which would make 5 x 27 planes in all ; but since each plane contains three of the lines, we have in obtaining this number reckoned each three times, hence the number of triple tangent planes is 45. Line of Striction. 461. To fnd the line of strict ion of a scroll. Let the equation of a generating line be ^ = ;..! + a, r=*'H/3, (1) where the constants are functions of one parameter 6] the equations of a consecutive generator corresponding to a value 6 + (16 of the parameter arc 77=(m + f7»0l+a + f^«, ^={n + dn)^-^^^d^. (2) Let P be a point in the line of striction ; PQ the shortest distance between (1) and (2) ; x^ y^ z and x-\-hx^ 2/ + %? ^ + ^^ the coordinates of P and Q. Since FQ is perpendicular to both generators, Zx + m'^y + n^z = 0, and Ix + [m + dm) By+{n-\- dn) Bz = 0] .'. dni8y + dnSz = 0. Also, by the equations (1) and (2), 8y — mSx = xd7n + d(x, hz — nBx = xdn + d^ ] hy Bz _ Bx ' ' dn — dm ndm — mdn _ By — mBx _ Bz- nBx (1 + m") dn — mndm — (1 + li^) dm + mndn ' /. {xdni + da) {(1 + n^) dm — mndn] + [xdn -I- ) sin 6'. Tiiis equation can also be written in the form r^ d — = -—[r {sin 6 cos d' - cos 6 sin 6' cos (0' - ^)]] r- cosec ^ sin^' s'm icb' — 6) . dq) 464. To find the perpendicular distance from the pole xqion the tanr/ent 2)lane. This may be obtained from the first three equations of the last article by squaring and adding, whence p' (m" + v' H- w" cosec* ^) = 1 , 1 , (duV {duV .a 312 POLAR EQUATION. 405. We may arrive at the al30ve result hy the l\)llo\vlng process, wliicli serves to shew the geometrical signification of the partial differential coefficients, and will be useful as an exercise. Let P be the point of contact, Fit a tangent line passing through OZj and FQ a tangent line in the plane through OF perpendicular to the plane FOZ; take E and Q points very near to P, and in OQ^ OF. take Op, Oj) each equal to OP; then F/) = r &m 6d(fi and Fp'^rcW ultimately, and Qp^—Fjy are respectively the values of dr due to changes of 6 and , considering the other constant, dr j:=-eotOPP, and = -?!'=- coiOFQ. r sin 6d(f> Fp Draw OF perpendicular to the tangent plane QFR^ and on a sphere, whose centre is P, let a8/3 be a spherical triangle with its angular points in FQ, FO, PP, join 8y, 7 being the intersection of FY and a/3, then By is perpendicular to a/3, and a8^ is a right angle, llencc cota8 = cotS7 cosa^Y, and cotyS5 = cot67 siuaS7; ASYMPTOTES. .".l.'> .•. cot'aS + cot'/SS = cot" 87 = ./ ; V 1 1 1 fd)-\' 1 {rh-\^ 2^ 466. Def. An asymptote to a surface is a straight line which meets the surface in two points, at least, at an infinite distance, while the line itself remains at a finite distance. An asi/mjytofic plane is a tangent plane whose point of ,con- tact is at an infinite distance, the plane Itself being at a finite distance. An asymptotic surface is a surfcicc which is enveloped by all the asymptotic planes to the surface. 467. General considerations on ayym2)totes. If we imagine any tangent plane to a surface, and consider the result of supposing its point of contact to be at an infinite distance, we shall be led to the following conclusions : Smce the plane at infinity intersects the surfiice in a curve, real or imaginary, there are generally an infinite number of directions in which a point of contact may be supposed to move off to Infinity ; to each of these directions will correspond an asymptotic plane. Each asymptotic plane is the locus of all the corresponding asymptotes, and these asymptotes will all be parallel, since they pass through the same point at infinity at which they are tangents. ►Since there are two tangents In every tangent plane at an ordinary point which pass through three consecutive points, viz. the tangents to the curve of Intersection at the point of contact, there are in each asymptotic plane two corresponding in- flexional asymptotes which pass through three points at an infinite distance. Since any plane which passes through an inflexional tangent intersects the surface in a curve which has a point of inflexion at the point of contact of such a tangent, the curve of Intor- .ss 314 ASYMPTOTES. section of the surface and any plane drawn through an in- flexional asymptote has a point of inflexion at an infinite distance. 4G8. The peculiarities which arise in the case of singular points at an Infinite distance can be examined without much difficulty by a comparison with what takes place at a finite distance. If, for example, there be a double point at infinity, in the place of the conical tangent at a finite distance, there will be a cylinder of the second degree formed by the asymptotes which correspond to the direction in which the double point lies. Of the generating lines of this asymptotic cylinder there are six which meet the surface in four points at infinity. The curve of Intersection with any plane parallel to these generating lines has a double point at infinity. The curve of intersection with any tangent plane to the cylindrical asymptote has a cusp at infinity. 4G9. To find the asymptotes to a given surface. Let F=F{^j ?;, ^) = be the equation of the given surface, (ic, y, z) any pomt ni an asymptote, --- = =- — =r its equations; and let F[\,fx^v) be arranged in a series of homogeneous functions of the degrees n, n- ], ... , so that F{X,fi, v)ee(I>,,-\- <^„_,-f...+ (^, + c. The points in which the asymptote meets the surface are given by the equation F{x + \r, y + fir, z + v7-) = 0, .,, T. 1 1 • (^ d d or, il JJ denote the operation a- ,— -Vy—j — V z — ^ >•>„ + r'-' [D^h,^ + «/,„.,) + r"-' {\D'4>^^ + i}>„_, + ,,_^ = r {n - 1) {D,^ + „_,) + ^r'n {n - l) <^,. = Ir'/i [n - 1) <^,, ; therefore, since ^,, = 0, (2) and (4) arc satisfied for all values of r. 470. Should the student be Interested in the discrimination of the various singularities which may occur, he will find a guide 31G ASYMPTOTIC SUIfFACES. ill two articles by rainvin,* who has nearly adopted our method of treatment, and has carefully followed out the consequences of supposing the conicold (4) to have the various forms of which it is capiible. 471. A singular asymptotic plane is one which touches the surface along a line at infinity, if considered as the limit of a tangent ])lane ; and if considered as the locus of asymp- totic lines, it is a plane such that lines drawn in any direction in it meet the surface in two points at an infinite distance. The analytical conditions are obtained by considering that the equation D(^^^ 4- „_, = must be independent of the values of X, yu,, V. Asyinjjtotic Surfaces. 472. To find the asymptotic surface of a given surface. The asymptotic surface being the surface enveloped by the asymptotic planes, which are tangent planes whose points of contact are at an infinite distance, is a developable surface cir- cumscribing the surface along the curve of intersection with the plane at infinity. The equation of an asymptotic plane is \ /i, V being connected by the equationa <^,^ = and \'' + /a"'+ v'=l. We shall write w, u'... for ,^ , t-^' , ..., and t/_, Z7_ ... for ~dx' ~dx' ••• • Considering a consecutive position of the asymptotic plane, we have the equations dP ,, dP , dP , dx'^^-^d/^'^^d'.'^'^''^ U_, dX+ K, dfx,+ Wdv = 0, \ dX+ fi d/ji+ V Jv = ; * Cl-elles JnnrnnK vol. «5. AS V MITOTIC SUHFACES. 31^ tlicrefure, by arbitrary imiltipliers, -''/' + -lII'„ + 77>' = 0, and, imiltiplying by X, /i, v, («-l)P+^l»(^,. + ^ = 0, .: B=0,' : 7//+U^„-, dv f, /i, ?;i c?// now the degrees o^ — ^vio-u"^ and U , arc 2(?t-2) and da ""' ^ ' ?i — 2; the degree of tlie equation is therefore 3?i — 5, and the number ot values of X, /a, v which satisfy this equation, and ^^_ = Is n (3/2 - 5), which Is the degree of the asymp- totic surface. 474. Or we may proceed thus : The asymptotic surface contains 3n [n - 2) lines in the plane at Infinity which are the Intersections of the planes of Inflexion of the cone ^„ = 0, and contains, moreover, the curve of the rj'^ degree. In which the plane at infinity Intersects the cone; hence, the number of points In which the asymptotic surface is met by an arbitrary line In the plane at Infinity = M [n - 2) + n = w (3» - 5). For limitations of the number arising from the existence of singular points, see Palnvlu's second article.* tl, vol. G5. Mirnion of ArruoxiMATiux. 319 Mdhod of Approximation. . 475. Although it is necessary to know general methods of hantlling the equations of surfaces, yet in order to find the shape at particular points or at an infinite distance, it is most in- structive for the student to employ peculiar methods to suit peculiar cases. The method of approximation by transferring the origin to the particular point in question, and rejecting all terms ■which can be shewn to be small compared with those retained, gives immediately conical tangents or any other form which nearly coincides with a surface in the neighbourhood of a singular point. The form of a sui-ftice at an infinite distance may be found by a careful consideration of the relative magnitude of the coordinates in the same manner as the author has treated the subject in his Treatise on Curve Tracing. The kind of con- sideration required may be seen by the following example. 476. To find the plane and parahoUc asymptotes of the surface whose equation is ck' -h ?/' -I- 2' - Zxyz - 3a {yz + zx + xy) = 0. The equation may be written ^lv - a {li' — v) = 0, where u = x + y + Zj v = cc'-i y' + z' - yz - zx- xy. If xl' and v be of the same order of magnitude when a?, y, z are very great, we have for a first approximation m = 0, and for a second t/a = 0, the plane asymptote touching along a circle at infinity. If w"^ be large compared with v, the first approximation gives v = aif, and the next gives v = a[u-a\ which is a para- boloid of revolution. The same results m.ny be obtained by making the line x-y = z one of the axes of coordinates, so that the equation becomes 320 PROBLEMS. in which, if x, ?/, z be of the same order of magnitude, \/(3) x-\- a = 0- and if x be large compared with 7/ and z^ The conical tangent at the origin is if -^^ z^ — 2x\ XVIII. (1) Prove that the tangent plane to the surface xyz = a^ forms with the coordinate planes a tetrahedron of constant volume. (2) Find the equation of the tangent plane at any point of the surface xyz + 2abc = box + cay + ahz, and find the conical tangent at (a, b, c). (3) If tangent planes be drawn at every point of the curve of inter- section of the surface a [ijz + ex + xy) = xyz, with a sphere whose centre is at the origin, shew that the sum of the three intercepts on the axes will be the same for all. (4) A surface is given by the elimination of « between the equations F{x, y, z, a) = and f[x, y, z, a) = 0; shew that the direction-cosines of the normal at a point {x, y, z) are in the ratio F'{x)f\a) -fU) na) : F'{y)fia) -f{y) F'{a) : F'{z)f («) -/'(=) F'i")- (5) The points on a conicoid, the normals at which intersect the normal at a fixed point, lie on a cone of the second degree, having its vertex at the fixed point, (6) Prove that the projections on the plane of .ry of the normals to the ellipsoid '- -^ ^.+ —^= I, at points whose distance from that plane is c coso, * a" 6' c* touch the curve (ox)^ + (%)■' = {a' - i'j* sin^«. (7) Given {x* + y' + z' + c* - a')' = 4c' (x* + »/'), find the points the normals at whLch make angles a, ft, 7 with the axes, and the loci of points for which (i) 7 is constant, (ii) a is equal to ft. (8) A chord of a conicoid is intersected by the normal at a given point of the surface, the product of the tangents of the angles subtended at the point by the two segments of the chords being invariable. Prove that, O being the given point, and P, P the intersections of the normal with two such chords in perpendicular planes containing the normal, the sum of the reciprocals of OP, OP' is invariable. (9) Find the tangent cone at the origin to the surface (x' 4 J/' + ax)* - (c' - a*) (x' + z') = 0; and shew that as n diminishes and ultimately vanishes, the tangent cone I'KOBLEMS. .'Jlil contracts, and ultimately becomes a straight line, and as a increases up to c, it expands, and finally becomes a plane. (10) Shew that the 27 lines in a general cubic surface intersect in 135 points. (U) Apply the method of Art. 448 to find the singular tangent planes of the wave surliice. (12) Shew that the normals to any scroll along a generating line lie on an hyperbolic paraboloid. (13) If tangent planes at two points on a generating line of a scroll be at right angles, prove that the rectangle under the distances of the points of contact from the line of striction measured along that generating line will be constant. (1-1) If a series of straight lines, generating a surface, be described according to a law such that the shortest distance between two consecutive lines is of a degree superior to the first, it will be at least of the third. (15) Shew that the lines of striction of an hyperbolic paraboloid V- = a: are its intersections with the planes ,-. -, = 0. be "■ b^ c^ (16) A straight line intersects at right angles the arc of a fixed circle, and turns about the tangent with half the angular velocity of the point of contact round tiie circle. Trove that the surface so generated intersects itself on a straight line, and find the tangent planes at any point of this line. Shew that the line of striction is a plane curve, whose plane is inclined to tiie plane of the circle at an angle tan'' 2. (17) Find the asymptotic planes and the asymptotic surface of the conicoid ax* + by* + cz* = 2x. (18) Shew that the coordinate planes are the three singular asymptotic planes of the surface rijz = «'. (19) From difTcrcnt points of the straight line - = ^ , z = 0, asymptotic x" ? ' z^ straight lines are drawn to the hvpcrboloid - + "^^ - -, = 1 ; shew that they o ^^ a* b* c* will all lie in the planes — ^ = ± - \'2. a b c (20) Shew that the asymptotic planes to the surface z (x* + t/) - ax' - by* = 0, are parallel to the plane xy, and that the locus of straight lines in these planes having contact of the second order at infinity is s = a, or 2 = i; and that the axis of r is an evanescent asymptotic cylinder. 322 PUOBLEMS. (21) If the cone of asymptotic directions have a double side, shew that the surface will generally touch the plane at infinity, and that the section by this plane will have its inflexional tangents in the intersection with the tangent planes at the double side of the cone. (22) Shew that the conicoid which determines the inflexional asymptotes of the surface, whose equation is x* - j/V - 2a^i/z = 0, is an hyperboloid of one or two sheets, the latter giving imaginary asymptotes. (23) Discuss the form of the surface s (x + j/)* - a (^' - t/*) + i'z = at an infinite distance. (24) Shew that the asymptotic surface of z (a; + yf - oa* + tj:* = is a parabolic cylinder. (25) Shew that there is a conjugate line in the surface a' {2 {xf + s') - x'f = {xf + c») (2/' + 2« - ay. (25) Shew that the surface {x" - 2') [x^ + 3!/' - 8* + 9a')« = {6a (a;* + ?/* - 2*j + AaJ has a conjugate hyperbolic line in the plane of sr. CHAPTER XIX. VOLUMES, AREAS OF SURFACES, &C. 477. To fnd the differential coefficients of the solid contained between a surface^ given in rectangular coordinates^ the coordinate planes^ and jjlanes parallel to them drawn through any point of the surface. Let ic, ?/, z and x + Aa?, y + Ay, s + As be the coordinates of two points P and Q upon the surface. Draw planes through P and Q parallel to the planes of yz^ zx^ and let V be the volume CRPSOM cut oft' by these planes from the given solid. If ^V be the increment of T, 324 VOLUMES, AREAS OF SUKFACES, &C. when X is, changed to a; + Aar, -while y remains constant, and a simUar interpretation be given to the operation A^^, the volume Pyjl/=A^F; also the volume PQNM^ which is the increment of A_^ F when ?/ changes to 2^ + Ay, =A|^(A^r), which is easily seen to be the same as A^(AyF). Let 2;,, 2^ be the least and greatest values of z within the portion of the surface PQ^ therefore PQNM lies between z^AxAt/ and z^AxAy, A F\ . /A..F> "K Ax J '\ Ay J lies between z, and z„. Ay Ax ' ^ Proceeding to the limit, in which z^ = z,^ = 2, we obtain cPV ^ cTF ^ dydx dxdy We may observe that, since the volume PrM is ultimately equal to the area RM x Aa;, the partial differential coefficient dV dV -^ represents the area RM^ and similarly -, the area SM. 478. The differential coefficient of the volume of a wedge of the solid contained between the planes of zx^ xy^ a plane through the axis of 0, and a plane parallel to yOz may be obtained as follows. Let Fbe the volume included between the planes zOx^ ^Oy^ the surface, the plane whose equation is ?/ = tx^ and a plane parallel to yOz through any point (a*, y, s), then A,Fis the in- crement of F when t changes to < + A^, x remaining constant, and is the volume which stands on a base whose area is \xAt.x) A^(A,F) is the increment of A, F when x changes to x + Ax^ and is the volume which stands on a base whose area is \ [x + Axf At - ix'Af. = {x + 1 Ax) Ax Af ; hence, as before, -^ ' is between z^ (a^ + ^Aa:) and z^[x+lAx), d''V and, proceeding to the limit, , -, = zx. VOLUMES, AliEAS OF SURFACES.ll&cT ' , /:325 ' , X ' // • /' ■ ^ 479. To find the differential coefficient ofthej>oiti6^of\/. surface given in rectangular coordinates, cut off by the coordi/iate j l)lanes, and plaiies parallel to them draion through any point of . the surface. ''. • Let P, Q be the points {x, ?/, z) and [x + Aa-, ?/ + Ay, z + Az), • S the surface PROS, cut off by the planes through P. A^, which is the increment of A^/S when y is changed to ?/ + A^/, and is evidently the same as A^ iA,^)- Let 7,, 7^ ^c the greatest and least inclinations of the tangent plane to the plane of xy for any point within the surface PQ. Therefore PQ is intermediate between AicA^/ sec7, and Ax^y 86072- Hence — — ^— or is intermediate between scc7, Ay Ax- and sec72, which are, in the limit, each equal to sec 7. T'"='''='''"-'=' 1^ "■• from the plane z O.r, and let rbe the volume of the wedge of a cone contained between the planes zOx and zOP, and the given surface, the axis of the cone being Oz, and 6 the scmi-vcrtical angle. OPRrS is the increase of the volume when 6 increases by A^, remaining constant, therefore OPRrS = A..V. 326 VOLUMES, AHEAS OF SURFACES, &C. OPSQT'is the increase of A, F when ^ becomes 4 A^, and therefore = A4. (A^ V), and shnilarly = Ag (A^ V). If OP, OS, OQ, OT intersect a sphere, whose centre is and radius OP, in P, s, q, t the volumes of OPSQT and OPsqt will be ultimately equal, and Ps = r/\d, Pt = r sin ^. A^, therefore A^ (AjF) is ultimately equal to -Jr" sin^ A0A^; 482. To /?«(/ f7//•,, and 1 : cos-v/r,, each of which becomes ultimately r : ^), where p is the perpendicular from on the tangent at P; .-. A* (A^.S') = - sin ei {x, ?/J - (f> [x, ?/,)} JXy 484. The student will have to determine in every particular case the best order in which to make the summation of the elements ; in some cases it will be advisable to take elementary slices of the surface, instead of the elementary parallelepipeds, as when the area of a plane section is known. Thus, in the case of an ellipsoid, the area of a section BPQ is irQN, RN, and a slice of the thickness dz = ~f- (c'-s')c/^, , , , . irah whence the volume is - .,- 1 [c' - z") dz = -^ irahc VOLUMES, AREAS OF SURFACES, &C 329 485. He must also judge wlietlier it is advisable to use other coordinates than those in -which the equation of the surface is given. Thus, the equation of an anchor-ring being {x' + f + z' + 6' - a^f - 4c' (ic' + y') = 0, if we make a? -^^ y" =■ r"^ ^ z^ = a' — [r — c)\ we can sum the elements which have their projections on the circular ring 2'iTrdr^ and the volume is [ ^irrdr V[a*-(r-cy-'] = [ 47r (r' + c) dr s/ {a' - r") = 27rc.7ra\ J c-a •' -a 486. To find the volume contained bctiveen the surface ichose equation is [x-\- yY = 4:az^ the tangent plaiie at a given j^oint, and the planes of zx and yz. Let the given point be (/, //, /<), the equation of the tan- gent plane is ^ -^ y = a/ [j] {- + h) ] the volume required is / (h\ ^x-\- y]' jjjdxdydz^ the limits being from ^ = a / (~) {•^+!/) - ^' to ^~T ' then from 2/ = to y = 2 \J{ah) — a*, since the tangent plane meets the surface where {x + y)'^ — 4 \/[ah) (a; + y) + 4a/i = 0, lastly from a; = to a; = 2 sj{ah). The volume is //: — \x^y-2,J{ttl,)Ydydx l-Jx.,^l,akW,uJl^-^f=yU- u u 330 t'OLUI^IES, AREAS OF SURFACES, &C. This result may be verified thus. Let A OB be the surface, ACB the tangent plane along the line AB^ ADB a plane parallel to xO>/, adh any section of the surface parallel to xOy. Then area adh : area ADB :: ad'' : AD' :: Od : OD ; therefore volume A OBD = I 2ah . j dz = alt' ; also volume A CBD = ^2ah . 2h = ^ali' ; hence the volume required Is -— . y' z' 487. To find the volume of the elliptic paraholoid ; + ~ = 2.r, cut off hy the plane Ix + my + nz =^?. Perform the integration in the order a*, y, 0, _ y' z' _p — my — nz ^'" 26"^2"c' ^'^" ~~l • For a given value of 2;, the values of y at the curve of intersec- tion are given by the equation x^ = ir.^, , 21)111 h ., 2h . . ^ ,,, or y -i ^y+-z --j{p-nz) = 0, (1) VOLUMES, AUKAS <>F SUKrACES, &C. 331 of which y,, ?/., arc the roots, aiul z must be taken between the limits which correspond to y, = y„, that is 2;,, z,^ are the roots of the equation -j^fJ"j^{l/-I/.)(i/.-y)dy^^^ by (1), = ^ /^^ £■' ((2/ - 2/J fy. - Z/.) - (y - !/X] ^^Udz therefore the volume = f ^ j (7' - w')' <^«, where 27 = z,^ - 2„ u = s - ^ (.2, + ^J 4 • ci 2* ' cos^dtW^ putting ?< = 7 sin I and i(^^_.~^)^= .^-+ I + -^-; .'. volume = - ^/{bc) ^-^ j^ . 4 6 The student may verify this result by the summation of elementary slices bounded by planes parallel to the given plane. 488. Tojind the volume contained hetwcen surfaces cos a COS0 V(l -cos'^'a sin*^) d(f> = — [C-sin '{cos a sln(C-y8)} - sin'' (cos a sln/3)] o since cos B = cos a sin ( C - /9) and cos A = cos a sin /3. VOLUMES, AREAS OF SURFACES, &C. 333 Wc have given this as an example of the determination of the limits in the case of polar coordinates, but the result is obtained immediately from the area of the spherical triangle, the volume required being the sura of an infinite number of pyramids whose vertices are in the centre, the volume of any one of which is ^adS, and the whole volume = ^a x area of the spherical triangle. 490. To find the volume of a icedge of a sphere cut off hy a right circular cylinder^ a diameter of xchose base is a radius of the sphere. Let the equation of the sphere be p^ + z'^ = a", and that of the cylinder p = a cos(/). ra ra cos (p The volume is I I 2p \/[d' — p') dpd

)d<^ = l«M«-ii;;(3sin/ dz__dzdy\^ fiv{X'-^'){X'-ry')[fj:'-v') •'• ^^ [dv df^ dv dfji) ~ I3y {y' - (3') ' ,,^ 1 fdi/ dz . dz dy\ 1 , '''=iUd-^-d-.di)'^'' - ^ X f^-i^'-^')i^'-y")(f^"-'l dud. _ X(m--v-)V{(X -- /3'-)(X---7^)1 . , y Vl(/^'^ - /3-^) (7-^ - /.^j {/:i-^ - O [i' - v')] ^ _ (/.'-OV{(V -A.-)(V--v T; ,7^,7, Art 286 The area of the surface cut off by four confocal hypcrboloids, for which /i = yti, and fx.,, v = v^ and v,^ Is j'^'fi'Mdfi X I"' Kdv-T'MdfM X [" v^NWv, where J/= y/f^. 1^.^ (y .y^ , -^= ^ (^-^Z. Vj-f/^T) ' 493. If the position of a point be given as the intersection of three surfaces F{x,ij,z) = a, G[x,y,z) = ^, and II{x, ij, z) = y, the expression for a volume may be obtained similarly as follows; when 7 is constant, the variation of a and /3 determines 33G VOLU^FES, AREAS OF SURFACES, &C, a surfiicc of which an elementary portion is {A' + B^ A-C'^)^ dad^^ and the equation of the tangent plane at this element is the perpendicular on which plane from a point determined by ot, ^, 7 + dy is . dx „ dii ^dz\ , dy dy dyj dx dy dz da da da do.' da' da dx' dy' dz dx dy r//3' dz d/3 and J' = d^ dx' d^ dy' d^ dz dx dy dz dy dy dy dy' dy' dy dx' dy' 'dz {A' + B'+cy- hence the volume of the elementary parallelepiped, whose opposite faces correspond to 7, 7 + dy constant, &c., is and the volume = / / jJdad^dy = I j ~ dad/3 dy, here \ J of tlie cone and ellipsoid being ellipses, and k given by the equation ffh ^ hf ^ fff _l gh - iif hf - vg fg - wh k* ' Quart, Jour, of Math., vol. ii. XX 338 PKOBLEMS. (10) Prove that the volume cut off by the i)lane rj = h from the surface «»x2 + iV = 2 {ax + bz) f is -^^ . (11) A cavity is just large enough to allow of the complete revolution of a circular disk of radius c, whose centre describes a circle of the same radius c, while the plane of the disk is constantly parallel to a fixed plane, and jjcrpendicular to that of the circle in which the centre moves. Shew that the volume of the cavity is ~ (3/7- + 8). (12) Two cones have a common vertex in the centre of an ellipsoid, and their bases are curves in which the surface is intersected by plHiies parallel to the same principal plane, prove thai the volume of the ellipsoid con- tained between the cones varies as the distance between the planes. (13) Prove tliatthe volume contained between the plane z = {c - x) cot « and the surface a-z' ^ {x - c) (j:* + »/') = is — (3 cot « cosec a - 2 cos^ a - 3 log cot ^a). (14) The volume contained between the surface c' & \a hj c \a bj ab and either of the planes yz or xz is „„- . (15) Shew that the whole volume of the surface whose equation is (x* + J/' + s*)' = cxyz is equal to ^-7- . ytio (16) Investigate the form of the surface whose equation is and shew that its volume between values of tan'' - from to 27r is -,7r^a^. X (17) Shew that tlie volume of the closed portion of the surface whose equation is 4a (?/' + z' - 40') + (x' - a") (3z + lOa) = is %\.\-rr {oaf. (18) If ^5 be an element of the surface of an ellipsoid at any point, and ui the area of a section by a jilane drawn through the centre, parallel to the tangent plane at that point, prove that the limit of 2 — =4, the summation being taken over the whole surface. Find AS in terms of a, /3, if a: = a cos a, y = 6 sin a cos/i, and z = c sin a sin/3. (19) If 5 be a closed surface, dS an element about P, at a distance r from a fixed point O, the angle which the normal drawn inwards n?akes I'KlMJLK.MS. 3.'}U with OP, shew tliat llic volume contained by the surface = ij J r cosr/)d>S, the summation being extended over the wliole surface. O being the centre of an ellipsoid, apply the formula to find its volume, interpreting geometrically the steps of the integration. (20) Shew that 1/ extended over the surface of an ellipsoid is equal to _ f 3 + p + -Jx volume of the ellipsoid. (21) Prove that the area of a closed surface, no plane section of which has singular points, may be expressed by the definite integral sin dfpdO u:"- p where p is the perpendicular from the origin upon the tangent plane. (22) If each element of a closed surface be multiplied by -^ cos 0, where r is the distance of the element from a point O, and is the angle between the direction of r and the normal to the surface measured outwards, shew that the sum of all such products is or 47r/t, according as O is without or within the surface. (23) If r be the distance from a point O of any element dS of a spherical surface, determine the form of the function/ (r) when / / (/5', the sum- mation being effected over the whole surface of the sphere, is constant for all positions of O within the sphere. (24) Shew that the shortest distances between generating lines of the same system drawn at the extremities of diameters of the principal elliptic section of the hviicrboloid, whose equation is — + — - ^ = 1, lie on the ■ ' a* h* c- . Prove also that the volume cj:?/ ahz surfaces whose equations are , ' , = + - ,. * X* 1 y* a - included between these surfaces and the hyperboloid ahc /a* - i' „ , a\ CHAPTER XX. TORTUOUS CURVES. CURVATURE. TORTUOSITY. 495. We have already shewn that curves may be con- sidered as the complete or partial intersection of surfaces, but in the investigation of the equations of tangents, osculating planes &c. we shall also look upon a curve as the locus of points which satisfy more general laws, the algebraical state- ment of which assumes the form of equations between the coordinates of any point of the curve and variable parameters, the number of equations being two more than the number of parameters. Instances of the latter mode of representation of a curve occur in dynamical problems, in which the curve is defined by equations between the coordinates of the position of a particle and the time of its arrival at that position. If the parameters were eliminated from the equations con- necting the coordinates and parameters, the result would be two final equations which would be the equations of two surfaces whose complete or partial intersections would be the curve in question. 49G. If the coordinates of any point on a curve can be expressed as functions of a single parameter t, so that for each value of t there is a single value of each coordinate, the curve is called umcursal. 497. As an example of an unicursal curve, Ave may take the Helix, which is generated by the uniform motion of a point along a generating line of a right cylinder as the gene- rating line revolves wltli uniform angular velocity about the axis of the cylinder. TOIiTL'OUS CURVES. 341 If \vc take the axis for the axis of z^ and tlie axis of x through tlie generating point at any initial time, 6 the angle through which the generating line has revolved when the point has moved through a space z on the generating line, we have, for the co- ordinates of the point, a being the radius of the cylinder, x = a cos ^, y = a sin d^ z = nad ; here 6 is the variable parameter, and the curve is the intersection of the surfaces x^ + ?/ = a^. and y = x tan — . 498. In order to explain the terms employed in the ex- amination of curves which are not plane, we shall consider such curves as the limits of polygons whose sides are indefinitely small ; and we observe that the plane which contains any two consecutive sides of the polygon of which the curve is the limit, does not generally contain the next side. The term double curvature, as is remarked by Thomson and Tait,* is not a proper expression, since there are not two curvatures ; and the property, that the plane in which the curvature is taking place at any point changes as the point changes, would be better represented by calling the curve tortuous and the measure of the corresponding property tor- tuositij. 499. Osculating plane. The plane containing two sides of the polygon of which a tortuous curve is the limit is in its ultimate position an oaculating plane of the curve. 500. Normal Plane. Any side of the polygon in its limiting position is a tangent to the curve, and a plane drawn per- pendicular to the tangent though the point of contact is a normal plane ^ being the locus of all the normals at the point. 501. Principal Xormal. The particular normal which lies in the osculating plane is called the pfrincipal normal. * Xatural Philo.iophi/. Art. 7. 342 TORTUOUS CURVES. CURVATURE. 502. Binormal* The normal wlilcli is perpendicular to the osculating plane is called a binormal, being perpendicular to two elements of the curve. 503. Polar Be.velojmble.-^ Let an equilateral polygon be inscribed in a curve, of which consecutive sides are FQ, QR^ BS, ST, and let p, q, r, s be the middle points of these sides. Let Aai), Bbq, Ccr be planes perpendicular to these sides, forming the polygon ABCD by their intersections. If the sides PQ, QR, ... be diminished indefinitely, their directions are ultimately those of tangents to the curve, the planes Aap, Bhq, ... are ultimately normal planes to the curve, the planes PQR, QRS, ... are osculating planes, and the surface generated by the plane elements Aah, Bhc, Ccd, ... is ulti- mately the developable surface enveloped by the normal planes of the curve, of which ABCD ... is ultimately the edge of regression. The developable enveloped by the normal planes is called the Polar Developable. 504. Circle of Curvature. A circle can be described con- taining the points P, Q^ R] when the sides are indefinitely diminished, this circle lies in the osculating plane, and its curvature may be taken as the measure of curvature of the curve in the osculating plane. Let the plane PQR meet Aa in J/, and let pU, qU be joined, then since PQ \s perpendicular to the plane Apa, it is perpendicular to ^;Z7, similarly QR is per- pendicular to q ?7, IT is therefore the centre of the circle through PQR. Therefore the centre of the circle of curvature Is the point of intersection of two consecutive normal planes and the osculating plane. 505. Polar Line. Draw pa, qa to any point in Aa, then, since Pp = Qp, a Is equally distant from P and Q, and similarly from Q and R, and therefore from every point in the circle of St. Venant. t Mon<'c. Tofac^p 3^2 ■ MunjJ/t k S^n, Uth TOKTUOUS CURVES. CUKVATUKE. 343 curvature. The line of luterscctlon of two consecutive normal planes is called by Monge the ^^olar Une. 50G. AtujU of Continfjence. The angle ^jL^*/, which is equal to the angle between the two consecutive sides PQ^ QR of the polygon, is ultimately equal to the angle between two consecu- tive tangents, and is called the angle of contiiigence. 507. Sphere of Curvature. Any point in Aa is equally distant from P, Q and i?; also any point in Bh is equally distant from Q^ II, and S] therefore their point of intersection is equally distant from the four points P, Q, P, S. Hence, it follows that a sphere can be described whose centre is P, and which contains the four points P, Q, P, JS, this sphere is ultimately the sphere which has the closest possible contact with the curve, since no sphere can be made to pass through more than four arbitrary points, it is therefore called the S2)here of curvature : the locus of its centre is the edge of regression of the polar developable. 508. Evolutcs. It has been shewn, Art. iiS, that, if a be any point in the intersection of the planes normal to PQ, QH, at their middle points p, q, ap and aq will be equal and will make equal angles with Aa. Produce qa to meet Bb in b] then a string, placed in the position baj), would remain in that position if subject to tension, since the tensions of the portions ab, ap resolved parallel to Aa would be equal, and, if its extremity were then moved from^> to q it would occupy the position baq. Similarly, if rb be produced to c in Cc, and if sc be produced to d in Dd. If we proceed to the limit, it follows that a string may be stretched upon the polar developable in such a manner that the free end, starting from any point in the curve, would describe the curve, if the string were unwrapped from the surface so that the part in contact with the surface remained stationary. The portion in contact lies on a curve called the evolute. Also, since the position of the line pa is arbitrary, the curve which is the limit of a,b,c,dy.. will change its position ac- 344 TORTUOUS CURVES. CURVATURE. TORTUOSITY. cording to the position of «, hence the number of cvoUites \s infinite. All the e volutes of a curve are geodesic lines of the polar developable. 509. Anyle of Torsion. The plane pUq perpendicular to A Ua contains the sides PQ^ QR^ and the plane q IV perpen- dicular to BVh contains the sides QB^ ES, and, since qU^ qV are perpendicular to the line of intersection QR of the two planes, the angle UqV is their angle of inclination. This angle, which is ultimately the angle between consecutive osculating planes, is called the angle of torsion. Also, since a circle goes round BVUq^ the angles UqV and ZTSKare equal, and the angle of torsion of the curve FQR^ is equal to the angle of contingence of the edge of regression of the polar developable. 510. Locus of Centres of Circular Curvature not an Evolute. Since q U will not, if produced, pass through F, because q U and q V include an angle in the same normal plane, the locus of the centres of circular curvature is not one of the evolutes. 511. Rectifi/ing Developable. If through every point of a curve a plane be drawn perpendicular to the corresponding principal normal, these planes will envelope a torse on which the curve will be a geodesic line, since its osculating plane will contain the normal to the surface at every point ; if therefore the torse be developed into a plane, the curve will be developed into a straight line. On account of this property the torse is called the Rectifying Developable. 512. Rectifying Line. The line of intersection of two con- secutive planes, enveloping the rectifying developable, is called the rectifying line for any point of the curve, being the line about which the curve must turn at that point in order to become straight, when the torse is developed into a plane. It may be observed that the rectifying line is not generally coincident with the binormal, which is the normal perpendicular to the osculating plane. TORTUOUS CUUVLS. CUUrATUKE. TOKTUOSITV. 345 In the figure at p. 3-42 the surface whose edge of regression is the limit of ABC... is the rectifying surface to the curve which is the limit of abc An is the rectifying line at a, and the binormal does not coincide with the rectifying line unless ^)rt be perpendicular to Aa^ or a be the centre of curva- ture of the involute of ahc... 513. If the polygon FQRS... were transformed into a plane polygon by turning the portion QBST... through the angle of torsion VqU about QB, and the portion BST... about BS through the corresponding angle of torsion, the inclination of any side ST in the new position in the plane of FQR would be inclined to PQ at an angle equal to the sum of the inclina- tions of the sides taken in order, and estimated in the same direction. Proceeding to the limit, we see that if, as a point moves along a tortuous curve, at every position which the point assumes the curve be turned about tiic tangent line through the angle of torsion, the curve will be replaced by a plane curve, such that the inclination of the tangents at the starting point and any other point will be the sum of all the angles of con- tingence ; if, therefore, s be taken for the angle between the tangents in the plane curve, dz will be the angle of contlngcnce corresponding to the extremity of the arc traversed by the moving point. 514. Bate of Torsion. The rate per unit of length of arc at which the osculating plane twists about the tangent line at any point, called the rate of torsion^ is measm-ed by the limit of the ratio of the angle of torsion to the arc at the extremities of which the osculating planes are taken. If, as we pass from PQ to QB^ see figure, p. 342, QB be turned in the plane PQB so that PQB is a straight line, and the plane QBS be then turned through the angle VqU^ the process being repeated along the whule of a given arc, the perimeter will become rectified, and the inclination of the last to the first position of the plane containing two elements will be the sura of all angles such as 176" between the extremities of the arc so rectified. Y Y 34G TANGENTS. Proccedinj^ to the limit, it follows that, if osculating planes be taken along the curve, and the elements of the arc be rectified in each osculating plane in order, the angle between tlic first and final positions of the osculating plane when the curve is so rectified will be the sum of tiie angles of torsion throughout the arc. If, therefore, t be this angle, ch Avill be the angle of tor.-^ion, corresponding to the point at which the last osculating plane is drawn. 515. Integral and Average Curvature:'^ The integral curva- ture of any portion of a curve is the angle through which the tangent will have turned as we pass from one extremity to the other, the average curvature is its whole curvature divided by its length. Let a sphere of unit radius have its centre at a fixed point, and let radii be drawn parallel to the tangents to the curve at successive points, the length of the curve traced on the sphere by the extremities of the radii measures the integral curvature oi" the portion of the curve considered, and the average curvature is the integral curve divided by the length of the curve. 51G. Integral and Average Tortuosltg. These are respectively the angle through which the osculating ])lane has turned in passing from end to end of any portion of a curve, and this angle divided by the length of the arc considered. On the sphere described in the last article let a curve be described by the poles of the tangents to the curve which measures the integral curvature, the length of this curve measures the integral tortuosity, and this length divided by the length of the arc of the tortuous curve the average tortuosity. Tangents. 517. Tangent to a curve at a given imint. Let s, s + A.s be the lengths measured along the arc of a curve from a given point to the points P and Q^ whose coor- dinates are x^ y, z and x + Aa-, y + A_y, z + A.:, and let c = chord rq. * Thomson and Tait. Not. Phil., Arts. 10-12. TANGKNTS. o47 As Q approaclics to ami ultimately coincides ^vItll I\ the cliord PQ and arc As become equal, FQ is the direction of the tangent at P, and the direction cosines of PQ^ viz. Ax Ai/ Az , 1.- . 1 (^-^ ^y (^z -, ■- , become ultimatelv -7-, , , -?- . c. c c • as ' (/s ' ds The equations of the tangent arc therefore dx dij dz Also since c' = [Ax^ + [AyY + (A^)^ (1) Let the equations of the curve be given in terms of a variable parameter ^, in the form x^4>[e), y = ^{d), z = x{0), then dx : d>/ : dz = f [6) : i|r' (^) : ;^' [0), and the equations of the tangent at a point corresponding to 6 arc ^ — x_r) — y_^—z (2) Let the equations be those of surfaces containing the curve F{^, 77, ^ = 0, and O (f , 77, ^ = 0. Then, at any point P of the curve, F'{x)dx-^F'{y)dy^-F'{^dz=^0, and (7' (.r) (/a; + (7' (3/) dy + 6-" (;r) c7^ = ; ■whence the equations of the tangent PQ may be written iT' (.,) [^-x)^F' iy) Iv -y) + F' [z) {^-z] = 0, and G' (x) [^-x)+ a [y] {ri-y)+ Cr {z) {^- z) = 0, ■which equations represent analytically the fact that the tangent to the curve lies in the tangent plane to each surface at the common point P. (3) If the surfaces, the intersection of ■\vlilch gives the curve, be cylindrical surfaces whose sides are parallel to the two axes of z and ?/, and their equations be v=f[^\ ?={l)> ^''^ equations of the tangent will be r-~~ = f(.r)fi-.r). 348 MULTIPLE ruINTS. These equations are the analytical representation of the fact that the projections of the tangent to the curve on the co- ordinate planes of xy^ zx are the tangents to the respective projections of the curve; which is obviously true, since the projections of P and Q have their ultimate coincidence simul- taneously with that of P and Q. 518. To find the directions of the branches of the curve of intersection of two surfaces at a multiple j^oint of the curve. The equations of the surfaces being i^(?,^,r) = 0, and 6^(?,^,r) = 0, and (x, i/y z) being a multiple point P on the curve, let tz^ = '^~y = ^^ = r (I) be the equations of a line through P; the points in which this line meets the surfaces are given by the equations F{x + Xr, y + /.u; .z + vr)=0) 2) and G{x + \r, y ^ [xr^ 2 + j/r) = 0j' there are an infinite number of directions which give two values of r equal to zero, since the curve has a multiple point at P; therefore the two equations must be one or both identically satisfied, or else they must not be independent equations, i. If only one of the equations (;}) and (4) be Identically satisfied, suppose this to be (3) ; then (.f, ?/, ~) will be a multiple point on the surface P(f, Vi ?) = ^ j ^^^^t i^' t''"'s be a double point, the line (1) must be one of the tangents whose directions arc given by MLi.Tii'Li: roiNTS. 349 and, since it lies In tlic tangent plane to G (^, 77, ^) = 0, These eqnations give the directions of the two tangent lines, which are the Intersections of the conical tangent to the first snrface with the tangent plane to the second ; and, similarly, for higher degrees of multiplicity. il. If (3) and (4) be both Identically satisfied, the line (I) ■Nvlll be in any of the directions of common tangents to i^(|, 7;, ^)=0 and G (|, ?;, ?) = ; the directions are therefore given by where s and t arc the degrees of multiplicity of the multiple points of the two surfaces at (cc, y, z). ill. If neither (3) nor (4) be identically satisfied, but the two equations be identical so as to be satisfied by an Infinite number of values of X : /i : v, there will be a surface AF-\- BG = 0^ which will pass through the intersection of F= and G = 0, for which (^4; 4 /^ f + vj]{AF+BG) = win be identically satisfied, if j be the value of the equal ratios -tttt \ i v ,- 1 i 1 ' A ^ G [x] G [y) In this case, therefore, \ : /z : v is determined by one of the equations (3), (4), and If in any of these cases two values of X : /i : v be equal, there will be either a point of osculation or cusp on the curve. 519. As an example of case HI. in the last Article, suppose we wish to find the directions of the tangents at the point 3,j0 koi:mal plane. (a, 0, 0) In the curve of intersection of the liypcrboloiJ and hyperbolic paraboloid, whose equations arc ?1 + -^ - - = 1 ,.2 ^2 and J -y =2(^-o). At this point the surfaces have a common tangent plane, whose equation is x = a\ the third surface, on which (a, 0, 0) is a multiple point, is in this case the cone X V /I IN, /I 1 and the direction cosines of the tangents to the curve are given by 520. Normal plane of a curve at a given point. The normal plane being perpendicular to the tangent to the curve, its equation is 521. To find the eage of regression of the polar developable of a curve. The edge of regression Is the locus of the Intersection of three consecutive normal planes to the curve. . The equation of the normal plane at (ic, y, z) is {^-x)dx + {'n-y)dyV{l:-z)dz^O, (1) that of the normal plane at a consecutive point Is found b}'- writing in this equation x + dx for a*, &c., the line of intersection of the two normal planes will lie in the plane {^-x)d'x+{'n-y)d'y^-[^~z)d'z- [dxY-[dyf- {dzY=0.i2) Again, writing x-\- dx for ic, &c., we obtain a plane In which the line of Intersection of the second and third normal planes lies, (^-x)d'x^{v-y)d'y + {^-z)d'.z - 3 {dxd'x + dyd'y + dz d'z) =■ 0, (3) OSCULATING TLANE. 351 and the coordinates of the point of the edge of regression satisfy these three equations. If we elaninate r, 3/, z from tlie equations (1), (2\ (3) and the equations of the curve, Ave shall obtain the two equations of the edge of regression. The line of which (1) and (2) are the equations is ^Monge's polar line, which is the axis of the osculating circle. The point given by the three equations (1), (2), (3) is the centre of spherical curvature corresponding to the point (ic, y, z) of the curve. b'1'2. To find the differential coefficient of the arc referred to 2)oIar coordinates. Transforming to polar coordinates x= r sin 6 cos 4> = P cos <^, y = r sin B sin 4> — p sin cp^ z =r cos 6 J p = r sin 6 J (dz\' (dp\' (d. 10 J -\d6j-^''^'-'''''"\d6 The equation is easily obtained geometrically by observing that ultimately {AsY = [Iry + [rAd)' + (/• smO A(f)j\ Also, \f J) be the perpendicular from the pole upon the tan- gent, and ^jr the angle between /• and the tangent, ^) = r sln^/r, A.s , ,.. . , fds' Ai and = sec-vlr ultiniateh', .*. ( ,- = „ .. \r ^ •" \drj r--p Osculating Plane. 523. Equation of the osculatincj idane. The osculating plane may be considered as the plane whicl passes through three consecutive points, whose coordinates are cr, y, 2! ; X + dx. . . . and x + 2dx 4 d^x. . . . Ml 352 OSCULATING TLANE. Let the equation of the osculating pLanc be .-. AJx + BcIi/-\-Cch = 0, and A {2clv + iVx) + B {2J>/ + d'y) + C [2dz + dh) = 0, or Ad'x + Bd'y + Cd'z = 0, hence the equation Is ^-x, 77-?/, ^-z I dx^ dy^ dz 1=0. d^x^ d^y^ d^z \ It may be noted that the equations of the tangent and osculating plane are of the same form, whether the axes be rectangular or oblique. 524. It should be obsei-ved with respect to the notation used above that if x^ ?/, z be supposed given as functions of ^, and we take points corresponding to values t^ < + t, ^ + 2t, which is the same as making t the independent variable, the values of X for t + T and t + 2t are dx d^x dt at 2 dx ^ d'x (2tY and.+ ^2r+^^i^-f...; and if the first be w^rltten x + Ax, the second will be x + Ax + A{x + Ax) or x + 2Ax + A''x ; hence if d be written for A, where t is indefinitely diminished, dx = — , T and d'^x = -,^ r^ ultimately. dt dr 525. As an exercise the student should find the equation of the osculating plane, considered as given by any of the following definitions : i. As a plane containing a tangent and a point In.definltcly near the point of contact. ii. As a plane containing a tangent and parallel to a con- secutive tangent. OSCULATING PLANE. 353 lii. As a plane which has a closer contact with the curve than any other plane. In employhig the definition ii. he may shew that the shortest distance between the tangents at the extremity of any arc (7s is generally of the order of (7/. 526. Direction cosines of the hinormaJ. The direction cosines of the blnormal, which is perpendicular to the osculating plane, are in the ratio dyd'z — dzd'y : dzd^x - dxd'z : dxd'y — dyd'x^ and the sum of the squares of these expressions =[[dxy^{dyY-^{dzY][[(rxy-^{d:-yy->r{' [x, y, z) — be the given equations, then, using the notation of Art. 420, Udx +Vdy +Wdz =0, U'dx + V'dy + W'dz = 0. II U V W 11 Let Z), -E", i^ denote the determinants ' ' ^y, ; dx dy dz , .-. -^ = -^ = -p = /c suppose, whence d^x = IcdD + Ddk^ d^y = hdE + Edh^ d'z=kdF + FdJc', .-. dyd'z-dzd'y = h'[EdF-FdE), hence the equation of the osculating plane is [EdF- FdE) [^-x)+...= 0. 530. Equation of the osculating plane in terms of the equations of the tangent planes to the su?faces. Employing the notation of the preceding article, we see that I)U+EV+FW=0; .-. UdD + VdE+WdF+DdU^ EdV^ FdW=0, THE INTERSECTION OF TWO SURFACES. 355 , ,^, , dU J dU , dU and dU= ax ~,~ + du -y- 4 «2 -7— dx '' dy dz ~ \ dx dy dz] dx if r denote the operation In the brackets, in the performance of which Z>, E^ F are considered constant ; .-. DdU+ EdV+ FdW= JcV (0) , hence UdD +VdE -^-WdF =-kr'{c{>), simikrly U'dD + V'dE+W'dF= - JcV' ((/>') ; .-. EDF- FdE=k[Ur-{4>']-UT' {c}>)], and the equation of the osculating plane becomes r{')[U{^-x)+v{v-y) + w{!:-z)} = r{ci>){u'{^-x)+...}. 531. To find the oscidating plane of the intersection of two concentric and coaxial conicoids. Let the equations be ax'' + by' + cs"* = 1 , ax' + l3y' + yz' = l, Z) = 4 (^7 - cl3) 7JZ = Ayz^ E= Bzx, F= Cxy, EdF- FdE = E\l f Ci = BCz'x'd (y (1) .E) \z, = BCx' [zdy - ydz) = hBCx' {Ez - Fy), and by (1) Ez -Fy = 4:{(x- a) x] .-. EdF- FdE= XkBC (a - a) x\ and tlie equation of the osculating plane is "-- %' (f - .,) + ^ - -S' (, - 2,) + ^- " .' (r- ^) = 0, which may be reduced to BG ,. CA , AB ,Y, . C^-Z') (7~o) ^^+(7^Kor:^j ^^ + (a-rO {0-b) " ^"^ ' 356 PKINCIPAL NOUMAL. 532. Or, by the method of Art. 530, since the equation may be written {D'a + E'^ + F'y) [ax^ + hi/v +cz^-l) - {B'a + K'b + F'c) (aa:| + ^i/rj + yz^- 1) = 0, and the coefficient of = BC [a- a) x^, as before. 533. To find the condition for a stationary osculating plane of the curve of intersection of two surfaces. The equation of an osculating plane is {EdF-FdE)[^-x)+...=^% the line of intersection of this plane with the next consecutive osculating plane is in the plane {Ed'F-Fd'E) {^-x) +...- {EdF-FdE) dx-...= 0', the last three terms are identically zero, since dx = kD^ and in order that the two osculating planes should coincide, Ed'^F- Fd'E _ Fd'B - Bd'F _ Bd^E- Ed^D EdF-FdE ~ FdB-BdF ~ BdE-EdB ' which are clearly equivalent to one distinct equation ; and each c ^. c c ' ix d'B\Ed'F-Fd'E\+... , ot the tractions is equal to —fr^rrnrr^ — frrW i the nume- ^ d'B[Ed]^ -FdE)-\-... ' rator of which vanishes, .-. d'B{EdF-FdE)+...= 0. Principal Normal. 534. To find the equations of the principal normal at any point of a curve. The principal normal is perpendicular to the tangent line and also the binomial, the direction cosines of which are pro- portional to dx^ dy, dz^ and dyd'^z-dzd'y^ dzd'x-dxd'z^ dxd^y — dyd^x respectively. Now we have identically d'x[dyc['z - dzd'y) + d'y [dzd'x - dxd'z'j + cPz {dxcPy-dyd'x = ; MEASUKE OF CURVATURE. ' 357 ' and if we make 5 the independent variable, d:'xdx + d^y dy -\-d^zdz = d^s ds = 0. These two equations shew that direction cosines of the principal , . , d^x d'y d^z , . . normal are proportional to -yj , -,% , --p^ , and with a general Independent variable, its equations are d ldx\ d fdy\ d fdz^ dd Us) dd \ds) J9 \dsj 535. If from any point in a curve equal distances he measured along the curve and its tangent^ the limiting position of the line joining the extremities of these distances is the principal normal. From the point (re, ?/, z) let equal distances a be measured along the curve and the tangent to the points Q^ T, Tiie co- ordinates of (2 are a; + ^ o- + [-— 4 £ ) ^ , &c. and those of T ds \ds J 2 ' dx ., .,..,,.. « + -,- 0", (xc, £ vanisliing in tlic limit. The equations of the line QT are f — X- dx -ds"" V -y -P ?-^ dz -ds"" d'x ds' + £ d:y ds' + £' d'z ds' + €" therefore the limiting position of QT'is the principal normal. Cauchy proposed, as a definition of the Principal Normal at any point, the limiting position of the line joining the points on the curve and tangent, whose distances from the point of con- tact measured along the curve and tangent respectively are equal, by which means the definition was made independent of the osculating and normal planes. Measure of Curvature. 536. To find the radius of curvature at any point of a tortuous curve. The reciprocal of the radius of curvature is the measure of curvatiu-e, or the rate per unit of length at which the tangent 358 MEASURE OF CURVATURE. to the curve clianges its direction. If p be tlie radius of curvature at a point P, and ch be the angle of contingence corresponding to the arc Js. - = -f . p as Draw Oj), Oq of unit length through the origin parallel to the tangents at F and Q the extremities of the arc c7.9, join jJ^', then, since the plane 2^^Q. '^^ parallel to the osculating plane, p^-, which is ultimately perpendicular to 0^?, is parallel to the principal normal. The cosines of the angle made by Ojj and Oq with the axis floe fi'JT Cl3(* of X are -r- and -r- +(? ,- , and, if /, ?n, n be the direction its els els cosines ofjoq, projecting OjjqO on the axis of x, we have dx , , /dx Tdx\ ,^ since j^q = dz ultimately. , d^x 1 . ., 1 (^^V ^'^ .-. l = p j^, , and, sunilarly, vi = p j^^ ,n = p ^^, ; I _ {^^ /^V I^S" •*• p' ~ V/.V "^ UsV ^ \dsV ' If G be the centre of curvature at P, the projection of OFG on Oic = a; + p/, hence the coordinates of the centre of curvature are 537, The radius of curvature may also be found without projections, as follows : Let \, /A, V and A, + A\, p, + A/z, i/ + Av be the dij-cction cosines of the tangents at the points Pand (), Avhose coordinates arc cc, 2/, 2 and ic + Aa;, 3/ + Ay, 2; + A;? ; and let As be the angle between the tangents cos A£ = X. (X -I- AX,) + /i (/i + Ayx) + v (v + Av), and (X + AX)' + (/i -t- A^)' + (v + Av)= - X'"' - ^a' _ v"' = ; .-. 2 (XAX + /iA/i + vAi') + (AX)" + ( A,a)'-' + (Av)' = ; MEASURE OF TORTUOSITY. 359 .-. 2 (1 - cos Ac) = (AX)"-' -f {AfjiY + (Av)"' ; .". ultimately, when PQ is indefinitely dimiuislied, {cky={cixy+{df.Y + {d.Y; If s be not the independent variable, 7 dx dsd'^x — dxd's „ .-. since [dxY + {dijY + {dzf = {d>>f, and dxd^x + dy d'^y + c?-3 d'^z = t/scf s ; i^* = [d'xY + (t/'y)'' + UFzf - id'sY. p 538. The student should, as an exercise, find the radius and centre of curvature, when the latter is considered as the point of intersection of two consecutive normal planes and the oscu- lating plane. Measure of Tortuosity. 539. To find the measure of tortuosity of a tortuous curve. Let Z, W2, n and l+dl, m + dm^ n + dn be the direction cosines of the binormals at two points P, Q^ whose distance along the curve is ds. Draw unit lengths Ojy, Oq parallel to the two binormals, and let X, //., v be the direction cosines of ])q ; the angle q Op = dr is the angle between the osculating planes, and -T- , the rate at which the osculating plane turns round the ds tangent line per unit of arc, is the measure required, which wc shall call — . a- Projecting Oj^qO on the axis of .r, l-\- dT.\-{I + dl)=0] .'. dT.\ = dl and X = a--j-: ds •■• a' ~ \ds) "^ [lis) "*" [TTs 360 GEOMETRICAL INTERPRETATIONS. which iDcay also be obtahicd as in Art. 537. a is sometimes called the radius of torsion at P, but It Is better to look upon - as the measure of tortuosity. 540. The measure of tortuosity may be expressed In another form. Since I '. m : n :: X : Y : Z, where X^dyd'z — dzd'y^ and similar expressions for i', Z^ Idx -i- mdy + ndz =0, Id^x + md:y + nd^z = ; .*. dldx + dm dy + dn dz = 0, and dl .1 + dm .m ■{■ dn .n = Q ] dl _ dm _ dn mdz — ndy ndx-ldz Idy — mdx _ dld^x + dm d^y + dn d^z Id'^x + md^y + n d'z _ Xd'x + Yd^'y + ZiF. IX+mY-vnZ X'+Y' + Z'' ' and {m dz - n dy)'' +... = (Z'^ 4. ,n^ + n') {{dx}' + {dyf + [dzf] - {Idx +...)' = ds' ; 1 _ Xd'x+Yd'y + Zcrz " a- X'+ Y' + Z' ' If there be no tortuosity, or the curve be plane, Xd'x+Yd'y-\-Zd'z = .at every point of the curve, as in Art. 527. Geometrical Intoyretations. 541. Saint Venant observes* that, if we take three con- secutive points P, (?, R for which | = ic, x + dx, x + 2dx-\-d'x respectively, the projections of P^, QR upon the axis of a; will be dx, dx 4- d'x ; and if the parallelogram PQRM be completed, by projecting the sides of the triangle PQM in order, since ♦ Journal de TEcok Pol., vol. 18. GEOMETRICAL INTEia'KKTATIONS. 361 rM= Qli, wc sliall have dc ■+ projection of Q21 - {Jx + cT'j:) = ; tlieretbrc d'x is the projection of QM. 542. In the general case, if a figure be drawn in which (Ix, d'x, (I>/^ d'y are all positive, the projection of twice the triangle PQM on the plane of xy will be easily seen to be dxd^ij + dydx — dy {dx + d'x) = dxd^y — dyd'x. Again, if Mm be drawn perpendicular to PQ^ PQ = dsj and ultimately m Q=QR-PQ = d's ; .-. M»r = QM' - Qnc = [d'xY + ((^y)' + [d'zy - [d'sf, - 1 ... ,(MoxV {d^xy + id'yY+id'zy-id'sY and - = limitof' ' _v y • -y \ 7 ^ — /_ If we make s the independent variable, this implies that QB = PQ, in which ease QM wnll bisect the angle PQB and be ultimately in the direction of the principal normal, the direction cosines of which will be as d^x : cFy : d'z. The radius of the circle circumscribing the triangle PQR po- will be ^\r', hence, if p be the radius of curvature, 1 _ ^^^-^V /"AY A^''^Y p'~ [dsv "^ U'v ^Uv ' 543. Oscidatinfj pJane^ hinormal and curvature of the luh'x. In the case of the helix, Art. 497, dx = — a sin 6 dd^ drx = — a cos 9 {dOy^ dy = a COS0 dd, d'y = — a sin 6 [dOy^ dz = 7m dd , d'z = ; dy d'z - dz d'y = nci^ sin d {ddy, dzd'x - dx d'z = - nd' cos e [doy, dxcT'y-dyd'x^ a' [ddy. hence the equation of the osculating plane is (I - x) {n sin 6) -{■ {v - y) (- n cos ^) + ^- .^ = 0, or n{^y-'nx)^a{i:-z) = 0. .'>G2 RADIUS OF CURVATURE. Tliis plane contains tlie point (0, 0, ;r), and tlicreforc tlie radius of the cylinder uliicli passes through the point (a-, ?/, .-), and this radius is the principal normal. The direction cosines of the binomial will be sin a sin ^, — sina cos^, cosa if a = tan"'/2 be the pitch of the screw. The measures of curvature and tortuosity arc respectively cos''' a , sin a cosa and . 544. To find the radius of curvature of a curve wJnch is the intersection of two surfaces whose equations are given; and to express it in terms of the radii of curvature of the normal sections of the two surfaces and the angle between thevi^ the i^lfine of each section containing the tangent to the curve. Employing the same notation as in Arts. 529 and 540, X=d>jd'z - dzd'y = F {EdF- FdE) = /c" { UV' ((/>') - UV ()}, and if p be the radius of curvature, {dsY ^ ^^., _^ ^., ^ ^, ^ ^., ^^ _^ ^,, ^ ^^.,^ , ,^,. ,, p - 2 ( uu' -f vv + WW) r (6) r-'(<^') + ( w'+ v"+ w") [r (') Y-:.. •'• p^- [D'-{E''+Fy Let 0) be the angle between the tangent planes to the surfaces at [x, ?/, z) and P^ ^U' + V'^ W\ ... UU' + VV' -^WW' = rP' (Msco, and D' + E' + F' ~ P'P" sin'^ w ; . \_ P' [r-\')Y-2PP' co^a>.r {4>')-rw + F-^ {r\^)Y ,.. •• p'^ P'P"s\u'(o{D'-\-E' + F'f " ' ^ ' As an example of the use of the preceding formulae, we shall obtain the radii of curvature r, r of the normal sections by replacing the equation of the surface = of a nonual plane ; in which case, if P,, P",, P,, and V^ be the OSCULATING CONE. 3G3 corresponding values of Z>, E^ F^ T, T," (0,') = 0, and, since the normal plane contains the tangent to the curve, D : E : F=dx: du : dz = I):E:F: \ I 1 •J ' licncc, since &) = |7r, wc obtain from (1) [r.»{<^)r {^'m' •■' ~ P' {d; + e; + F;y r [W + e' + Fy ' 1 r[4>')Y anu, sun I'^'b'j ,:^- p-^D'->rE' + l^y' I _ 1 / 1 1 2 cosci) p'' sin'^o) V'*^ ?•■' »•'■' which result will be obtained in the next chapter by ^Icunicr's theorem. 545. To find the vertical angle of the oscidatuKj cone of a curve. L,Qt 2>Oo^ qPp^ rQ'i bo three consecutive planes which become ultimately the osculating planes of a curve OPQR] these planes intersect in P. Take P as the vertex of a circular cone which touches each of the phaups, this cone Is in the limit, the osculating cone of the curve at P. Let PII be its axis , op, 2n^ v the sections of the three planes made by a plane perpendicular to the axis, and /, u the points of contact wlth^>(/, (ji-. Draw ///, J/// perpendicular to the planes pP>i, 'lQ''\ ^''^" tlie angle tl[u will be the angle of torsion, and pPi the angle 3G4 EADIUS OF SCREW, of contlngence, and we shall have dr = ,| and ch = ,, ulti- mately ; therefore, if 2\/r be the vertical angle of the cone, , lit (h (7 tan V" = -rr = T" = - • ^ rt ih p 546. 77ie rectifying line is the axis of the osculating cone at any poi»f of the curve. For each of the planes through the tangent lines PQ^ QR perpendicular to the osculating planes lyPq^ qQr ultimately contains the axis PII of the cone, 547. To find the rate per unit of length at which the angle between principal normals increases. Let PQ^ rQR^ sRS be the directions of the sides of a polygon ■which are ultimately tangents to a curve. In the planes PQr, rPs respectively draw QU, QV perpen- dicular to rQR^ sRSj these arc ultimately in the direction of consecutive principal normals. Draw QU' in the plane QRs, perpendicular to QR, so that UQU' is the angle between the consecutive osculating planes = Jt, and U' Q V is the angle between consecutive tangents = dz. Let dx be the angle between the lines QU, QV, and let these lines and QU' meet a sphere of radius unity, whose centre LOCUS OF CENTRES OF CURVATURE. VjGo is In Q, In U, V, U', the angle VU' U is a right angle ; there- fore, we have ultimately or {dxT = (dry + {ch)% ^X the rate per unit of length of the increase of the angle between principal normals is the reciprocal of what we shall call the radius oi screw ; let k be this radius, 1 I 1 then ., = .. + - . «" cr' p' It may be noticed that (7t = ^7;^ cos \/r and (/3 = (7% sin-v/r, which represents that the angular velocity round the rectifying line, which is normal to the plane UQV^ is the resultant of two angular motions round the tangent line and the binomial which produce the rectification. 548. To find the anfjle of contingence^ and, the element of the arc^ of the locus of the cenf)-es of curvature of a given curve. Let ()7?, ES be sides of an equilateral polygon wliii-h arc ultimately tangents to the given curve; let BV be the in- 4G6 RADIUS OF SPIIEKIAL CUKVATURE. tcrsectlon of planes pcrpciullcular to Qli., RS through q^ r their middle points; BV is therefore ultimately a tangent to the edge of regression of the polar developable; let BU^ CW be the tangents preceding and succeeding BV. From q draw qU^ <^F perpendicular to BU an^ BV, join rl\ which will be perpendicular to BV^ and draw rW erpcndlcular to CW. If iri' be produced to 2V, UVw and UV will ultimately be the angle of contingence dt and element ds required. Let B be the radius of spherical curvature at q^ and <^ the angle which it makes with the polar line BU. Since circles can be circumscribed about the quadrilaterals BVlTq^ CioVr^ qVU=qBU=cjy, and VBU= VqU=dT, ultimately, also r Vic = rC W= r C V + VC W= (/> + J + dry. By drawing a diameter through V to the circle BVUq^ it is easily seen thjit VU=R smVBU, and therefore that dn'^Bdr. 549. To find the radius of spherical curvature. If in the figure BVl/q^ UM be drawn perpendicular to QV^ VM= dp ultimately ; .-. {ruiTY = ipdry-,{dpY; also Bdr cos^ = VM= dp ; therefore the distance between the centres of circular and . 1 -n , '^P (^P spherical curvature = ii cos(^ = -7- = tr- . ^ dr ds 550. To find expressions for the radius of curvature of the edqe of repress ion of the polar developable of a curve. COOKDINATLS IN TERMS OF ARC. 3<)7 These arc readily obtained by a method su^^gested by Routli,* which can be exphiined by the last figure. Considering the curve BCD^ U and V are the feet of the perpendiculars from q on the tangents to the curve; and substituting the corresponding letters in the known formnlEe 7; + -7 : , , ?* -r- 1 we obtain two expressions for the radius of a-, Oy the principal normal, Oz the binomial, the planes of yz, zx^xy being the normal, recti- fying, and osculating planes, and let s be the distance of g. point (a*, y, z) from (9, measured along the arc. Then, at the origin, ^-1 ^^-0 ^'''-0 ds~^' ds'^' ds~^' d'x ^ (Fy cFz p ,2=0, Px^ = l, p 7. = 0, '^ ds ' ds ' "^ ds these quantities being the direction cosines of the tangent and principal normal. If a be the angle made by the tangent at dx (.r, y, z) with the tangent Ox^ , = cosa ; d'x [da.\' . (Pa ''•d? =-''''''■ \ds} —ds^ therefore, at 0, since doL is the angle of contingcnce, and a = 0, ds'~ p'' ♦ LliinrU'rli/ ,/ounifil, vol. Tit., p. 42. 368 COORDINATES IN TEKMS OF AKC. Again, If jB be the angle made by the principal normal at [x, y, z) with (9y, d^y ^ dp d'y rfy . _^y3 da ds ds ds ds tl)crefore, at 0, 1 dp d'y ^ p ds '^ ds ' and if 7 be the angle made by the binormal at (a*, ?/, z) with the principal normal at the origin, (dz d^x dx d^z\ dp fdz (Px dx d^z\ fdz d^x dx d^z\ _ . dy ''• 'tb \ds ds' ~'d's~div'^^\dsi?~'ds d?) "~ ^'"'^ ;^' therefore, at 0, p -y^^ = sin y -r = - , since dy is the angle ds ds a ' of torsion ; therefore, by Maclaurin's theorem, ^ = ^-(J' s' s' dp y~2p Gp'ds' s^ ~ 6/30- ■ 552. To find the avrjle hetween two consecutive 2^rinc>j)al nortrnds. The direction cosines of the principal normal at (.r, y^ 2), .9 1 s a point near the origin, are as - ^ : - : , and the secant . P' P P^ of the angle between this normal and that at the origin is /- I +.sM - + — ji ; therefore, the angle required is ulti- mately.y(l-fi); • 1 - 1 1 /c" p" O"' ' where k is the radius of screw, as in Art. 017. COORDINATES IN TERMS OF ARC. 3G9 553. To find the shortest distance between consecutive irrincipal normals and its position. The equations of a principal normal at (a:, y, z) are ap- proximately - P [^ - x) = s {v - y) = the plane xy being the plane of reference. The differential equation is, in this case, ^^dx-^,dy = 0; .'. a" logx -Z'Mog?/ = log(7; .-. x*= Cxf. The lines of greatest slope are the intersection of the cone with the cylinders represented by this equation ; and it may be observed that no generating line, except those in the prin- cipal planes, is a line of greatest slope, unless the cone be a right cone. Line^ Surface^ and Volume-Integral. 660. We give here two theorems relating to line, surface, and volume-integrals, which are of great importance in certain problems in Electricity and Conduction of Heat, and which serve as illustrations of the subjects of this and the preceding chapter. Line-Integral and Surface- Jyitegral. 561. Def. If R be any quantity having direction, called a vector quantity, and s be the angle between Its direction and that of the tangent to a curve at any point (.r, y, z) taken in a definite direction, the Integral jR cossds is called the line- integral of Ji along the line s, s being measured from a fixed „„ . . , , . ff dx dy dz\ , pomt. ihe nitegral may be written 11 " -, + ^ j^ "^ ^;rj ""') w, V, v) being functions of ic, y, z. If rj be the angle between the direction of R and a normal to a surface at any point (a;, y, 3), the integral jjR costjdS is called the surface-integral, the summation being taken over SURFACE AND VOLUME-INTEGRAL. 373 the whole of a surface S. The Integral may be written JJ{Ul+ Vm+ Wn)dS, U, F, IF behig functions of {x, y, z), and /, 7«, 71 the direction-cosines of the normal to the surface at [x, y, z) measured in a definite direction. 562. To shew that a line-integral tahen round a given closed curve can he represented as a surface-integral of a surface bounded hy the given curve. Suppose the closed curve L to be filled up by any surface S^ and suppose S to be divided into an infinite number of small elements, one of which is o- bounded by the line \. If we take rals for two o jtimated in the portions of the sums taken over fi will be taken in opposite directions, and being of the same magnitude will vanish ; those lines X which abut upon L are the only portions which will not be traversed twice ; hence the sum of all the line-integrals for the elements \ will be that of the line L. The proposition will, therefore, be proved if we shew that it Is true for any elementary line \ and corresponding surface cr. Let [x^ y, z) be any point on o-, and (^ + f , y + Vi ^ + H any point on A,; the line-integral for X, u^ y, lo being given at (^, 2/, ^), = IK" + -^ ^ + ^ ^ + ^' ^) ^^^ +•••} ultimately. Since \ Is a closed curve, Jd^ = 0, /|^/^ = 0, and if we sup- pose the summation taken in the direction from x to ?/, hence the line-integral for X is -^ ( -r -j-] n +...[ a. [\dx dgj J The line-integral of L is, therefore, equal to the surface- integral JJ{Ul + Vni + Wn) ds, when U= -. , , &c. 563. The surface-integral of a directed quantity or vector^ taken over a closed surface, may be expressed as a volume' integral of a certain function. We observe that if the theorem can be provcil for an elementary portion of the volume enclosed within the surface, 374 SURFACE AND VOLUME-INTEGRAL. ■within which tlie directed quantity is supposed to be continuous, the general theorem will follow, as well as Its modification, when the enclosed volume Is intersected by surfaces across which the directed quantity changes discoutlnuously. For, if fj, u^ be two elementary volumes enclosed by the surfaces o-^, o-.^ to which a portion a' is common, the normal components along o-', which belong respectively to cr, and o-„, being In opposite directions and of the same magnitude, will disappear In the summation. If, therefore, we sum for all elements within a volume F, throughout which the value of the vector changes continuou^^ly, the only points of the resolved vectors which are not destroyed are those which belong to the points of the elements which abut on the enclosing surface S. If the vector change discontinuously In passing surfaces 2„ 2.^, &c., the theorem will hold for the portions F,, V^... into which they divide F, and the volume-Integral over F would be equal to the surface-integral over S^ together with the surface- integrals over S,, 2,^; the differences of the vectors on opposite sides of these surfaces replacing the vectors in the first Integral. Let an elementary volume v be Inclosed by the surface o-^ (a:, ?/, z) being any point within u, and let (•« -f- ^, ?/ + »7, 2 + ^) be any point upon o-, and ?<, v, w the components of the vector at (a*, ?/, z) parallel to the axes. The surface-integral for a is // |(„ + ;^«j+...),+(,,+J,+...)„,+..,jrf,, hence the surface-integral for o- = (-- -j- -. -f- . ) uultimatclv: ® \(/^ di/ dzj ' ' therefore, if u,, ?*,' be the values of ?/ on opposite sides of 2,, //(,..™..„),;«=///(,t.|.|>r + " f which represents the theorem in Its most gencfal form. PROBLEMS. 375 XX. (1) Tlie equations of the tangent to the curve of intersection of the surfaces ax* 4 by* + c;' = 1 and bx* + ctj* + «2* = 1 are ab - c' be - a* ac - b* The tangent line at the point x = y = z lies in the plane (a - 6) x + (6 - c) y + (c - fl) z = 0. If ac = b*, the tangent lines trace on the plane of xy the two straight lines whose equation is -, — r, = -,——r. • ^ r - 6' a' - 6^ ('2) The equations of a sphere and cylinder being X* + y* + z* = 4a' and x" + z' = 2ax, prove that the equations of the tangent to the curve of intersection at the point (a, ft, 7) are (a - a) r + 73 = aa and (3y ^- ax = a (4a - «), and that the equation of the normal plane is ^^(■-")(^l)• (3) Find the tangent line of the intersection of the surfaces 2 (x + z) (x - a) = a' and z (y + z) {y - a) = a^ and ehew that it consists of plane curves. (4) The paraboloid whose equation is ax* + by* ,= 4z has traced upon it a curve, every point of which is the extremity of the latus rectum of the parabolic section through the axis of z ; shew that the tangent to the curve traces upon the plane of xy the curves whose equations are r sin 20 = + (« - b). (5) Shew that the equation of the normal plane at any point /, y, h on X* y* z* the curve defined by the equations -: + Vi + -^ = ^ i x' + v* + z' = tf' is ^ a' b* c' a*{b*-c*) b*{c*-a*) c*{a*-b*) „ f 9 A (6) A point moves on an ellipsoid so that its direction of motion always passes through the perpendicular from the centre of the ellipsoid on the tangent plane at the point; shew that the curve traced out by the point is given by the intersection of the ellipsoid with the surface x'"""2/""'2'"' = constant, /, m, n being inversely proportional to the squares on the semi-axes of the ellipsoid. 376 PROBLEMS. (7) If the osculating plane at every point of a curve pass through a fixed point, prove that the curve is plane. Hence prove that the curves of intersection of the surfaces whose equations are a;* + yl + s' = a' and x* + ?/* + 2* = Ja* are circles of radius a. (8) Prove that if every pair of consecutive principal normals to a curve intersect, the curve must be a plane ; and find/ {0) so that the curve, whose coordinates are given hy x = a cos6, y = b sinO, z=/ (0), may be plane. (9) A curve is traced on a right cone so as always to cut the generating line at the same angle ; shew that its projection on the plane of the base is an equiangular spiral. (10) If a string be unwound from a helix so that the straight portion is a tangent to it, shew that any point on the string will describe the involute of a circle. (11) Prove that the locus of the centres of curvature of a helix is a similar helix ; and find the condition that it shall be traced on the same cylinder. (12) When the radius of curvature is a maximum or minimum, the tangent to the locus of the centres of curvature is perpendicular to the radius of curvature. (13 j Coordinates of any point in a curve are a: = 4a cos^0, y = 4a sin'6>, z = '6c cob'O; find the equations of the normal and osculating planes; and find the relation between c and a that the locus of the centre of the sphere of curvature may be a curve similar to the original curve. (14) A straight line is drawn on a plane, which is then wrapped in a cone ; shew that the radius of curvature of the curve on the cone varies as the cube of the distance from the vertex. (15) If a tortuous curve be projected on a plane, the normal to which is inclined at angles a, /3 to the tangent and binomial at any point, the curvature of the projection will be to that of the curve as COS7 : siu'a. (16) If ^ be the radius of curvature of a curve, then that of its projection on a plane inclined at an angle a. to the osculating plane is p sec a if the plane be parallel to the tangent, and p cos' a if it be parallel to the principal normal. (17) If the measures of curvature and tortuosity of a curve be constant at every point of a curve, the curve will be a helix traced on a cylinder. (18) If - , - be the measures of curvature and tortuosity at any point p '{'-'-QT- The equations of a cylinder and cone are r sin = a and coiO = ^ (c^ - c"*). If Ju A J, and A-^ be the areas of the cone reckoned from = to (f> = fi-a, ft and ft ^ a respectively; then will yi, + A3 = (e" + e"*) At. CHAPTER XXI. CURVATURE OF SURFACES. NORMAL SECTIONS. IXDICATRIX. DISTRIBUTION OF NORMALS. SURFACE OF CENTRES. INTEGRAL CURVATLTIE. DIFFERENTIAL EQUATION OF LINES OF CURVATURE. UMBIUCS. 5G4. In this chapter we proceed to examine the curvature of surfaces, and to explain how the amount of curvature may be estimated. If the student will consider the simpler surfaces with which he is familiar, such as a sphere, an ellipsoid, or an hjperboloid of one sheet, he will have examples of the kind of flexure which may take place at an ordinary point of any surface. Any point of a sphere or a pole of a prolate or of an oblate spheroid is an example of a point of a surface from which, if we proceed along any section made by a plane containing the normal ; the curvature is the same. Any point of an ellipsoid is an example of a point on a surface at which, if a tangent plane be drawn, the surface in the neighbourhood of the point lies entirely on one side of the tangent plane ; such surfaces are called Si/7iclastic. If a tangent plane be drawn at any point of an hyperboloid of one sheet, the surface will Intersect the tangent plane, and bend from it partly on one side and partly on the other ; such surfaces are called Antidastic. 5G5. Let two planes be drawn through the same tangent line at any point of a sphere, one containing the normal and the other inclined at an angle 6 to the normal, the sections made by these planes will have their radii In the ratio 1 : cos^. This simple relation between the radii of curvature of a normal and oblique section, containing the same tangent line, will be proved to be true for any surface at an ordinary point. 380 NORMAL SECTIONS. The student may for an exercise prove it when the tangent line is drawn through the extremity of a principal axis of an ellipsoid parallel to another principal axis. 566. Consider next tlie curvature of the sections of an ellipsoid made by planes passing through OA, the normal at Aj an extremity of one of the principal axes; if AP be one of these sections intersecting the principal section BC, perpen- dicular to OAy in P; OP and OA will be its semiaxes, and its radius of curvature at A = -p^—r ; also -p^-.- » -rr-r '^vill be the UA (JA (J A radii of curvature at A of the principal sections BA and CA^ and since OB^ OP, OC arc in order of magnitude for all positions of OP, we see that of all the normal sections through yJ, the two sections which have their curvature a maximum and minimum have their planes perpendicular to each otlicr. This property of- normal sections will be found to hold for any ordinary point of all surfjvces. rjQ7. If POB=^ ^^ _- + ^^^ = __ ; hence, if p, p' be the radii of curvature of the sections AB, A C, and B that of the .„ cos'^ sin'^ 1 section AP. i , - = t, • ' /? p L This relation between the radii of curvature of the normal sections of least and greatest curvature, called principal sections, and that of any other normal section inclined at an angle 6 to one of the principal sections will also be found to hold for any surface. 5G8. These three properties which are true for all surfaces will enable us to determine the radii of curvature of all plane sections through any point, when those of any two sections, not containing the same tangent, arc known. AWmal Sections. 569. To find the relation between the radii of curvature of sections made hy planes containing the normal at any point of a surface. NORMAL SECTIONS. 381 Let the surface be referred to the normal at the point as the axis of ;:, and the tangent pLine at as the pLane of xt/. (h , (h — and -y- dx ay s, t be the vaUies of , ., , 7 , , , .. , dx dxc/i/ dy z^\ {rx^ + Isxy + ty^] + «S:c. Let tlic surface be cut by a plane passing through (9.r, and inclined at an angle Q to the plane of zx\ at every point of this plane x = u cos^, y^xi sin Q ; .-. z = \ [r cos'^ + 2s cosfi* sin^ + t ain'^) x^ (1 + s) where e vanishes in the limit. If R be the radius of curvature of this section, 1 . . 2.r -75 = limit —., = ;• cos"^ + 2s cos d sin d + t sin""'^. li II' The directions of the normal sections of -svliich the curvature is a maximum or minimum are given by the equation -{r-t) sin2^ + 2s eos2^ = 0. If a be one solution, the rest will be included in the formula |«7r + a, hence the sections of maximum and minimum curvature are at right angles. These sections are called the Fn'ncqyal Sections of the surface at the point considered. If the planes of the principal sections be taken for the planes of zx and yz, a = 0, and therefore s = 0, and the expression for the curvature of any section will become ^ = r cos'^^ + t sin'6 ] let p, p' be the radii of curvature of the principal sectionSj I 1 1 then - = ?• and — = t. P P 1 _ cos^^ sin'^ •'• 7Z - "7" "^ 'p" ' also, if i?, W be the radii of curvature of any perpendicular normal sections, 1111 11 li P P These theorems arc due to Eulcr. 382 THE INDICATRIX. The Indicatrix. 570. Euler's theorems and other theorems relating to the curvature of plane sections of surfaces are deduced with great facility by means of a curve called the indicatrix, employed first by Dupin for this purpose. Def. The indicatrix at any point P of a surface is the section made by a plane parallel to the tangent plane at P and at an infinitely small distance from it. In cases in which,, as in antlclastic surfaces, the curve of intersection extends to any finite distance, the name of indicatrix only applies to the portions of the curve which arc infinitely near to P. 571. At any ordinary point of a surface the indicatrix is a conic. Taking the axes as in Art. 5G9, the equation of the surface is of the form z = ^ {rx^ + 2sxy + ty^) +&c.^ and by transfor- mation of axes the term involving xy may be made to disappear, so that z = ax^ + hy^ + terms of higher dimensions. If the surface be cut by a plane parallel to the tangent plane and very near to it, for which s = A, in the neighbourhood of the point of contact h = ax' + hy'^ ; the indicatrix is therefore a conic whose centre is in the normal. 572. Pendlebury has noticed^'- that the indicatrix may be, at particular points of some surfaces, of any form, and the- number of directions of principal curvature for such points may be any number, in fact, equal to the number of apses in the Indicatrix. He gives as an example a surface x' + y'^ = azj) f-j generated by a parabola revolving round its axis, its latus rectum increasing or decreasing with the angle through which its plane has revolved ; such surface would look like a paraboloid with ridges and furrows radiating from the vertex. * Messenger oj Mathematics, vol. I., p. M8. THE INDICATIilX. 383 573. The radius of curvature of a normal section of a surface varies as the squaix of the corresponding central radius of the indicatrix. Let CP be the central radius of tlie indlcatrlx which lies in any normal section whose radius of curvature at is i? ; tlien 2^ = limlt— — -, (see figure, p. 386); therefore, since OC is constant for all normal sections It x CP'. Ilcnce all theorems in central conies which can be expressed by equations homogeneous in terms of the radii and axes, can be replaced by corresponding theorems in curvature. Eulcr's theorems follow at once, and if i?, R' be radii of curvature of normal sections inclined to a principal section of a surface at angles 9^ 6', such that tan tan 6' = -—, P then ii + ii' = p -f p', and BR' sm'^d' - 6) = pp. 574. AVhcn the Indicatrix is an ellipse, the surface is syn- clastic at the poiirt considered. A point of a surface for which the indicatrix is a circle is called an umhilic^ the curvatures of all sections made by planes containing the normal at an umbilic being equal. 575. AYhen the form of the indicatrix Is hyperbolic, the surface Is antlclastic at the point considered ; In this case the radii of curvature of normal sections containing the asymptotes arc Infinite, such sections pass through the Inflexional tan- gents, and their directions arc given by tan''^= — , p, p being the absolute values of the radii of curvature. In order to deduce theorems from geometrical properties of the hyperbola, It may be necessary to suppose two Indicati-Ices, one on each side of the tangent plane at equal distances from It. If p = p and R be the radius of curvature of a normal section inclined at an angle 6 to a principal section 7? cos 2^ = p. 384 THE INDICATRIX. 576. When the section made by the plane parallel to the tangent plane is a parabola, the part of the section which is called the indicatrix is two parallel lines which become ulti- mately, as in the case of a developable surface, two coincident lines. Such points are called FarahoJic iwints^ sometimes also Cylindrical jjoints. As an example of a parabolic point, take a point of the '2 2 2 cone -^, -^ 'jr. = —, ^ at a distance I from the vertex In the a c generator - = -; transform the axes so that the normal at this point is the axis of ^r, and the generator the axis of .r, the resulting equation of the cone \?> h = —yr^y^ — zx — s^, let z = h and a = ctana, then ?/ = ■ — hix -\- 1- h coi^y). the section by a plane parallel to the tangent plane is therefore a parabola, the distance of whose vertex from the normal at the point considered hl — h cot2a, and since this remains finite, when h is made indefinitely small, the degeneration into two nearly coincident parallel lines in the neighbourhood of the point is explained. T2T The finite principal radius of curvature Is — . 577. The intersection of tivo consecutive tangent ^ylanes and the direction of the line joining the points of contact are p)(tralkl to conjugate diameters of the indicatrix. Let CP, CD be conjugate semi-diameters of the indicatrix for the point of a surface ; since the tangent plane to the surface at P contains the tangent to the indicatrix at P, its intersection with the tangent plane at is parallel to CP, and proceeding to the limit, when OC vanishes, the proposition follows. Def. Tangent lines at any point of a surfiice drawn parallel to conjugate diameters of the indicatrix, are called Conjugate Tangents. OBLIQUE SECTIONS. 385 578. It follows from this property of consecutive tangent planes, that if a torse envelope any surface the directions of the generating lines at any point of the curve of contact aro conjugate to the tangents to the curve. 579. To find the relation between the radii of curvature of a normal and oblique section of a surface made by planes passing through the same tangent line. Let the tangent line through which the planes are taken be the axis of x, anS, S' \ and let 7>'/, (ja be normals to the surface 6', which, since they intersect, must Intersect in the polar line alJA^ perpendicular to the osculating plane pU(i\ similarly qah^ rb, normals at q and r, intersect in the polar line bVio the consecutive osculating plane, and brV=bqV=aqU-\- UqV. Let a', b' be corresponding points for the surface S' ; .-. b'rV^-a'qU-itUqV; .'. b'rb = a'qa\ 396 LINES OF CURVATUBE. hence, normals to SS' at consecutive points ^, r are inclined at the same angle, therefore the surfaces cut one another through- out at a constant angle. Reciprocally, if tioo surfaces cut one another at a constant angle^ and their curve of intersection he a line of curvature on one surface^ it will he a line of curvature on the other also. Let it be a line of curvature on S^ and let the normals to S' at 2 and r be qa!'h' and rh" meeting Ua in a" and Uh in h\ h" ; then since the angle between the normals to S and S' at 2', r are equal, hrh" = bqb', and hrh" = hqh'\ therefore b' and b" coincide, and the normals to S' at q^ r intersect; that is, the curve is a line of curvature also on S'. Cor. If a line of curvature be a plane curve, its plane will cut the surface at a constant angle. 602. The analytical proof given by Bertrand is very simple. Let P, Q be consecutive points on the curve of intersection of surfaces >S', S' ; x, y, z and x + dx^ y + dy^ z + dz their co- ordinates ; Z, 9n, 71 and Z', wi', n the direction-cosines of the normals at P to S and S'. If the curve be a line of curvature on S, the normals at P, Q will intersect ; .*. X — lp = x + dx— [l + dl) pj dl _ dm _i^n _1 . . ' ' dx dy dz p' ^ Since PQ is perpendicular to both normals, ldx + 9ndy-i-ndz = 0j ,^x and I'dx + m'dy + n'dz = 0. i. If the curve be a line of curvature on both surfaces, ldr + mdm' + nd7i=0. , ,,. , ,,,x .'. d [W -f mm' 4 nn) = 0, or the cosine of the angle between the normals is constant. ii. If the curve be a line of curvature on /9, and the surfaces cut one another at a constant angle, I'dl + m'dm + n'dn = 0, dupin's theorem. 397 and d {W + mm + nn) = ; /. Ml' + mdm' 4- ndn = 0, also I'dl' -f in' dill + n'dii = ; therefore, by (2), — = -—=-., tlic condition tliat the curve shouhl be a line of curvature on S'. 603. When three series of surfaces cut one another orthogo- naJly^ the curve of intersection of anij two of them is a line of curvature on each. Let the origin be a point of intersection of three surfaces, one of each series, and the tangents to their lines of Intersection the axes. The equations of the three surfaces may then be written x^- o^f + lhyz-^cz' +...= % (1) y + a'z' + 2h'zx + cV +. . .= 0, (2) 5; + a'V+25"a'_y + c"/+...= 0. (3) At a consecutive point on the curve of intersection of (2) and (3), we have 2/ = 0, z = 0^ x=x'^ and the equations of the tangent planes are, ultimately, X .2c' x -{■ y -\- z.2h'x' = 0^ X . 2a" x' + y . 2l}"x' + 2 = 0, and since these also are at right angles, Aa'c'x"' + 21)' X + 2yx = 0, or, ultimately, I' + h" = 0; similarly, Z»" 4 ^ = 0, Z» -h h' = 0, wlilch can only be satisfied by Z» = 0, Z»' = 0, i" = 0, and therefore the axes are tangents to the lines of curvature on each surface. Hence, the tangent lines, at any point of intersection of three surfaces, to their curves of intersection, are tangents to the Hues of curvature of the three surfaces through that point, and, con- sequently, their curves of intersection must coincide with the lines of curvature. This Is Dupin's tiieorem. A proof is given by Cayley,* which puts in evidence the geometrical ground on which the theorem rests. Quurttrly Juumal, vol. XU., p. 186. 398 gauss' measure of curvature. Measure of Curvature. 604. Gauss gives the following definition of Integral and Total Curvature. Def. The Integral Curvature of any given portion of a curved surface is the area enclosed on a spherical surface of unit radius by a cone whose vertex is the centre, and whose generating lines are parallel to the normals to the surface at every point of the boundary of the given portion. Horograph. The curve traced out on the sphere as described above is called the horograph of the given portion of the surface. Average Curvature. The average curvature of any portion of a curved surface is the integral curvature divided by the area of the portion. Specijic Curvature. The specific curvature of a curved surface at any point is the average curvature of an infinitely small area including the point. This is the measure of curvature which was shewn by Gauss to be the reciprocal of the product of the two principal radii of curvature at the point considered. 605. To shew that the reciprocal of the product of the principal radii at any point of a surface is a proper measure of the curvature. Let an elementary area QRS be described including the point P of a surface, and let a series of lines of curvature divide this area into sub-elementary portions, such as p)qrs^ and let /3„ p,' be the principal radii of curvature at p in the directions p>1-)PS] the horograph for ^jjrs will bo a small rectangle whose Bides are — and — , , and area = ^ -'— . Px Pi PxPx CURVATURE. 399 But, if p, p be the principal radii of curvature at /', where s vanishes in the limit ; therefore the specific curvature = lini.-££i:= \. This expression is independent of the form of the clcmcntaiy portion including P, and is analogous to the measure of cur- vature in plane curves, the solid angle of the cone corresponding to the angle between the normals to a plane curve at the extremities of the small arc on which a point of the curve lies. 606. To determine the radius of curvature of the normal section of a surface through a given tangent line at a given point in terms of the coordinates. Let the equation of the surface be /''(|, 77, ^)=0; and let (a:, ?/, x) be the given point P, (X, fx, v) the direction of the given tangent ; also let [x + dx, y + (7y, z + dz) be a consecutive point Q taken in the normal section, so that ultimately dx : dy : dz = \ : fi : v. Then, if QIi be perpendicular to the tangent plane, P the radius of curvature of the normal section will be the limit °^ 2QB- The equation of the tangent plane is U{^-x) + V{v-2/)-\-W{^-z)^0', .__ Udx + Vdy + Wdz •'• ^^^- ±F ' where r' = U' + V'+U'\ ]>ut, Q being a point on the surface, Udx + Vdy + Wdz+l [m(^7x)» + ...+ 2u'di/dz+...]=0j neglecting terms of degrees higher than the second ; 400 CURVATURE. • ' - 2{Udx^Vdy-{-Wdz) = + Since we have the conditions U\ + Vix + Wv = () and X''' + /i"-^ + v' = l, the problem of finding the directions of the principal sections and the magnitude of the principal radii of curvature is the same as that of finding the direction and magnitude of the principal axes of the section of the conicoid, ux^-\-...+ 2uyz-\-...= \^ made by the plane Ux-\-Vy-\- Wz = 0. 607. To determine the ijrincipal normal sections^ and the radii of principal curvature at any ])oint of a surface^ in terms of the coordinates of the point. The radius of a normal section containing the tangent whose direction is (X, n,. v) is given by p uX' + V+ lov' + 2i(>v + 2 vVX + 'iw'\^l - -p (>'' + /*' + r) = 0, ( 1 ) where f/X -f T>-h TFv = 0, (2) and when B. is given, the corresponding tangent lines are the lines of intersection of the cone and plane represented by these equations, X, /i, v being considered current coordinates. When iZ is a maximum or minimum, these directions coincide, and the plane is a tangent plane of the cone ; hence the direction of the principal sections are given by (m - cr) X + w yi. -f vv _ lo'X + {v — a) fjb + u'v Ij - V - w ' wncreo--^, whence we obtain X U[[v-a-){io- o-)-ti'"'} H- V[i(!v'-io'[io-a)] i-W[w'u' - v'{v-a-)] CURVATURE. 401 which, by the equation U\ + T> + Wv = 0, leads to V [[v - a)[w - a) - ?<"-'}+...+ 2 FIF{yV-?/(?f-a)]+...= 0. (3) This equation gives the values of the principal radii of curva- ture, and the values of \ : /* : v, corresponding to each ruot, arc given by the preceding system of equations. Cor. The product of the roots of (3) is IP {vw - It") +...+ 2 VW{v'to - vu') i-.. . U' + V' + W = [U^ -{V- + Tr") X measure of curvature. Since the measure of curvature vanishes at every point of a developable surtacc, the numerator equated to zero is the condition that a surface should be developable. 608. We cannot help calling attention to another form of the quadratic giving the principal radii, which was set in an examination paper for Clare and Cains Colleges in 1873. Since 2V]ViJ,v= U'-X'-V'fM'- W'^v'^ &c., the expression P for p can be put into the form AX' + Bfi^ + Cv\ where A^u-k- -r/TTr ( ^^'^' ~" ^'^■' ~ ^^"^io") , &c. Construct the conicoid A^- + Bif -\-CX'' = P, having its centre at the point (.r, y, z) of the surface, the directions of the axes of the section made by the plane f'^ + !'»; + ir^= are the directions of the tangents to the principal sections of the surface, and the corresponding values of It will be the squares of the section. Hence, by Art. 234, AR-p^ nn-p'^ en -I' ' a quadratic giving p, p the principal radii of curvature. Also the direction cosines of the tangents to the lines of curvature are as J^_^: ^/_ ^ : ^,^._^„ where p., p.^ are to be written for R. ¥ V F 402 UMBILTCS. G09. To determine the conditions of an unibilic. At an urabllic R retains a constant value for all directions (X, yti, v), satisfying the two conditions (1) and (2). Ilcnce at an unibilic the cone (1) must break up into two planes, one of which is the tangent plane (2). The left-hand member of equation (2) must therefore be a factor of the left-hand member of (1), and the other factor will therefore be ^^^{v-a)-\-^^[iv-a). Multiplying the two, and equating coefficients, V , -a)=2u\ T^/ ^u^''- -^)+^rb'^ -a) = 2v', ^(.- a) = 2w, which, on eliminating a, lead to the two conditions W'v + VSo - 2 VWu' Uho + W'u-2 WUv r^-fir • W'-fU' _ V' i + ir'v-2UVio' tf^+v " ' These two equations, together with the equation of the surface, will, in general, determine a definite number of points, among which are included all the unibilici. It may happen that a common factor exists, so that the three equations are satisfied by the coordinates of any point lying on a certain curve. Such a curve is called a line of spherical curvature. It should also be observed that fZ, F, W have been assumed to ha finite in the foregoing investigation. Should one of them, as C/", vanish, we must have, in the same manner, V^i-\Wv a factor, and must therefore have (?<-0-; V+...+ 2(/yUV+... = ( T> + Wv) |/.-\ + (. - t dx is written for X, &e,, and ^ = a,- + 6'j,, &c., cr a 0z=d>/ + -dv - r -, */o-, dy, dV, F ! = 0, (7^, dW, W ' which is the differential equation of the lines of curvature. Expanding dUj dV^ dW^ and eliminating dx, dy, dz, and da, It — cr, ^c' The coefficient of U'^ is - w, V- a, u, V V, n\ ?/' — cr, W V, V, ir V— (J. u = 0. u , w — a- , whence we obtain the quadratic given in Art. G07. 614. The foregoing equations for determining the principal curvatures undergo a considerable simplification, if the equation of the surface be of the form 406 CURVATURE. Wc shall then have ?/, v\ xo all zero ; the equation giving the length of the radius of curvature of any normal section, whose tangent line is (\, /a, v\ will be P the quadratic equation for the principal radii of curvature will be Ru-P'^ Rv-P"" Rw-P~^' the diflferential equation of the lines of curvature will be U[v — w) dydz + V{io — u) dz dx + W[u — v) dxdy = 0. The conditions for an umbilic in this case reduce to ic = v = io when U, F, IV are finite, but since this is the exceptional case mentioned in Art. 609, in which ?(', v\ and lo' vanish iden- tically, there are other umbilics which are given by U=0 and {v — u) W^ -^ [w — u)V^ — 0, and similar equations when V= and W'—Q. The whole number of umbilics is therefore, as before, 71 [{n - 2Y + 3 (n - I) (3n - 4)| ^w (lOn' - 25m + 16). 615. To obtain the differential equation of the lines of cur- vature^ and to find the centres and radii of jirincipal curvature when the equation of the surface gives one of the coordinates explicitly in terms of the other two. Let the equation of the surface be t=/(|, 77), and let P, Q be consecutive points on a line of curvature whose coordinates are a;, 3/, 2;, and x + dx^ y + dy^ z + dz^ then the normals at P, Q intersect ; and if (|, 17, ^) be their point of intersection, ^-x^p[K-z)=^0, and 7;-7/ + ;2(e-^)=0, (1) but f , 77, ^ remain the same when x + c?.r, y + <7y, z 4- dz are written for ic, 3/, 2; ; therefore d2){X-z) = dx^ 2)dz^ and dq {^-s)= dy + qdz ; (2 ) rdx + sdy _ dx + p{ jjdx + qdy) ^ sdx + tdy dy + q [pdx + qdy) ' .-. {(1 -f q') s -pqt] {dyY + 1(1 + ql r - (1 +/) t] dxdy CURVATURE. 407 ■wliicli is the dilToivntlal equation ot" the projection of the lines of curvature on the pKinc of xy. Let p be the radius of curvature of tlie principal section through PQ^ hence by (1) p''= (1 +/ + -?')(::- TA therefore, writing in (2). for. -r or ^^^^^._^^.^ , {rJx + sdy) a + dx +p [ikIx + qihj) = 0, .-. [r'f) dx = 0] ... [ra- + 1 +/) (f(r + 1 + 2') " {^<^ + M)* = », or (r^ - s'-'} 0-- + { ( I + 5-) r - 2/) js + ( 1 + 2>') t ] a + 1 + ir + (f = 0. (4) Cor. Gauss' measure of curvature Is J_ _ 1 I _ _ rt-s" . . ^ pp ~ aa 1 -^f + q' " (1 -^f + iJ ^ which vanishes for a developable surface. G16. To find the umlillcs of the surface z =/(j', ^). Since the normals at points passing in any direction from an umbilic intersect the normal at the nmbilic, neglecting small quantities of the third order in dx and f/y, the equation (3) must be true independently of the value of dy : dx^ and this condition is satisfied by 1+;/ _V3_ ^ 1 + 7' . r' ~ a t "* these equations, with the equation of the surface, determine the nmbilics. Curvature of Com'coids. G17. To find the radii of principal curvature at any point of a ci ntral conicoid. Let P be any point on the conicoid, supposed in the figure to be an ellipsoid, POP' the diameter through /', CL the radius parallel to the tangent at P to any normal section whoso radius of curvature Is rcfiuircd, PQL the central section having the 408 CURVATURE OF same tangent. Let a plane be drawn through a pohit Q near F parallel to the tangent plane at P, meeting CP in F, and let 2>j "^ be the perpendiculars on the tangent plane from C and Q, so that -ct : ^; : : VF: CF. The radius of curvature of the normal section is the limit of — — or -^ — , and 2-BT 2ot ' QV''.CUy.PV.VP''.CP'' :: -UT.VF' : 2}.CP=2'S} :p ultimately, hence the radius of curvature = . P If a, /3 be the scml-axes of the central section parallel to the tangent plane, — and — will be the principal radii of curvature, which we shall call p and p. 618. To find the coordinates of the centres of curvature. Let the equation of the conicoid be ?' + 2^' + - = 1 (1) ^8 + j. + ^. -I, u; and let (/, ,7, h) be the point P, then f , 7;, ^, the coordinates of the centre of curvature corresponding to /?, satisfy the equations d' W c' CONICOIDS. 409 and by Art. 28'), if the equations of oonfocal conicoiils through Pbe a* and /3' are respectively a* - a'"' and «" - (i"* ; tliercforc the coordinates of the centres of curvature arc fcr gV" lu-"^ fa" g}/' he' a' ' />" ' c'" ' a' '' b' ' c" ■ CAd. If three confocal conicoids (^), (5), (C) intersect in P, the centres of principal curvature of {A) at P are the poles with respect to [B] and [C] of the tangent plane to [A] at P. Let the three conicoids {A\ (P), and [C] be a" y' z" , a:" v' ^^ , , ^* '/' -' , « 6 c a ^ c a b c intersecting in (/, 17, //). The coordinates of the centre of curvature of the normal section containing the tangent to the intersection of {A) and {D) are ., , ^z- ^ ^1 and Its pohir, with respect to [C], is a b c 1 + t« ■«" ~5 = ^) *^^ tangent pLane to {A) at P. a o c SlmUarly for tlic other centre of principal curvature. This proposition is due to Salmon. 620. The curve of intersection of tia confocal conicoils is a line of curvature on each. Let PT be a tangent at P to the curve of intersection of two confocals S and .S", P^V, PN' normals at P to S and 5' ; and suppose a central section of S made by a plane parallel to the tangent plane .V'PP, and therefore to the indicatrix to S at P. Now it is shewn (Art. 285) that PN' is parallel to one axis of this section ; therefore PT is parallel to the other axis • hence, the tangent to the curve of intersection of S and S' at any point is parallel to an axis of the Indicatrix of cither surface at that point, and the curve Is a line of curvature. uou 410 LINES OF CURVATURE 621. At any point in a line of curvature of a conicoid^ the rectangle C07itained hy the diameter parallel to the tangent at that point and the perpendicular from the centre on the tangent plane at the point is constant. Let the line of curvature on the conlcoid S be the curve of intersection with /S", and let FT be a tangent to it at any point P; PN^ PN' normals to S and S' at P; then, if a, /3 be the semi-axes of the central section parallel to N'PT^ the tangent plane to /S, which are parallel respectively to PT and PN\ it is shewn (Cor., Art. 285) that ^ is constant, and if p be the perpendicular from the centre on the tangent plane, pa/3 is constant, therefore pa. is constant. 622. The following proof is independent of the properties of confocal surfaces. Let P, Q be consecutive points on a line of curvature, the corresponding centre of curvature, p the radius of curvature, p the perpendicular on the tangent plane, a, /3 the semiaxes of the section parallel to the tangent plane, a being parallel to P^, and let CP=r, then by the triangle OOP 0C' = p' + r'-2pp, and since, for a change from P to Q, 0, C are unchanged in position and p is unaltered, rdr = pdp. But, by Art. 273, (e + ^'' + r' = ci'^V'-^c% a^p = ahc] .-. ada + (3d/3 + rdr = 0, and ^ + ^ + ^^' = 0, multiplying the last equation by a'^ orpp, and subtracting the preceding, we obtain (a'^ — ^'^) c?/3 = ; therefore /3 is constant, unless a = /3, which is only true at an umbilic, therefore ^)a is also constant. 623. To shew that the curves of intersection of a given conicoid with all confocal conicoids ivhich intersect it satisfy the differential equations of a line of curvature. Let the equation of the surface be a c ^ ^ ' OF CONICOIDS. 411 then the differential equations of the lines of curvature arc ^dx+^^d^+'-oints intersect. If the axis of s be a normal at an umbilic, the equation of the surface is of the form ^= a {^"^ + tj') + u^ (1 + e), where u^ is of the third degree in | and 77, and t vanishes in the limit ; the equations of a normal at (ic, 3/, z) are but if this normal meet that at the umbilic, the equations are satisfied byj|^ = 0, 97 = ; du du dy ^ dx ' which gives three directions in which the point (x, y, z) must be taken. 628. To find the three directions for which normals to a conicoid intersect the normal at an umbilic. Let a^"^ + br)^ +c^'^ — 1(1) be the conicoid, (a, 0, 7) the um- bilic, (a + X?-, /Mr, y-{- vr) a point adjacent to it in the direction (\, /A, v), the equations of the two normals are ^ — a — \r 7) — fxr ^—y—vr « (a 4- \r) bfir c (7 + vr) ' aoL cy one condition that they may intersect is /a = 0, one direction is therefore that of the principal section containing the umbilic ; for the other conditions f - a = \r- -v (a-l- \>-) and ^- 7 = I'r— - (7 + vr) ; .'. cy [b — a)\ = aa{h-c)v'j or, since b — a c — b THROUGH AN IMnil.lC. 415 aoik + 07^ = ; (2) and, by (1), n (a + X?f + bfi'r' + c[y + vr' = 1 ; (3) .-. aX' + bfjL' + cv' = 0', (4) (2) and (4) give the two other directions for wliich the normals intersect ; and, since : 3) is satisfied for all values of r, they are the directions of the imaginary generatrices through the umbilic. G29. ^Ye may observe also that since «'-'a-'\-' = cV^'^ (i - «) X" = (c - J)" ; .-. a\' -f bf^' + c'v' = b {X' + /i'-' + v') ; .-. X' + fi' + v' = by (4), which shews that these generatrices pass through the imaginary circle at infinity. (Since the argument of Art. G28 is independent of the mag- nitude of r, it is true that all the normals at points along one of the umbilic at generatrices intersect, and they have therefore this character of lines of curvature, but Cayley has remarked in a note on a paper upon " the direction of lines of curvature in the neighbourhood of an umbilicus,'"* that they are the envelopes of the lines of curvature, and belong to the singular solution of the differential equation of these lines, as oppeai-s from Art. 624. 630. In the note referred to above, Cayley remarks that since, at an umbilic, -^ is determined by a cubic equation, there are generally three directions of the line of curvature, which may arise from three distinct curves, or from n curve with a triple point at the umbilic ; and, referring to a paper by Serret,t he states that the lines of curvature on the surface xyz = 1 are its intersection with the series of surfaces /* = [x" + wy" + wV)* + (a;" + o»y + ctz') «, • FroBt, Quart. Joum. of Math., toI. x., p. 78, and Oaylej, ibid. p. III. t Liout. Jaum., t. 12 (1847), pp. 241— -.'64. 416 LINES OF CURVATURE. where to is an imaginary cube root of unity ; now at the umbilic (1, ], 1), corresponding to which h = 0, {x' + a>if + <^'zy = {x' -f wy + (ozy ; .-. X^ + (Jiy^ + Oi'z^ = X' + Co'y + Oiz\ or = (u (a:"" + w'Y'' 4- (nz^)-, or = o)'^ {x^ + wy -f (w^'") ; .'. 7f = z\ or x^ = y\ or 3^ = a;"''; hence, through the umbilic (1, 1, 1) three distinct lines of cur- vature pass, viz. the curves y = z^ xy^ = 1 ; x=y^ zx^ =^\] and z = x^ yz^ = 1. 631. The differential equation of the line of curvature of xyz = 1 is xdydz [y^ - z') -f ydzdx [z^ - ^) + zdxdy [x'' — y') = 0. Multiply by xyz^ and let x^ = j?, y' = q^ z' = /• ; .*. p[g_- t) dqdr + q{r- p) drdp + ^{p — q) dpdq = 0. (1 ) Again, if h={p + a>q+ toV)^ + (/> + (o^q + (or/ [dp + codq + (o'^h-y [p + coq + co'^r) - [dp + (o'^dq + codrY [p -f sin' \

' + 2-^> ^ Cyp f ^> 4 -gg -t- C ^ J3 4 Cg* f 2^'g r a " « * (19) Shew that the projection, on the plane of xy, of the indicatrix at any point of the surface : = (e" -f e ") cosj is a rectangular hyperbola. (20) Shew that the indicatrix at any point of the surface y = x tan - is the part of a rectangular hyperbola which lies near the point. Prove that the section by the tangent plane near the point is the generating line and a portion of a parabola. (21) Deduce the conditions for an umbilicus from the equation giving the radii of ciu'vature, by making the roots of the equation equal. (22) Shew that a sphere whose centre is the origin, and the reciprocal of whose radius is - + r + - touches the surface whose equation is (23) Prove that the radius of curvature of the surface x" + j/"* + s"* = o" at an umbilic is x 3 "" . in - 1 (24) Prove that the measure of curvature at any point of an ellipsoid is proportional to /)', p being the perpendicular from the centre on the tangent plane. (25) Prove that the measure of curvature at any point of the paraboloid ?L + 1 = X varies as (-) , p being the perpendicular from the origin on the tangent plane. (26) Prove that the measure of curvature at every point of the cllipUo b paraboloid 2: = - 4 '^ where it is cut by the cylinder ^, + |. =" ^ "'* *<1"*^ 420 PROBLEMS. (27) Shew that the specific curvature at any point on the surface xyz = abc, varies as the fourth power of the perpendicular from the origin on the tangent-plane at the point, and that at an umbilic it is I (^abc)'^. (28) If a plane curve be given by the equations - = cosO + log. tan 10, - = sinO, the surface produced by the revolution of this curve about the axis of x will have its measure of curvature constant. (29) In a surface, generated as in (15), if = log taniO, the measure of curvature will be the same at corresponding points on the fixed line and on the circle. (30) The integral curvatures of the portions of the ellipsoid — + |j -t- , = 1 x* v^ z' cut off by the cone -r + ^r - -r = are in the ratio of -/S - 1 to V2 + 1« ^ a* b* c* (31) Shew that the integral curvature of the whole surface (32) Shew that the integral curvature of the portion of a surface of revolution cut off by any plane perpendicular to the axis of revolution is Att sinVo, where a is the angle which the normal to the surface at any point on the curve of intersection of the plane and surface makes with the axis. (33) If any cylinder circumscribe an ellipsoid, it divides the ellipsoid into portions whose Integral curvatures are equal. Hence, if three cylinders circumscribe an ellipsoid, the integral curvature of the portion of the ellipsoid cut off is tt - POQ - QOR - MOP, where O is the centre, and OP, OQ, OR are the directions of the axes of the cylinders. (34) Prove that the integral curvature of the portion of the surface of 3 ellipsoid -.+ ?-,+ -,= hyperboloid of one sheet X v z the ellipsoid —. + ?;+ -^ = 1, bounded by its intersection with the confocal '^ a* b' c* + ~-^, + -^^. = 1 . _ c» /r(a*-\«)(6«-X«)\ x' v^ 2' (35) Find the umbilic of the surface — + ~ + — •= k^, and shew that, at ^ ^ abc the umbilic, - = ^ = - , the directions of the three lines of curvature are abc , . ^. clx dtf ihi (h . (h dx ^. , given by the equations — = t" > t" = — > ^"^ — = — respectively. PROBLEMS. 421 (3G) If the inclination of two surfaces at any point of their curre of intersection be (', 5 the arc of the curve of intersection, p^, p, tlie principal radii of one surface ; « the angle between the tangents to the curve and to a principal section, and />,', i^.', a corresponding quantities for the other ^ do Bin 2a /I 1\ sin 2a' /I 1 \ ,, . surface, prove that -r = — :z— I .,-—-—)• "ence, shew as a \pi pj 2. \p^ Pi/ that if two surfaces intersect always at a constant angle, and the curre of intersection be a line of curvature of one surface, it will also be a line of curvature on the other surface. (37) If one series of lines of curvature on a surface be plane curves, lying in parallel planes, the other series will also be plane curves. (38) The planes drawn through the centre of an ellipsoid, parallel to the tangent planes at points along a line of curvature, envelope a cone which intersects the ellipsoid in a sphero-conic. (39) On an umbilical conicoid, the projections of the lines of curvature on the planes of circular section, by lines parallel to an axis, form a series of confocal conies, the foci of which are the projections of the umbilics. (40) Find the differential equation of surfaces possessing the property, that the projections, on a fixed plane, of their lines of curvature cross each other everywhere at right angles. Prove that it is satisfied by surfaces of revolution whose axes are perpendicular to the fixed plane ; and obtain the general solution. (41) Prove that the three surfaces yz= ax, VC** + y*) + -/(*' + «*) = *. v'(x' + y*) - ;{x* + s*) = c, intersect each other always at right angles ; and hence prove that, on a hyperbolic paraboloid, whose principal sections are equal parabolas, the sum or the difference of the distances of any point on a line of curvature from the two generators through the vertex is constant- (42) In the helicoid, whose equation is t/ = z tan- , the lines of curva- ture are the intersections of the helicoid with the surfaces represented by the equation —^ — — ^ = ca"' f - «'" for different Talue* of c. Also, prove that the principal radii of cur\ature are, at every pomt, constant, equal in magnitude, but of opposite signs. (43) Tangent planes are drawn to a scries of confocal conicoida- from a fixed point on one of the axes, the locus of the points of contact it a surface ; prove that three such surfaces corresponding to three poinU ooo on each axis cut one another orthogonally. 422 PROBLEMS. (44) Prove that the lines of curvature on the surface a ax - O* + i"* aX - a + C are two systems of circles, ^vho8e planes are parallel to the axes of y and 2 respectively, and pass each through one of two fixed points on the axis of z. THE END OF VOL. I. PRINTED BY W. METCALFE AND SON, TEINITY STREET, OAMDUIDGE. UNIVERSITY OF CALIFORNU LIBRARY This book is DUE on the last date sta„,ped below Reschedule: 23 cents on fi^t da, overdue '--^..>. *^? 50 cents on fourth day feverdile. ' * "^ '' One daiftx oa seventh day overdue. OCi ?947 (^^^^ f M JUN20I976 1O , ^Q^M^ fi£C.cjfi. 2^76 FEB 7 t955 J ^f^||;°1987 3War'55JP ' ^ 195510 JAN AUG 1 1 2003 LD 21-100to-12,'46(A2012816)4120 riov y RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY BIdg. 400. Richmond Field Station University of California Richmond. 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