ew BCR... bal J Se ke iF) LP ? Te i Ce RK! sa Male a 4 . “4 0 8-4. teen ae Lee ye % ) 16 Ty ACO AE 4 rete ra , f Fs Wd it beer hia WTR On CHE Bi tea Oe UFR Oa ie ate 2 Lapetetsgee 8 teed dl he}, Perea ire 1a eee th Lars aame en is4 “yee 4H eG te hee Oe od cu a peers UCL Me ae nba be dd Gee aed ates Hn seH Ue SPafups a bhip ot eet A * - a4 ) Oe et NG 8 Ale ti if eth Cow Pi ieatatarn'd Pace! ere: DC ee Ware, oo bn mPa Dai b> Wak COW INO, We LTT ek Wake SPR CEL eo Bh ny OW hog eM AB HEA BM Miph sy Pe 1' Hes SUP eerie’ Saar Riksta Paeaes bel hs po Waa ae, Way. vis aye tere te Li RRR bbl wee RES RUNG Hi Boat eT ee eh HG Se ada Wiss BiH i) 4 54 i hoy ie eae et Ne 8 Mire 33 Sree it Le adh oh ; J ae ye LPM nour are Merl scree Se STE “9 sath Neh, ‘ ain {Orden See 4 2 bo Ant : ‘4 ‘ } io m ‘ ; se ; fat ye | MER Ae yd i wash ’ a4 ange 1 } y rs a 4 J i S, + - MW KE + sien ; P| Ue Rea im ir eam a : Perit Yow te th) 4 ipherer +4 is sm i i A Cee SORE RCM ke ae ee ee OSG Au hen it ak yeast / ; Gh ee te ee eg WH at cen Sc ibyth ited hae tts: to Aaa DPR ae ere aN ae PiN ete ; era Hoos) be ro fri Jee ae GH de eS f ne ase Wed tH te wave 49) bee i , Re y ¥ 4 JE 4 iu 3 Hr nh f bd at Sa , ti Ci renus Laat ee j it hat Pe \ VA ity . aU iy ts a iia : ST MAS a noun ea i evened teed wt a Wine, |) te 1 ee. phd ih Winters sta s ¥; Vad oy 4 eine Me ba hy oe | PT OOM OC EA Eas eA ey ee (ey te Cn ea) Ma Maas ait re 4 CE Tha Uri er ar aren WW WM YC ES Bd Fy Brey ashes ae dae Baa i rule se ys WS ae VE a hear wh 4) Boe aA J feet WE i ete by Fi * ' Oa ee NO 4 Fade dss Vas ¢ rf y , tata } as elds if ie snag vs he ea ae Bade aL Fenris sacir arses { pei waa ian Sa Ah Ms ae " ine ANY et 18 When ares Kile wt uf Se Heit Mat ear ure hay Fup i ron y y, ran uta iy It Ldaw oat wad ¥ li es i, ait i) #0) Ne ares ee hs) ' i lagi tiy all ne i ive My an " 4 ing i ai i, ‘Fe (mates et a i on Th See au 4 hay 4a Eley teh to IE Waites 1 a) (Hels 44-4 4 j 034 Perea a fang ot ay itd at atl Ao) Sah a = ease eat i i yon a apt Le : hae Sora aes Se =e <= ya ahs + iit 1K oe ol ea eee Poder ara Mire 41 i a si oh ea pom e dane el aR REM A RE Nee he git a1 Wodacuna i PAA i aan i dul anya ite tied Ys iva “f vo a v 7 ate if ih araeMat : 3 i Lg +f Hint qh ah wat We aa hina i Pe A Rt iNet A iB iad Vadaats rae Mt vihtat “iy ys ticited Dah), aii URE belie SHinthag AZAR. staat auton ato pala tha de ai ete P hatasicaanst fyeg) eMaled ‘ RL EULA RG ran $y ; ¢ ' : ae 4 ‘ i f Syugpaewe Mathys 7) : 4 4% Drie Peary betty ‘ Poe Bi 1 Riqawalt Qa4 BLUES aie: ie i t oe \akeporrihy tas Asya) ' m4 i) , PY rr TO Sih Ag pt ap heated AB Geek 6 ‘ 4 ia eae yy Lae x SiN WV Return this book on or before the Latest Date stamped below. University of Illinois Library L161—H41 oy ae nd it VL Ft el Se ia He : P Ape A TREATISE ON THE APPLICATION OF ANALYSIS TO SOLID GEOMETRY. ~ 9 / € ‘ ~ Tov del OYTOC N YEWpETPLKH yvwoic EoTLY. TlAarwr. COMMENCED Penal G REGO RY. se Ma As LATE FELLOW AND ASSISTANT TUTOR OF TRINITY COLLEGE, CAMBRIDGE. CONCLUDED BY W. WALTON, M.A. TRINITY COLLEGE, CAMBRIDGE: CAMBRIDGE: DEIGHTONS: LONDON: WHITTAKER AND CO.; SIMPKIN AND CO. 1845. ne * Ay a oy . vu ‘ef . ae By SG: fil oe o yes i SV] om “ \ = P ‘ ay Re eee ee ) ¥ x : ‘ . ae bee ye aw ¢. farts j ed t i cami | * + f i : J sc} a tos YE Ss 7? e \ - (a “ Ste = NAG 2 as eee we oe rege CAMBRIDGE: PRINTED BY METCALFE AND PALMER, TRINITY-STREET. Tero Lo S* << PREFACE. THis work was commenced by Mr. Gregory in the course of the year 1842; and inthe Autumn of 1843 had received his final revision as far” as. the. end of. Chapter x1.: he had likewise made numerous See from various sources pre- paratory to writing Chapters xu1., xtr., xIv., xv.: and had arranged a collection of Problems which, with slight additions, forms the subject of Chapter xv1., the last of the Treatise. His further progress in this work was unhappily arrested by death. Having, in accordance with the last wishes of my most valued friend, undertaken the completion of this work, I have fulfilled my task to the best of my ability. It is hoped that the natural difficulty of brmging to a conclusion a treatise commenced by another, will secure for me the indulgence of the reader. The principal object of this Treatise is to develop a system of Solid Geometry, in a form suitable to mathematical students, > by means of symmetrical equations. The general advantage . of symmetry in this branch of mathematics is so striking, that 2 eye ° . athe utility of such a work will be at once recognized. There vun 147194 620666 lv PREFACE. are undoubtedly many cases in which unsymmetrical methods have the advantage of brevity; a bigoted adherence to sym- metrical investigations has therefore been avoided. It is scarcely necessary for me to state that I have derived great assistance from Leroy’s Géométrie-des Trois Dimensions, Moigno’s Caleul Différentiel, Gregory’s Examples of the Pro- cesses of the Differential and Integral Calculus, and from several articles in the Cambridge Mathematical Journal. My numerous obligations to my mathematical friends have been acknowledged in the course of the work. WILLIAM WALTON, M.A. Trinity College. CAMBRIDGE, January 1845. ARTICLE 1-6. 11. 12-13. 14. 16. 16. 17. 18. 19. 20. al, 22. 23. 24-27, 28. 29. CONTENTS. CHAPTER I. Exposition of Principles and Fundamental Theorems. Elementary notions Interpretation of equations Theory of Projections Distance between any two ne in rao of fan Pectin co-ordinates Distance between two Borie in erie of ASFERE co- mehtnrtn Relation between the direction-cosines of a line : Square of any plane area equal to the sum of the squares of its projections on three planes at right angles to each other Cosine of the angle between two straight lines in terms of the direction-cosines of the lines Projection of any finite straight line on anOUeE cea line found by first projecting the line on three rectangular axes and then taking the sum of these projections projected on the second line Area of a triangle expressed in eee of ihe co- verde of Re angular points Volume of tetrahedron is sti in terms of the co- drdhviniee of its angular points Analytical Theorems CHAPTER ILI. Of the Straight Line and Plane. Two methods of applying Analysis to Geometry Equations to a straight line To find the angles which a given pouight the makes mith the co-ordinate axes ° To find the equations to a straight ine which passes Thaar two given points ° ° Page 15 15 16 17 18 21 24 Vi CONTENTS. ARTICLE Page 30. To find the condition that two straight lines may intersect, and the position of the point of intersection ; 26 31. To find the angle between two straight lines of which the equa- tions are given : 27 32. To find the conditions that two echt lines shi be raraltel or perpendicular to each other : ; ; 28 33-38. Equation to a plane : _ 39. To find the angles which a given nist makes ith thee co-ordi- nate planes : 33 40. To find the angle between the een line Ain plane, et equations being given — 41. To find the angle between two plates of sath the equations are given : 34 42. To find the conditions that oo nian may be pernilel or per- pendicular . _— 43. To find the equation ca a plane ar a aeher chrodeu three given points ; 35 44. To find the equations to the ine of miseenttne of ie alates 36 45. To find the conditions that a straight line may be perpendicular toaplane . 37 46. To find the condition amt a straight line may ‘ibe parallel to a plane : -- 47. To find the sonmaone that a given auaient ine! may lie ina given plane sa 48. To find the length of the serieaconle drawn fone a given point on a plane, the equation to which is given — 49. To find the length of the perpendicular from a given pote on a given straight line 4 38 50. To find the perpendicular distance betweed two dtraipht ae not in the same plane 39 51. To find the shortest distance Parveen two straint weet : 40 52. To find the equations to a straight line which cuts at right angles two given straight lines ; 41 53-58. Problems relating to the straight line and abies sotenrerl to oblique co-ordinates . : : : 42 CHAPTER III. Transformation of Co-ordinates. 59. ‘To change the origin of co-ordinates, the axes remaining parallel to their original position ; ° 48 60. To pass from a rectangular system to any ‘other the origin remaining the same ; - ‘ : 49 ARTICLE 61-64. 65. 66. 67. 68-69. 70. tt. 72-73. 14, 75-76. 21-18. 79-88. 89-91. 92. 93-94. 95-99. CONTENTS. To pass from one system of raion co-ordinates to another also rectangular Euler’s Ernie for Sate from one eae of perenrclae co-ordinates to another . To pass from one system of oblique co-ordinates to Bieter also oblique Degree of an egos not aiieciad by Pigeoretian of co- entte nates . Intersection of a surface ie a plane Polar co-ordinates CHAPTER IV. Reduction of the General Equation of the Second Degree. General remarks : : : ° Process of Reduction ° Possibility of the roots of the Hough llieating cubic The general equation of the second degree always reducible to the form Px? + Py? + BP"? + 2Qx + 2Q°y + 2Q"2+ H=0 Separation of surfaces of the second degree into different classes CHAPTER V. Interpretation of the Equation of the Second Degree. Central Surfaces : : Surfaces without a centre Surfaces having a line of centres Remaining surfaces of the second order CHAPTER VI. Theorems relating to Surfaces of the Second Order. Diametral Planes 100, 101. Principal Diametral Planes 102. Form of the equation to the surface aant a dearer) plane is taken as the plane of zy, and the axis of z parallel to its chords 103-105. Conjugate diametral planes 106. 107. To find the relations between oblique eeeste qameier and principal diameters of central surfaces Remarks on surfaces of the form Py’ + P’2? = 2Qzx Vii Page 50 52 53 57 58 62 63 67 74 17 78 81 85 86 87 88 91 Vill ARTICLE 108. 109, 110. Lil; 112. 113. 114-119. 120. 121. 123-126. 127, 128. 129-132. 133-142. 143, 144. 145. 146-150. 151-153. 154. 155-158. 159. CONTENTS. Page Similarity of surfaces : 92 Conditions that two surfaces of the sotond price may ne similar . = If two similar surfaces “of the adhe ae cut aren other, their line of intersection is a plane curve : 94 If four similar surfaces intersect each other, the six planes of intersection pass all through one point 95 If two surfaces of the second degree intersect in a plane curve, their second intersection (when they have one) is also a plane curve ; : ; ; — Of plane sections 5 96 The sections made by parallel ance in a Ree of the second order are all similar curves 100 To find the locus of the centres of sections of a paras of the second order made by a series of parallel planes. 101 To find the axes of a section of the ellipsoid 2 2 2 =+5+5-1, made by a plane lz + my + nz = : : 102 Circular sections , 103 Any two circular sections a Barina to dierent series lie on the surface of the same sphere . 107 Conditions that the equation of the Heese degree ana represent surfaces of revolution ; : 108 Rectilinear generating lines : ‘ : 112 CHAPTER VII. Of Curves in Space. Preliminary remarks . 3 119 Analytical distinction between plane curves and Ther of double curvature : : : 120 Properties of the intersections of aaties : 121 Curves on the surface of a sphere. Equable spherical spiral. Spherical ellipse : . ; . 124 Helix : : : : : 127 CHAPTER VIII. Of Tangents and Normals to Surfaces. Tangent lines and tangent planes. 129 Property of the intersection of a surface at its sancene plane. : : 3 f 132 ARTICLE 160. 161-164. 165-167. 168-175. 176, 177. 178-179. 180. 181-183. 184-186. 187-195. 196, 197. 198-202. 203-2085. 206-211. 212-216. 217-220. 221-224. 225-233. 234. 235-238. 239, 240. 241-243. 244, 245, CONTENTS. Skew surfaces and developable surfaces ‘ Equations to the normal. or of the tangent plane. surfaces Curve of contact Examples CHAPTER IX. Of Tangents to Curves, Normal and Osculating Planes. Equations to the tangent line Equation to the normal plane. gent and normal plane Equation to the osculating plane Examples Line of greatest slope. Examples CHAPTER X. Of the Generation of Surfaces by the Motion of Curves. General observations . Edge of regression or cuspidal eure of deeelanenle pareacet Cylindrical surfaces Conical surfaces Conoidal surfaces Skew surfaces having more ‘than one pecs Developable surfaces Surfaces of revolution CHAPTER XI. Envelops to Surfaces. Envelops; equation to the surface which envelopes a series of surfaces drawn after a given law; characteristic and edge Envelops to surfaces hich have sas sadenendent hare meters Developable surfaces Developable surface circumscribing i given ani Tubular surfaces Example of an envelop to a aes tia two oderanseat parameters. Wave surface Direction-cosines of the Sonat Tangent plane to homogeneous Direction-cosines of the tan- 1x Page 133 134 136 138 145 146 147 149 152 154 159 160 163 166 169 173 175 179 184 185 188 190 192 x CONTENTS. CHAPTER XII. On Partial Differential Equations to Families of Surfaces. ARTICLE Page 246, 247. General principles. : : : 195 248. Cylindrical surfaces ; ; : ; 197 249. Conical surfaces : : : : 198 250. Conoidal surfaces ‘ ; : ; 199 251. Surfaces of revolution : z 200 252. Ruled surfaces having a director ako : : 201 253. Developable surfaces . ° : : — 254. Tubular surfaces . . 202 255. Symmetrical equation for deaelonabla surfaces : 203 256. Symmetrical equation for tubular surfaces ; 204 257. Transformation from symmetrical to unsymmetrical seine 206 CHAPTER XIII. On Singular Points and Lines of Surfaces. 258. General method of investigation. Examples . : 208 259. Condition that a plane may touch a surface ina curve line 214 CHAPTER XIV. On the Curvature of Curves in Space. 260. Measure of the rate of curvature ; 217 261. Radius of absolute curvature. Angle of contingene : oo 262. Measure of the rate of torsion . 218 263. To calculate the angle of contingence at any point of a curve in space. Magnitude of the radius of curvature : — 264. To find the meni and position of the radius of curvature 220 265. Another method of finding the magnitude and position of the radius of curvature : 223 266. To calculate an expression for the angle of torsion ; 225 267-271. Points of inflected torsion. Points of suspended and of in- finite torsion. Points of inflected curvature. Points of suspended and of infinite curvature. = 272. Distinction between plane curves and curves of double curva- ture in regard to their radii of curvature . 228 273. To shew that a curve of double curvature has an infinite number of evolutes , ; “4 229 - ARTICLE CONTENTS. CHAPTER XV. On the Curvature of Surfaces. 274, 275. Normal plane to a surface. Radius of curvature of a normal 276. 277, 278. 279. 280. 281. 282. 283. 284. 285. 286, 287. 287. 288-291. 292-295, 296. 297. 298. 299, 300. 301. section To obtain an expression for the Ghat of siaGinGs ioe from second differentials Remark on the double sign in the expression for the -nifox of curvature Unsymmetrical form of the expression ee the “ttt of cur- vature Examination of the oeene of the eit of eet A any point of a surface in relation to the tangent plane To find the greatest and least radii of curvature of the normal sections at any point of a surface To shew that the normal sections of greatest and tenet curva- ture are at right angles to one another. Principal sections. Principal radii of curvature Principal radii of curvature of an ellipsoid Another demonstration of the perpendicularity of the principal sections To prove that the oratire of any ieee aecnan is eal to the sum of the curvatures of the two principal sections multiplied respectively by the square of the cosines of the angles which the principal planes make with the normal plane. : Condition that two of the co- mnate aires may coincide with the principal sections at any point of a surface Another demonstration of the theorem of Art. 284. Utility of this theorem To shew that the sum of the mba of any two ‘tse sections at right angles to each other is constant at a given point of a surface Remarks on the theorem of Art. 284 Conditions for the existence of umbilici Another investigation of the problem of Art. 280 Cauchy’s demonstration of Meunier’s theorem Lines of curvature on surfaces Lines of curvature at any point of a surface are it right apis to each other, and coincide with the directions of greatest and least curvature Simple method of finding the Prana pectont and Rema radii at any point of a surface of revolution ; x1 Page 230 231 232 234 239 257 259 x CONTENTS. ARTICLE Page 302. Lines of curvature on an ellipsoid . ; 259 303. Dupin’s Theorem : 4 ; : 263 CHAPTER XVI. i PROBLEMS ; : 265 ERRATA. PAGE LINE CORRECTIONS. 1, Note. Not any ronan on the subject of this Note have been found among Mr. Gregory’s MSS. 2, 10from bottom. Read Oz instead of Ox. 16, 5 from bottom. The figure ought to be so drawn that JZ, shall lie within the area OMM,. 31, 13. Read r-being the ratio of, the distance between the points xyz, Lyy,%,, to that between the points 2,y,%,, %%%,; mstead of y being’... (5/992): 35, 8 from bottom. Read - y,z, instead of — yz, . 40, 13. or (a~a')X+(B~B) e+ (y -y)», Read t+ {(a-a@’)X 4+ (B-P) ut (y-Y) 4. 40, 15. Instead of the expression for 6, take rene (mn' — m'n) (4 — a’) + (nl' — nl) (B - B’) + (in - I’m) (y - rae sin 0 66, 4 from bottom. Read P"y + Q" mstead of Pa+ Q. 74, 11. For (y, 2) read (a, y). 92, 15,17,18. For 7 and 7’, wherever they occur, read 7, and rj’. 98, 9. For condition read function. 98, 11 from bottom. By this cutting plane, 7s meant a plane of parabolic section, which in this Article is proved to be parallel to one, and to one only, of the generating lines of the asymptotic cone. 125, 13. cos® should be replaced by sin ©: corrections of consequent errors must be made throughout Art. (152). 137, 4. Insert =0 just before the comma. 149, 9. There are errors in Art. (181) consequent on those in Art. (152). 161, 7. For a, B, y, read x, y, 2 APPLICATION OF ANALYSIS TO SOLID GEOMETRY. CHAPTER I. EXPOSITION OF PRINCIPLES, AND FUNDAMENTAL THEOREMS. Elementary Notions. Art. (1). Ir is necessary for the Application of Analysis to Geometry that we should have the means of expressing by symbols, not only the absolute magnitudes of geometrical quan- tities, such as lines, areas, angles, &c., but also the positions of points. Our habit of denoting arithmetical quantities by a single symbol naturally leads us also to denote the simplest geometrical magnitudes by a single symbol, and thus we represent straight lines of different lengths by such symbols as @, b, x, y, a, 3. Then, in virtue of a principle we shall here | basa as known, an area is denoted by the product of two, and i solid by that of three such symbols considered as numbers.* Miigtes, being a species of geometrical magnitude not homoge- aeous with straight lines, we shall denote also by single leHers ising generally ite Greek letters X, u,v, 0, ¢, ~. Functions of ingles, such as sines, cosines, &c., we shall often denote also by imgle letters: they may always be considered as the ratios of wo of the symbols of straight lines. * The reason of this symbolization will be found in the Appendix. B wo ELEMENTARY NOTIONS. (2) The position of a point in space is determined by referring it to three fixed lines intersecting each other in one point. And the mode by which this is done is the characteristic feature of the Application of Analysis to Geometry. Any fixed line is called an aais, and the three fixed inter- secting lines are called the co-ordinate axes, their point of intersection being named the orzgin. Each of these lines may be considered as determined by the intersection two and two of three planes. Thus, in fig. (1), if Oz, Oy, Oz be the three co-ordinate axes, O the origin, we may consider the axis Oz as the intersection of the planes zOz and yOz; the axis Oy as the intersection of zOy and xOy, and the axis Oz as the intersection of Oz and yOz. ‘These planes are termed the co-ordinate planes, and may be used as fixed planes to which the position of a point in space may be referred. In speaking of these planes we shall call xOy the plane zy, yOz the plane yz, and xOz the plane zz. (3) To shew how a system of co-ordinate axes may be used for determining the position of a point in space, let P (fig. 1) be a point situate within the solid angle Ozyz, and through P draw PA, PB, PC parallel to the three co-ordinate axes Oz, Oy, Oz respectively, and meeting the co-ordinate planes in 4, B,C’; then the position of the point P is known if we know the lengths of the lines PA, PB, and PC, which are called the co-ordinates of P. For if along the line Ox we measure a length OD equal to PA, and through D draw a plane parallel to the plane yOz,| every point in this plane has a line equal to OD or PA as its co-ordinate parallel to Oz. In like manner, if we measure along. Oy and Oz lengths O# and OF equal to PB and PC respec- tively, and through # and F' draw planes parallel to zOz and) xzOy, every point in the former has its co-ordinate parallel to Oy equal to OF or PB, and every point in the latter has its co-ordinate parallel to Oz equal to OF or PC. Hence, the point which is determined by the intersection of these three planes has for its co-ordinates parallel to Oz, Oy, Oz, the lines PA, PB, PC respectively. In other words, the position of the point P is determined by the preceding construction, and there- ELEMENTARY NOTIONS. 3 fore the position of a point may be considered as known when the lengths of its co-ordinates are given. It is easy to see from the figure that the intersections of the three planes drawn through D, #, and F, with each other and with the co-ordinate planes, determine a parallelopiped, of which O and P are opposite solid angles. Hence we may obtain the point P by a simpler construction ; for, since OD= PA, DC= OE= PB, if along Oz we measure OD = PA, and at D draw DC parallel to Oy and equal to PB, and through C draw CP parallel to Oz, making it of the given length, the point P will be determined. We might of course equally well begin by measuring the first co-ordinate along either of the other axes. The co-ordinates PA, PB, PC of a point P being different lengths of straight lines are, according to the explanation in Art. (1), usually represented by the symbols z, y, z, when they are indeterminate, and by other letters, as a, b,c, or a, 3, y, when determinate values are assigned to them. (4) In what precedes we assumed that the point P is within the solid angle Ozyz, and that the co-ordinates z, y, z are measured along Oz, Oy, and Oz in one direction only: we have therefore as yet the means of determining the position of a point only within a limited portion of space. For, since the lines which intersect at O may be considered as infinite in length, the three planes, which by their intersection determine these lines, divide space into eight solid angles, of which we have considered but one. If we indicate by 2’, y/, 2’ arbitrary ‘points in the prolongation of the axes, these eight solid angles may be denoted by Oxyz, Oz'yz, Oxy'z, Oxyz, Ozy'z; Oz'yz, Oz'y'z, Oxz'y'z'; and for each of these divisions we should require to use a separate set of symbols to indicate in which octant the point under consideration is situate, so that eight sets of formule would be required in discussing the position of a point in all possible positions. The artifices of analysis fortunately enable us to avoid this complexity by reducing all these sets to B 2 4 ELEMENTARY NOTIONS. one ; and this is done by the aid of the algebraical symbols + and —, in the following manner. (5) We agree, as may be done consistently with the proper- ties of the symbols, that, when starting from a given point a straight line of given length is considered as positive, a line of the same length measured in the opposite direction is to be reckoned as negative. ‘Then, if lines measured from O towards x (fig. 2) be positive, those measured from O towards z’ are negative; and if lines measured from O towards y be positive, those measured from O towards y' are negative; and if those measured from O towards z be positive, those from O towards 2’ are negative. Now the co-ordinates of a point P in the octant Oxyz are measured along Oz, Oy, Oz, and are therefore by agreement all positive. But the co-ordinates of a poimt P’ in the octant Oxyz' are measured along Oz, Oy, and Oz' ; consequently the first two are positive and the third negative. Hence, any formula involving the co-ordinates of P may be transformed into one involving those of P’ simply by putting — z for z, or changing the sign of z. In like manner, if we have a point P” in the octant Oz'y'z, its co-ordinates are measured along Oz’, Oy', Oz, consequently the first two are negative and the third ‘positive; so that a formula involving the co-ordinates of P may be transformed into one involving those of P’ by changing the sign of z and of y. In a similar manner we may proceed for all the octants according to the following scheme :— In the octant Ozyz the co-ordinates are +2, +y, + 2, es «ee 6 eeeee? Ox'yz oeeteeeeeeeeeetes# @ a akig + Ys + 2, Se eer 5 OLY BOs vents sein as ee ee ye cher Oe ONY 2 ign sas iis ca ns oie ale oho act ee ee asia ay oe 8) ESO OE RNA AL TIO OP Mc fore, ge OE aes Oz' U Ss) 9.0.0. 0's. 6 6 '« “Ys eeceeever eer ees ee eee ily p Ys = hs ede Theale O50 Yf' 2 ren iase sola tating a. the Boul ge tee tame ea Ox'y'2' Sept oDy, et BY Lic ia ears sata al ks eile aia ae a sen It appears then that by supposing each of the quantities x,y,z to be both absolutely positive and absolutely negative, and by combining these in all possible ways, we can represent INTERPRETATION OF EQUATIONS. 9) the position of a point in any octant, that is, in any part of space. (6) In defining the co-ordinate. axes we made no restrictions as to the angles at which they are inclined to each other, but it is usually most convenient to use as co-ordinate axes three straight lines which are at right angles to each other: such a system is called a system of rectangular co-ordinates. Interpretation of Equations. (7) The results of the applications of Analysis to Geometry are expressed in equations involving the co-ordinates which have been denoted by z, y, 2; we must therefore, before proceeding further, consider what is the geometrical interpretation of such equations. Let us take a single equation, such as tee Us this may be considered as a relation which enables us to deter- ‘mine any one of the variables when the other two are given, ‘two being always arbitrary. Let these be z and y, so that the equation is equivalent to another of the form z= oY); ‘then we are at liberty to assign arbitrary and independent values to z and y, and for every such pair we obtain from the equation a definite value for z. Now to every pair of values of x and y there corresporids a point in the plane of zy; and if through this we draw a line parallel to the axis of z, and ‘measure along it a length equal to the value of z given by the equation, it 1s clear that we shall in that way obtain a series of points constituting a surface, not forming a solid, since we take only one point in each co-ordinate parallel to the axis of z, which is drawn through every point in the plane of zy. We here sup- pose that the equation z= $(2, y) gives only one value of z for each pair of values of x and y; but if it should give several values, the only difference is that in each co-ordinate parallel to z we must take a determinate number of points, and these taken together will constitute a surface of several sheets. 6 INTERRRETATION OF EQUATIONS. It is to be remarked, that though we spoke of assigning arbitrary values to 2 and y, they must be such as will give only possible values to z; that is to say, will affect it with the signs + — and — only, for we confine our interpretations to such results. If the equation cannot be satisfied by combinations of possible values of the variables, its interpretation does not come within: the scope of our present purpose. Should however it be pos- sible to satisfy the equation by dividing it into a system of two or three other simultaneous equations, it will then represent a limited number of lines or of points, according to a principle of which we shall speak immediately. ‘Thus the equation (w - af + (y - BY + (2- of = 0, which is satisfied by no possible values of the variables, except C0, ey On ei e represents a point. If the equation be satisfied by several dis- tinct independent equations, it represents as many distinct surfaces. (8) If the equation involve only two out of three of the variables, it still represents a surface, but one of peculiar kind. — Thus, if we have the equation Wk; Yy) = 0, it is satisfied by certain values of x and y, independently of z. Here z and y are no longer both arbitrary, but one is given in terms of the other by the equation ; to each pair corresponds a point in the plane of zy, and the series of such points con- stitutes a curve in that plane. If through each point in this curve we draw a co-ordinate parallel to z, every point in that 7 co-ordinate has the same values of z and y as its co-ordinates | parallel to these axes; and therefore the equation Sz; y) = 9 is true for every point along each co-ordinate parallel to z drawn through each point in the curve. That is to say, the equation S(&, y) = 0 represents a surface such that every straight line drawn parallel to z through a certain series of points in the plane of zy les wholly in the surface. Such surfaces are called cylindrical, the INTERPRETATION OF EQUATIONS. 7 common right cylinder having been the first of the kind of which the properties were known. If the equation contain only one of the variables, so that it is of the form f(x) = 0, it can always, by the theory of equations, be decomposed into simple factors of the form z-a. If the second term of this factor, or a, be a possible quantity, the equation z-a=9 indicates a series of points of which the co-ordinates parallel to z are equal, that is, a plane parallel to yz: if the second term be not possible we do not interpret the equation. Hence, the equation f(x) = 0 represents as many planes parallel to yz as it contains possible linear factors of the form z- a. Thus we see that in all cases when a single equation is interpreted, it represents a surface of some kind or other. (9) When two simultaneous equations are given, as SI, Y> z)= 0, S{2, Y> z) = 0, each of these represents a surface, and when they are combined the co-ordinates z,y,z must belong to points common to the two sur- faces, that is to say, to the line of intersection of the two surfaces. Hence, two simultaneous equations represent a line which will be in general a curve of double curvature, unless either one of the equations be that to a plane, or the combination of the two lead to the equation to a plane. Since two equations may be combined in an infinite number of ways, the result of any such combination is the equation to some surface which passes through the intersection of the two given surfaces. Any such result may be used instead of one of the given equations, if such a change conduce to simplicity. Thus, if we combine the equations so as to eliminate any one of the variables, the resulting equation may be used instead of one of the given equations. Suppose that 2 is the variable which is eliminated, so that ; p(x, y) = 0 is the resulting equation. This, by Art. (8), is the equation to ~ aaa 8 INTERPRETATION OF EQUATIONS. a cylindrical surface parallel to the axis of 2; and as we may obtain similar equations for each of the other axes, it appears” that any line in space may be considered as the intersection of | two cylindrical surfaces parallel to two of the co-ordinate axes. (10) If we wish to determine the curve in which a surface is cut by one of the co-ordinate planes, as that of zy for instance, we must combine the equation to the surface : S(@y,2)=0 with z=0, . as for all points in the plane of zy the co-ordinate z is zero: these two equations taken together determine the curve of” intersection, or, as it is called, the trace of the surface on the plane of zy. Even though the equation do not contain 2, it must be combined with the equation z= 05; since, when taken by itself, an equation of the form S(@y) = 0 . represents a cylindrical surface as we have just seen. | If, instead of supposing z = 0, we combine Sesy, 2) = 0 “withez =a, | we determine the intersection of the surface with a plane of which every point is at the same distance from the plane of LY, that is, which is parallel to it. The substitution of @ for z in the : equation to the surface gives vA (z, Y; a)= 0, which, considered by itself, is a cylindrical surface, and when combined with z = 0 it gives us the trace of the cylinder on the plane of zy, which is clearly the same curve as the intersection 3 Posy sz) i= 0 ewith 2-220, Of the intersection of a surface by other planes we shall speak elsewhere. (11) When three simultaneous equations are given, it is easy to see that they are sufficient for determining absolutely the values of the three variables z, y, z, and consequently that they must represent one or more points. Ca Gam) Fundamental Theorems. (12) Theory of Projections. When a point is referred to a plane by means of a straight line drawn parallel to a fixed axis, the point where the line meets the plane is called the projection of the point on the plane. Thus in fig. (1) A is the projection of P on the plane of yz, B is the projection on the plane of zz, and C'that on zy. If a series of points, forming a line, be in this way projected on any plane, their projections constitute a line which is called the projection of the line on the plane. When one line or several lines connected together enclose a plane area, the area enclosed by the projection of the lines is called the projection of the first area. If the plane on which the projection is made be perpendicular to the fixed axis, the projection is called orthogonal, and it is this kind which we shall have chiefly to consider: unless, therefore, the contrary be expressly stated, the projection is always to be considered as orthogonal. This idea of projection may, in the case of the straight line, be somewhat extended ; for if from the extremities of any terminated straight line we draw perpendiculars to a line fixed in position, the portion of the latter intercepted between the feet of the perpendiculars is also called the projection of the former line on the fixed line. From this definition, combined with what has been said in Art. (3), it is easy to see that the rectangular co-ordinates of a point are the orthogonal projections on the co-ordinate axes of its distance from the origin. (13) The general property of all orthogonal projections of bounded straight lines or plane areas, is that the projections are equal to the original line or area multiplied by the cosine of the angle between the straight line or plane area, and that on which it is projected. This must be proved separately in each case. Ist, When a straight line is projected on a plane. Let PQ (fig. 3) be the given terminated straight line, ABCD the plane of projection: draw PM, QW perpendicular to it; then IZN is by definition the projection of PQ on ABCD. Since PM and 10 FUNDAMENTAL THEOREMS. QN are both perpendicular to the same plane, they are parallel to each other; in the plane therefore in which they le draw PR parallel to WN, and meeting QN in R, so that PR is equal to MN. Now the inclination of a straight line to a plane is the angle which the line makes with the intersection of the plane and a plane perpendicular to it passing through the line. Since, then, PM and QW are perpendicular to ABCD, the plane of PQMN is also perpendicular to it, and the inclination of PQ to the plane ABCD is measured by the angle between PQand MN or the equal angle QPR. Let this be 0, then in the triangle PQR Re RQ cos; and therefore MN = PQ cos 8, as was to be proved. It is to be observed that we consider the inclination of the straight line to the plane to be the acute angle which it makes with its projection. 2nd, When a straight line ts projected on another straight line. Let PQ (fig. 4) be the terminated straight line, 4B the line on which it is to be projected, and which is not necessarily in the same plane with PQ. In such a case, since the lines do not meet, their inclination is measured by the angle between one of them, and a parallel to the other drawn through any point in it. Draw PM, QN perpendicular to 4B; then, by definition, MN is the projection of PQ on AB. Through QN draw a plane perpendicular to AB, and let F& be the point where it is met by a parallel to AB drawn through P: join QR. Then, since a straight line which is perpendicular to a plane is perpendicular to every straight line in the plane, the angle PRQ is a right angle ; and therefore Deer GCOS a) Late But since PRMN is a rectangle, PR = MN; so that, calling 0 the inclination of PQ to AB, we have MN = PQ cos 6, as was to be proved. ‘The angle 9 is, as before, supposed to be the acute angle which PQ makes with AB. * FUNDAMENTAL THEOREMS. 11 If instead of two fixed points PQ, connected by a straight line, we have any number of points PQ P,Q, (fig. 5) connected by straight lines PQ, QP,, P,Q, and if from P, Q, P,, Q, we draw on AB the perpendiculars PM, QN, P.M, @N,, the whole line AZN, is composed of the projections MN,, NM, 1 MN,. But MN, may be considered as the projection of a single line PQ, connecting P and Q, ; therefore the projection of any single line connecting two points is equal to the sum of the separate projections of any number of connected lines which join the same points. Such a series of lines may be called a broken line ; and we may thus say generally that if any two points be connected by a straight line or by any series of broken lines, their projections on any line are equal. ‘This is a propo- sition which we shall frequently have occasion to use. In the figure we have supposed all the separate projections to be additive ; but if one of the points, as Q,, were in the posi- tion Q,, the projection of P,Q, must be subtracted: we may, however, get rid of the necessity of attending to this. For if we consider the angles as measured by the inclination of lines estimated all in the same direction, as for instance the inclina- tion of PQ to AB and of P,Q, to AB and not to BA, it is clear that the latter will be an obtuse angle whenever by the position of Q, the projection of P,Q, is to be subtracted; and hence the sign of the term is given by the sign of the cosine. 3rd, When a plane area is projected on a plane. We shall begin with a triangle of which one side is parallel to the plane of projection. Let ABC (fig. 6) be the triangle, A’B'C’ its pro- jection, of which we suppose the side B’C" to be parallel to BC. Through 4.4’ draw a plane perpendicular to BC and BC", which therefore cuts the triangle and its projection in the lines AD and A'D’ perpendicular to BC and BC’. The area of the triangle ABC is then equal to } BC.AD, while that of A'B'C' is equal to B'C'.A'D' or | BC.A'D’. But A'D’,, being the pro- jection of AD on the plane, is equal to AD cos 9, if @ be the inclination of the plane of the triangle to the plane of projection, or of AD to A’'D’. Hence we have A'B'C' = ABC cos 0. # 12 FUNDAMENTAL THEOREMS. If one of the sides of the triangle be not parallel to the plane of projection, we may draw through one angle, as B, (fig. 6a) a plane parallel to the plane of projection and meeting the plane of the triangle in some line BD. Then by what has preceded, as A’BD is the projection of ABD, BC'D of BCD and A’BC' of ABC, we shall have A'BD = ABD cos 0, BC'D= BCD cos 0; since the side BD of each of these triangles is parallel to the plane of projection. Hence, subtracting the latter from the former, _4’BD - BC'D =(ABD - BCD) cos 6, or A'BO' = ABC cos 6, as was to be proved. Since every polygon may be divided into a number of tri- angles, of each of which the preceding proposition is true, it applies also to the sum of the triangles, that is, to the polygon. Also, the theorem may, by the method of limits, be extended to curvilinear areas, since they may always be considered as the limits of polygons of which the proposition is true. By means exactly similar to those employed in the case of several series of lines terminated at the same point, we may shew that if any number of straight lines be connected either by one plane area of which they are the boundaries, or by any number of plane areas having common edges, the projections of — all on any plane are equal. (14) Yo express the distance between any two points in terms of ther rectangular co-ordinates. Let PQ (fig. 7) be the two points, and assume Oz, Oy, Oz as rectangular co-ordinates; draw PN, QN’ parallel to Oz, and NM, N'M' parallel to Oy; then OM, MN, NP are the co- ordinates of P and OM’, M'N' and N'Q are those of Q. Let OM=2z, MN=y, NP =z, OM'=2,, M'N'=y,, NQ=z,; let PQ =r, and let A, uw, v be the angles which it makes with axes of z, y, and z. On PQ, as a diagonal, construct a rectan- gular parallelopiped, the sides of which parallel to the axes of L,Y, z are equal to L,-2, Y,-Y;> a — &. FUNDAMENTAL THEOREMS. 13 Now if we project on PQ the broken line PTSQ, we have r =(z, — x) cosr+ (y, — y) cos pw + (2, — 2) cos ». Again, projecting PQ on the axes of z, y, and z, we have 2—-£=7cosrA, y¥,-y=rcosp, 2,-2=7 cosy; multiplying these equations by z, — z, y, — y, 2, - 2 respectively, and adding, we have, by the previous equation, r= (x, = a) ai (Y, a y) a (2, - z)’, which is the required expression. If the point Q be the origin, then z,= 0, y, = 0, z,=0, and we have for the distance of P from the origin, OP =(+y' + 2), (15) Yo express the distance between two points in terms of their oblique co-ordinates. Make a construction similar to that in the last article, and let a, (3, y be the angles between Oy and Oz, Ox and Oz, Ox and Oy: then, using the same notation as before, and projecting the broken line PTSQ on PQ, we have r =(x,-— 2) cos A + (y, — y) Cos we + (2, — 2) Cos ». Again, projecting PQ and the broken line PTSQ on the axis of zx, the two projections are equal because PQ and PTSQ have the same extremities ; hence rcosA=2,-2x+(y, - y) cos y + (Z, — 2) cos BB, as y is the angle between Oz and Oy, and (3 that between Ox and Oz. Similarly for the other co-ordinate axes we have y cOS w=Y,-Y + (2, —2) cosat (x, - X) Cosy rcosvy =2,-2+(#,- x) cos +(y, — y) cosa. Multiplying these equations by z, — 2, y, — y, 2, — 2 respectively, and adding, we have, in consequence of the preceding equation, r= (a, ia ay +ly, pr a eg hat 2 (a, — x)(Y, * y) ny + 2(x, - £)(z,— 2) cos B + 2(y, - y) (2, - 2) cos a. It is obvious that this gives us the expression for the length of a diagonal of a parallelopiped in terms of the sides and the angles which they make with each other. 14 FUNDAMENTAL THEOREMS. (16) To find the relation between the cosines of the angles — which a straight line makes with three rectangular azes. Taking the origin O (fig. 8) in the line, let POz =a, POy= 3, — POz = y, and let z, y, z be the co-ordinates of any point P in the line; then if the distance OP be 7, we have, by Art. (14), Party +2; but since z, y, z are the projections of r on the co-ordinate axes, — we have z=rcosa, y=rcos z=rcosy; therefore, substituting for these quantities, we have 9 7” = 7’ (cos’a + cos} + cos*y), or cos’a + cos’3 + cos’ y = 1, a very important relation, to which we shall frequently refer. © The cosines of the angles which a straight line makes with the co-ordinate axes are quantities which we shall often have occa- — sion to use, and as they serve to determine the direction of the line, we shall call them the direction-cosines of the line; and — when we wish to speak of a straight line with reference to its direction-cosines, which we may call J, m, n, we shall name it the © line [/, m, 7]. (17) The preceding theorem enables us to prove a very important property of the orthogonal projections of plane areas. For since any two planes make with each other the same angle as two lines respectively perpendicular to them, if we have a plane area perpendicular to the line of which the direction- cosines are Cos a, cos (3, and cos ¥, its inclinations to the co-ordi- nate planes of yz, zz, zy are a, 3, y respectively. Therefore if the magnitude of the area be denoted by A, and those of its projections on yz, zx, and zy by A,, A, A,, we shall have A,=Acosa, A,=AcosP, A,= A cosy. Squaring and adding these, and observing that by the preceding theorem cos’a + cos’3 + cos*y = 1, we have A= AP + At+ A}; or the square of any plane area is equal to the sum of the squares of its projections on three planes at right angles to each other. > Seared —_ se EE ee , — FUNDAMENTAL THEOREMS. 15 (18) Zo express the cosine of the angle between two straight lines in terms of the direction-cosines of the lines. If the lines do not meet, the angle between them is found by drawing through any point in the one a line parallel to the other. Take this point as the origin O (fig. 9); let a, 8, y be the angles which OP, and a,, B,, y, those which OQ makes with the rectangular axes. Take in OP any point P of which the co-ordinates are z, y, z, and in OQ any point Q of which the imco-ordinates are z,, y,, 2,, and jon PQ; let OP =r, OQ=7r,, PQ = 6, and POQ (the angle between the lines) = 9. Then, in | the triangle POQ, we have o=r +77 - 2rr, cos 0; but if we express 6 in terms of the co-ordinates of its extremities we have, by Art. (14), o =(%, iol +, Sid ce me) =2, 4+y (+27 +et+y+ 2-2 (xu, + yy, + 22) =P 47 — 2 (Lx, + YY, + 2%). Equating these two values of 0’, we have rr, cos 8 = xx, + yy, + 22,3 Duta w= cos a,'€-yis ricos 3} 0'z =r'cos:s a Sr cosa, ¥, = 7, cos P,, \2.= 7, 60s yy; therefore cos @ = cos a cos a, + cos (3 cos (3, + cos y Cos ¥,. (19) This theorem proves the following proposition: the projection of any finite straight Ime on Bhs may be found by first projecting the line on three rectangular axes, and then taking the sum of these projections projected on the second line. For if 7 be the length of the finite line, and a, (3, y the angles which it makes with the axes, its projections on them are | rcosa, rcos, 7 cosy. Then if a,, 3,, y, be the angles which the second line makes with the axes, the projections of the preceding quantities on the second line are rcosacosa,, 7 cos cos ,, cos y cos y,, and their sum is r (cos a Cos a, + Cos [3 cos 3, + Cos y Cos y,) = 7 cos 8, § being the angle between the lines. 16 FUNDAMENTAL THEOREMS. The same proposition is applicable to a plane area; so that to find the projection of any plane area on a plane it is sufficient to project it on the three co-ordinate planes, and then to take the sum of these projections projected on the second plane. (20) To express the area of a triangle in terms of the co- ordinates of its angular points. This may be most conveniently done by first finding the projection of the area of a triangle, one of whose angular points is at the origin. Let AOB (fig. 10) be such a triangle, and MOM, its projection; then if OM=r, OM,=1r,, MOz = 0, WOT 0% area MOM, = 3 rr, sin (6.- 0,). Let the co-ordinates of M be 2, y; and of M,, z,, y,; then area MOM, = 3 (wy ~ zy,), which is the expression for the projection of 4OB on the plane of xy. Now let ABC be the triangle of which the area is to be determined *' 2, 7} i220 i, Yin 2 yas 50s, ste: CO-ordinatesaon A, B, C; join its angular points with the origin, so as to form the three triangles AOB, BOC, COA. Then, by what has preceded, projection on zy of AOB =$ (xy — zy,), THE SOLE of BOC =3 (wy, - xy,); BTR 12). G0TL CO Ate (ee ore may But, by Art. (138. 3), since the triangle ABO, and the three triangles AOB, BOC, COA, are terminated by the same lines AB, BU, AC, the projection MMM, of ABC on zy is equal to the sum of the projections of the other triangles on that plane. Hence, calling that projection .4_, we have A, 2 (ay FG oY Ys 1 Ua LY) since it is clear from the figure that the projections of BOC and COA must be subtracted ; and in like manner, if A, A, be the projections of ABC on the planes of xz and yz, we es A, =) (e2,- 42+ 4,2 — 42,4 2,2, — £2), 1 ~ ae (zy, Wie otk ae ho ae Ye + Y.%, — Y 2p) FUNDAMENTAL THEOREMS. 17 But since the co-ordinates are rectangular, we have (Art. 17), if A be the area of the triangle ABO, A’= A? +A? + A}, and thus the area is expressed in terms of the co-ordinates of the angular points of the triangle. From the nature of projections it appears that the cosines of the angles which the plane of 4 BC makes with the co-ordinate planes of zy, zz, and yz, are Ane Ane A. Te ae respectively. (21) To express the volume of a tetrahedron in terms of the co-ordinates of its angular points. Take for simplicity one of the angular points as origin, and let OABC (fig. 10) be the tetrahedron; then if OH be drawn perpendicular to the plane of ABC, and be put equal to h, the volume of the tetrahedron V=4Ah, A being, as in the last article, the area of ABC. Now since OH is perpendicular to the plane of ABC, it makes with the co-ordinate axes of z, y, z the same angles that the plane of ABC makes with the planes of yz, xz, and zy ; hence, by the last article, if a, (3, y be these angles, ae eee sae: COs ie cosa = —*, = : a i A But if we project the broken line ONMA on OF, the sum of the projections is equal to OH, since OH, being perpendicular to the plane of ABC, is perpendicular to every straight line in it, and therefore to 4H. Hence we have h = a 2 fea. A, therefore substituting for A, mi A, “ae, values previously found, and cancelling the terms Ahith destroy each other, we find p= (xz,y, — £LY,2, + L,Y2, — LLY2, + @,2Y, — L,2Y,)- If we wish to introduce the co-ordinates 2,, y,, Z, of the fourth point, we have merely to substitute x-2,, Y- Ys. 27 %> w- 2, Y,- Y3 0 % — 2%, ©, —- Xe5 Y2- Ys % — %s for the simple co-ordinates. 18 FUNDAMENTAL THEOREMS. (22) As we shall have frequent occasions to use several analytical theorems which are perhaps not generally known, it will be of advantage to premise them here. I. If is Sas hg Bea each of these ratios is equal to 1 2 2 2 2 (a + a," + a, + &c.) 1? 2 2 2 2 (6° + 6° + 6 + &c.) na + n,a, + na, + &e. and to ESSE EE en a he De : nb + nb, + ,b, + &e. n, n,, n,, &c. being any quantities whatever. For assuming each of the ratios equal to r, we have (ME RD Us rh eae AR Cees Squaring and adding, e+a/ +a, + &.= 7 (0 + b+ 67 + &e.); whence, extracting the root and dividing, 1 (+a +a, + &e) a A, i (0° + b? +b, + &e.)y Again, wa=77b, na,=7n,b., na, = 77.0, , xc. By addition, (na + n.d, + n,a, + &c.) = 7 (nb + 7,6, + nb, + &e.) Na+ na, + n,a, + &e. aa W hence Pe nae AES wh tel Sd LE GA NS nb +n,b, + n,b, + &e. b b II. If we wish to determine the variables from three simulta-_ neous equations of the form ALA OY + C2 = Aig. in (A) ax+by+ez=d,......(2), act by + 625d... 3), instead of eliminating first z and then y, in order to determine x, we may eliminate both at one operation by the following rule: Multiply (1) by 4,¢, — ¢,b, ; (2) by eb, — be, ; (8) by bc, — b,c, and add: it will be found that the coefficients of y and z are identi- FUNDAMENTAL THEOREMS. 19 cally equal to zero, and we have r= d(b,c, a c,6,) i d, (cb, x bc,) a d, (bc, e b,c) 2 a(b,c, — ¢,b,) + a, (cb, — be,) + a, (be, — b,c)” with similar expressions for the other variables. If d = 0, d, = 0, d, = 0, the equations contain only two independent variables, since we may divide all by any one of them ; and the condition that the equations should coexist is a(b,c, — ¢,b,) + a, (cb, — be,) + a, (bc, — b,c) = 0. We shall frequently refer to this process under the name of eross-multiplication ; and the student is recommended to make himself familiar with the forms of the multipliers, as a ready use of the process will be of great service to him. III. The sum of three squares of the form (ay — bry + (cx — azy + (bz - cyy may be put in a shape which is very convenient, especially in geometrical investigations. For if we add and subtract from the preceding expression the three squares Cu. b*y’, C2, the expression may be transformed into (+0 +0) (2+ y' +2) - (ax + by + cay, 5 ; 7 ( by + cz) y or (+840) (ty +2) 41 Baer MORAY chs GN weal | ( Bt ¥ vii Ce ENC re eal) _ Now, if a, 4,¢ be taken as proportional to the direction- cosines of some one line, and z, y, z of another, the expression ax + by + cz (P+ B+ (ery +2) js equal to the cosine of the angle between the lines: let this be 6; then the sum of the squares is equal to (2? +04 c)(a’ + y’ + 2’) (sin OY. IV. If we obtain as the result of any process that a function of x is equal to a function of y in which y is involved in a man- ner similar to that in which z is involved in the other, then, as there is nothing to distinguish one co-ordinate from another Cc 2 20 FUNDAMENTAL THEOREMS. when they are symmetrically involved, we may say that each of these functions is equal to a similar function of z, and this is the consequence of the general symmetry of our expressions. Thus, if we have two equations lx + my +nz= 0, la + my +n'z=0, and eliminate z between them, we find (In' — Un) x + (mn' - m'n) y = 0, S Y or vant — ins Ta here the two sides of the equation are symmetrical, one with respect to 2 and the other to y. We may therefore say that each is equal to itm , this being the corresponding sym-— metrical expression with respect to z. CHAPTER II. OF THE STRAIGHT LINE AND PLANE. (23) There are two methods which we may use in applying analysis to Geometry: either we may assume equations of dif- ferent forms, and then determine their geometrical meaning ; or we may define lines and surfaces by their geometrical properties, and from the definitions determine their equations. We might pursue either of these methods exclusively, and so build up a system on an uniform plan ; but we shall find it more convenient to use sometimes one and sometimes the other method. In treating of the straight line and plane we shall use the second system, because their geometrical definitions are so well known, and their chief geometrical properties are so familiar to us, that it seems more natural to translate these into analytical language than to adopt the inverse process. The surfaces of the second order will be treated by the other method. In the following investigations the co-ordinates are considered as rectangular, except when the contrary is expressly stated. The Straight Line. (24) To find the equations to a straight line. Take A a fixed point in the indefinite straight line AP (fig. 11), and let a, B, y be its co-ordinates. Let xz, y, z be the co-ordinates of any other point P in the line, and let /, m, m be the cosines of the angles which the line makes with the three axes, or the drection-cosines of the line. Then if 7 be the length of the portion AP of the line, z- a, y — 3, 2—y are the projections of 7 on the axes of £,y, % respectively. But by the nature of projections f—a-ir;,) y—p=mr, 2— +4 = nr. pd THE STRAIGHT LINE. Hence we have Bole abe Vedio (1). soeoeosvoeoeveteeev eee These three equations are equivalent to two only, as any one may be derived from the other two; and as they express two relations between the co-ordinates of P which is any point in the straight line, they are the equations to the line. As these formule express the equality of three ratios, it is often convenient to denote each of them by the single quantity ¥, to which they are each equal. (25) It is evident that we may write the equations to the straight line under the form if L=kl, M=km, N=kn, k being an arbitrary multiplier ; that is, if Z, M, N be proportional to 7, m,n. Consequently if we have given equations of the form (2), they represent a straight line, and the quantities LZ, M, N are proportional to the direc- tion-cosines of the line. The quantities a, 9, y are always the co-ordinates of a point through which the line passes. (26) There is also another form in which the equations to the straight line may be written: for if we combine the first and third members of (2) and also the second and third, we have L M smplcmanraGams oP ee oe ie which may be put in the form t= ae + py -Yy= be Ges ee if The form (3) is that which has been usually employed by writers on this subject, but it is not so convenient as (1) and (2), because the expressions are not symmetrical with respect to the three variables z, y, z. We can easily interpret the meanings of the constants in equations (3); for, considering the left-hand equation, we see THE STRAIGHT LINE. 23 that it is the equation to a straight line in the plane of zz, which is evidently the projection of the given line on that plane. Now ais the tangent of the angle at which this projection cuts the axis of z, and p is the portion of the axis of z intercepted between the origin and the projection. In like manner 6 is the tangent of the angle which the projection of the given line on the plane of yz makes with the axis of z, and g is the portion of the axis of y which is intercepted between the origin and the projection. (27) The position of the line will vary according to the yalues of the constants in its equations, and some of the more important cases we shall here consider. In equations (2), if a=0, 8 =0, y=0, the line passes through the origin, and its equations are then y eo Gate ee hig CIO Oe ro Ce 4). Te VAGON) (4) If any one of the denominators vanish, as if Z = 0, then, in ° ° ° H bs ~e order that the three equations may exist, the ratio = must not become infinite ; and this can only be avoided by making it , ; 0 ; : indeterminate, or of the form re that is, by putting z-a=0. Now Z = 0 implies that 7 = 0, or that the direction-cosine with respect to x is 0; that is, that the line is at right angles to the axis of x, and therefore parallel to the plane of yz. Hence, equations of the form Ee Oa iene ann ary ches rare erat (5), represent a straight line parallel to the plane of yz; and simi- larly for the other co-ordinate planes. If two of the denominators vanish, as L=0, M= 0, it follows that Were a5 a {5 =O. fate.) 28 Ntrts wz. (0)3 which in this case are the equations to the line. Since Z = 0, M = 0 imply that the line is perpendicular both to the axis of x and that of y, it must be parallel to the axis of z. A line, therefore, which is parallel to one axis is represented by making 24 THE STRAIGHT LINE. the co-ordinates with respect to the other axes each equal to a constant. If we consider the form (3) of the equations to the straight line, p = 0, g = 0 imply that the line passes through the origin, in which case its equations are Cie, SY 02. If @ = 0, the equations are C= p, Y= 02+ dq, which being of the same form as (5) represent a straight line parallel to the plane of yz. If @=a, which corresponds to N= 0 in the form (2), the equations (8) fail, and we must combine equations (2) in a different way. Let us then combine the first and second and the second and third, when we have y=B+>(@-a) z=7+ 2 @-a); in these, if V = 0, they become YEP ts heey» : M M if a,- = and p,= PB - +a. These equations, by what has just been said, represent a straight — line parallel to the plane of zy. Hence, if in equations (3) either of the coefficients of z becomes infinite, the equations represent a line parallel to the plane of zy. We now proceed to apply these equations to the solutions of — problems relating to straight lines. (28) To find the angles which a given straight line makes with the co-ordinate axes. Let the equations to the line be then if A, w, v be the angles which the line makes with the axes, we have by Art. (25) L=kecosA, M=kcosp, N=k cos »v. Squaring and adding these equations, we have, since cos’ \ + cos’ w+ cos’ v = 1, L? + M+ N=, THE STRAIGHT LINE. 29 and consequently celts Se cos = poe te ae cos ee Bey aS VE M4N?)? VMN)? V0 MN) It is to be remarked that each of these expressions admits of two values in consequence of the double sign of the radical in the denominators, but as the same sign must be taken in each case, there are only two sets of values for the cosines, corresponding to the supplementary values of the angles A, u, v, made with the positive axes of x, y, z by the two portions of the line measured in opposite directions from any point in it. It is necessary to make some convention respecting the mode in which the angles A, 4, v are to be measured, and that which is always used is that they are the angles made with the axes by that portion of the line which makes an acute angle with the positive axis of 2. This implies that cos v is positive, and therefore that the radical has the same sign as NV. cos A= (29) To find the equation to a straight line which passes through two given ports. Let the co-ordinates of the points be z,, y,, 2,, %5Y5%> and the assumed equations to the line be TASES OPS EY L M N in which a, 3, y, L, M, N are to be determined in terms of @>Y.> 29 Lys Yo» %- In order that this line may pass through one of the given points, as the first, it is sufficient to make a=z,, B=Y,, y=%» a8 a, f, y are the co-ordinates of some point through which the line passes. The equations then become wL— xX, en ee ie L M IN But if the line is to pass through the point (2,, y,, 2,) these quantities must satisfy the preceding equations ; therefore L,— @, CEU Lad Lied L M N Eliminating LZ, M, N by dividing the first set of equations by 26 THE STRAIGHT LINE. the second, member by member, we have peer ee oe 1 ET ee ae ee ee L-X%, Y,-Y, &_— % as the required equations. (30) To find the condition that two straight lines may inter- sect, and the position of the point of intersection. Since two straight lines in space are not necessarily in the same plane, and since two lines which intersect must be in the same plane, some relation must exist between the constants in the equations to the lines, in order that they may intersect, and the condition must be also that which holds in order that the two lines may be in one plane. Let the equations to the lines be Tatty Rime ate eae Rad i = m n! l,m, n, l,m’, n' beimg the direction-cosines of the lines. These may be written in the forms e@=a+dr, Vy [oe mr, ee = renee Gay xz = a’ + ee y eS B + m'r', a = y + n'ir eon ee (25: If the lines meet, the co-ordinates x, y, 2 of the point of inter- section must satisfy the equations to both lines. Hence, z, y, z are the same in (1’) and (2°); therefore subtracting each equation of (2') from the corresponding equation of (1'), we have a-a+lr—Tr'=0, Dern 77 = 0,75 ae eee y-y +nr—n'r' = 0. These three equations contain only two variables, 7 and 7’, and therefore, in order that they may co-exist, there must be some relation between the constants, which is the condition we are seeking. On eliminating r and 7’ by cross-multiplication (Art. 22), the condition is found to be (min - mi’) (a— a’) + (' — En) (B - B') + Em - Im’) (y - y')= 0...(4). THE STRAIGHT LINE. QT To determine the position of the point of intersection, elimi- nate 7 between each pair of equations (3), which gives (lm —Im')r' = m(a-a)-1(3 - B), (n' —Un)r' =l(y-y')-n(a-@), (m'n - mn')1’ = n(B - B’) - m(y- y). Squaring and adding these, we have yn (0-6 +(B- BY +(y—7V- {Ua-a!) + m(B-B)+2G—V)F (l'm — ln’ + (In' — ny + (m'n — mn so that 7’, the distance of the point of intersection from the point (a', 3’, y'), is determined. If : = a = = , the equation of condition (4) is satisfied inde- pendently of a, 3, y, a’, (3', y’, but then the value of r’ becomes infinite, showing that the lines intersect at an infinite distance or are parallel, as will be seen in Art. (82). If the equations to the straight lines be given under the forms z=az+p, y=bz+4q, e=aztp,, y=bz+q', the condition that they may intersect is a-ad b-Bb' 5, 4 es 6 ustele.e 01.6 6 xfs sine isp 5a (5), ar Mae De 7. (31) To find the angle between two straight lines the equations to which are ihe aac Oh iy ce ee imi By (18), if 0 be the angle between the lines, and A, u,v, A’, uw ”” be the angles which they make with the axes, cos # = cos X cos X’ + cos ~ Cos wu’ + COS ¥ COS 4’. Substituting for the cosines their values given by (28), we have LL + MM' + NN’ cos 9 = a: GE? eM? + Ny! (L? + M” + N*y 28 THE PLANE. from which we also obtain {(L'M - LM’Y +(LN'- L'NY +(M'N- MN'Y\}* (Li + Mi + NL? + M? + NY The expression for the cosine admits of two values, positive and negative, corresponding to the acute and the obtuse angles which the two lines make with each other. The value of the sine must always be taken as positive, since the angle between the lines can never exceed two right angles. sin f = (32) To find the conditions that two straight lines may be parallel or perpendicular to each other. If the two lines be parallel, their direction-cosines must be equal; and as L, M, N are proportional to the direction-cosines of the one line, and L’, M', NV’ to those of the other line, these quantities must be proportional ; or are the conditions of parallelism. If the lines be perpendicular to each other, the cosine of the angle between them must be equal to 0, which, by the last article, gives LL + MM' + NN’ =0 as the condition of perpendicularity. If instead of L, M, N, &c. we use the direction-cosines l,m, n',U, m',n, we may put the condition in either of the forms i’ +mm'+nn' = 0, or (Um — bm’) + (in' - Unf + (m'n - mn'f = 1, the latter being derived from the expression ‘for sin @ in Art. (31). The Plane. (33) Zo find the equation to a plane. For the purpose of investigating the equation to a plane, it may be defined as the surface traced out by a straight line which moves in such a manner as always to pass through one given straight line and to remain parallel to another. The moveable straight line is called the generator, and the fixed straight line through which it THE PLANE. 29 always passes is called the director.* Let the equations to the director be LEN! GUAGE seman Ee ee ceyy Since the line (G) remains always parallel to a fixed straight line, while it passes through (D), its direction-cosines /', m', n’ remain constant, while the values of z’, y', 2’ vary in such a way as to satisfy the equations (D): this we may denote by putting x,y, 2 for x, y, z in those equations. We then get e—-a=l%, y -—-B=mr, 2-—y=n7, | ~» ~ ~_ 2. Baie. RU Ye Ub Tey eee em Adding these equations, we have Pa UI LIP Mee: fale Saree cree CL), Gots Stee TP CENT wowe 2 (2), Za ILE? ee e's ones (3). In these equations 7 depends on the particular point in (D) through which (G) passes, and 7’ on the point in (G) which is under consideration ; but we wish to find a relation between 2, ¥, 2, Which shall be true for all points of (G) in every posi- tion. Such a relation it is plain we shall obtain by eliminating rand 7 between equations (1), (2) and (3), since the result being independent of 7, 7’ (the only quantities which particularize the position) must be true for all positions. The elimination is readily effected by cross-multiplication, which gives us (m'n — mn') (x — a) + (In' — Tn) (y - 3) + Um — lim’) (2 - y) = 0..-(4). This is a relation subsisting between the co-ordinates of every point of (G) in every position ; in other words it is the equation to the plane which has been defined as the locus of (G). * It has been usual, following the French fashion, to give these words a feminine termination, and to call them “ generatrix” and “directrix;” but as it is not the custom of the English language to acknowledge the distinctions of gender in inani- mate objects, I have taken the liberty of so far deviating from ordinary practice as to use that form which admits of a plural termination in our language; such words as Generatrixes and Directrixes being scarcely admissible. 30 THE PLANE. (34) We may also adopt a more convenient method for the elimination of 7 and r’ founded on geometrical considerations. Let cos A, cos ~, cos v, be the direction-cosines of a line perpen- dicular to the plane containing, and consequently perpendicular to, both these lines, so that by (32) they satisfy the conditions ZcosX +m cos w+ cos v= 0, I’ cosA + m' cos + cos v= 0. Then if we multiply (1) by cos A, (2) by cos u, (8) by cos », and add, the second side of the equation disappears in conse- quence of the preceding conditions, and there remains (x — a) cosA +(y — 3) cos w+ (e-y)cosv=0.... (5), as the equation to the plane. It may also be written in the form xcosA+ YyCcosu~t+zcosy=acosA + cos w+ ¥ COS ». Now cos X, cos pw, cos v, are the direction-cosines of a line perpendicular to the plane, and a, (3, y are the co-ordinates of some fixed point in the plane. Hence the second side of the equation is the sum of the projections of these co-ordinates on a line perpendicular to the plane. But this sum is equal to the perpendicular from the origin on the plane, since one extremity of the broken line a+ (3+ y, being the origin, coincides with one extremity of the perpendicular, while the other extremity is projected on the perpendicular by a line lying in the plane. Hence, calling the perpendicular from the origin 6, we have a cosA + (3 cos w+ y cos v=6, and 2 cos.N'+'¥ cos + 2:Cosiv’= Oy). Jou. (6), which is one of the most convenient forms of the equation to the plane. As we shall have frequent occasion to speak of the line which is perpendicular to a plane, it will be convenient to have a dis- tinct name for it, and we shall call it the normal to the plane, while we shall also call the direction-cosines of the line, or cos X, cos 4, cos y, the direction-cosines of the plane, since they determine the position of the plane as much as that of the line. (35) It appears then that the equation to the plane is of the first degree in 2, y, 2, and conversely we may show that the THE PLANE. 31 general equation of the first degree can represent nothing but a plane. The geometrical idea of a plane is that it is such a surface that a straight line which passes through any two points in it lies wholly in the surface. Now let gy Ge aD) re sien sc ts (1) be the general equation of the first degree, which by Art. (7) must represent some surface or other ; and let z,, y,, 2, 25 Yos 2 be the co-ordinates of two points in the surface, and which there- fore satisfy the preceding equation. Then the equations to a straight line passing through these two points are, by Art. (29), r being the length of the lme between the point (a, y, z) and (z,,Y,,2,). Substituting in (2) for z, y, 2, their values from (2), we have {A (w,- 2,)+ B(y,-y,)+ C(z,-2)} 7+ Az, + By, + Cz, = D...(8). But since (Z,, ¥,, 2,) (Z,» Y,» 2,) Satisfy the equation (1), we have Az, + By, + Cz,= D= Az, + By, + Cs,; hence equation (3) is satisfied identically, and therefore, what- ever be the value of r and therefore of x, y, z, these quantities satisfy the equation (1). But (x, y, z) are the co-ordinates of any point in the line (2), consequently every point in the line hes in the surface represented by (1), which is therefore a plane. (36) If the equation to the plane be given under the general form Az + By + Cz =D, it is easy to determine the geometrical meaning of the constants by comparing it with xz cos A +Y COS w+ ZCOS v= 6. This comparison gives | Az=kcosX, B=kcosp, C=kcosy, D=ké, or pee i to wll oD \ COSA UECOS Luk) .COs}y 8 That is, the coefficients of the variables are proportional to the oo THE PLANE. direction-cosines of the normal, and the constant term is propor- | tional to the perpendicular from the origin on the plane. (37) Let us now discuss the equation to the plane for different values of the constants. If D = 0, the equation is Ax + By + Cz=0, which is satisfied by 2=0, y= 0, 2=0, or the plane passes through the origin. If A = 0, the equation is By + Cz = D. Now if 4 =0, cos A = 0, or the normal is perpendicular to the axis of z, and therefore the plane itself 1s perpendicular to the plane of yz. In like manner, if either of the other coefficients of the variables vanish, we shall have an equation to a plane perpendicular to the co-ordinate plane containing the variables remaining in the equation. From this it is easy to see that the second form of the equations to the straight line given in (26) is equivalent to assigning the equations to two planes perpendicular to two of the co-ordinate planes, and so determining the position of the line. If B = 0, C= 0, the equation is reduced to Az = D or x =a a constant. Since B = 0 and C= 0, cos uw = 0, cos vy = 0, or the normal is perpendicular to the axis of y and that of z, and therefore to the plane containing them: hence the plane is parallel to the plane of yz. In like manner for the other axes we see that y =) represents a plane parallel to zz, and z = ¢ one parallel to zy. From this it is easy to see that z=0,y=0,2=0, are the equations to the co-ordinate planes of yz, xz, xy, respec- tively. (38) If in the general equation of Art. (86) we make one of the variables vanish as z, we obtain the equation to the intersec- tion of the plane by the plane of zy. ‘This equation is Az+ By = D, showing that the intersection is a straight line. os ws THE PLANE. If we make y = 0, z = 0, we have Az = D, as the equation for determining the point at which the plane cuts the axis of z. Let p be the distance of this point from the origin, then D Piz aae In like manner if g, 7 be the corresponding quantities for the other axes, D D Tiigaas ME Hence the equation to the plane may be put under the form ze ey, TEE ORE which is often very convenient in practice. The quantities p, g, r are called the ntercepts on the axes. (39) To find the angles which a given plane makes with the co-ordinate planes. These angles are the same as those which the normal makes with the axes. If then the equation to the plane be Az + By + Cz = D, the equations to the normal are ppp since A, B, C are proportional to the direction-cosines of the normal. If A, u,v be the angles which this line makes with the axes of 7, y, z, we have VA B 0%)’ OF BS ABO)’ tT (CAE BALO?Y? which expressions, therefore, determine the angles which the plane makes with the co-ordinate planes. (40) To find the angle between the straight line and plane, the equations to which are cosA = Az + By + Cz = D, 6) 34 THE PLANE. By the angle between a line and a plane is meant the least | angle which the line makes with any line in the plane; that is, | the angle between the given line and its orthogonal projection on the plane. Hence, the given line, its projection, and the normal to the plane, lie all in one plane, and the angle between » between the straight line and the normal. If this angle be om we have, by Art. (31), “ cos ¢ = Al+ Bm + Cn, so that if @ be the required angle, sin 9 = Al+ Bm + Cn. | (41) To find the angle between two planes, of which the equations are Ax + By + Cz =D, | Ae’+ By + Cz= 2D. The angle between two planes is the same as the angle between two lines drawn perpendicular to them; that is, it is equal to the angle between their normals. But the direction-cosines of the normals being proportional to A, B, C, A’, B’, C’, we have, if” 6 be the angle between them, | AA'+ BB'+CC cos § = ———__________________ , (A? + B+ CY (A? + B? + OY which expression therefore determines the angle between the two planes. j (42) To find the conditions that two planes may be parallel or perpendicular. If the planes be parallel, their normals must also be parallel, and therefore, using the equations of the last article, we have, by Art. (32), AyniaBt VG —= «= so —_- SO as the conditions of parallelism. The condition of perpendicularity is at once obtained from the value of cos 0, for if the planes be perpendicular to each other, 9 is a right angle, cos @ = 0, and therefore | AA's BB «CC =0. THE PLANE. 35 (43) Zo find the equation to a plane which passes through three given points. Since the equation to the plane contains only three indepen- dent constants, the three conditions of making the plane pass through the three given points are sufficient for determining the constants in the equation; or, in geometrical language, the position of a plane is determined by making it pass through three given points. Let the co-ordinates of the three points be (z,, y,, 2), (X,5 Yo» 2) (Lz 5 Yz» 2), and assume the equation of the plane to be Ax+ By+ Cz=D; it is required to find A, B, C, D, or rather the ratios of any three of them to the fourth in terms of the nine co-ordinates. Now if the plane passes through the point (z,, y,, 2,) those quantities must satisfy the equation to the plane, so that we have the condition Ax, + By, + Cz,=D; similarly Az, + By, + Cz,=D, and Az, + By, + Cz, =D, from which = 2 and 4 are to be determined. ~~ B C : Eliminate D and D between these equations by cross-multi- 2 plication, so as to find cep similarly for ee and : the common D D D denominator of the fractions is D= LYK 3 — UYs%_ + LLY%, — VoYj2%, + L3Y,2%, — LY,%, » and the numerators are A = Y.2, — Yshy + Ys%y — Yo%1 + YF — Y2% > B= 23,57 €2, + 2,3, = 2,2, + £2, — U355 C= LY. — LY, + LLY, — LY. + LY, — LYs- The results of Chap. 1. Arts. (20) and (21) enable us to assign geometrical meanings to these expressions; for if V be the volume of the tetrahedron, of which the origin is the vertex, and the three given points are the other angular points, then D=-6V. D2 36 PROBLEMS RELATING TO Also if A,, A,, A, be the projections on the co-ordinate planes of the cries ‘of which the three given points are the | angular points, : A=-2A,, B=-2A,, C=-24,; hence the equation to the plane may be put under the form Az+Ay+Az=3V. Problems relating to the Straight Line and Plane. (44) To find the equations to the line of intersection of two planes. Let the equations to the planes be le+my+nz=6 le+my+nz=6. When these planes intersect, the co-ordinates z, y, z of the points — of intersection are the same for both, and therefore the two equations may be taken as simultaneous, and combined accord- ingly. Eliminating then, first y and then z, we have (U'm — Im') x = (m'n — mn’) s + m& — m'S, (Um — Im') y = (' —Tn) z+ U8 - 08. These equations being of the form (3) of Art. (26), shew that the line of intersection is a straight line. If we wish to put the — equations under the symmetrical form, let mn-mn'=XA, n-lTn=u, Um-Im' =». Then eliminating z between the last two equations, we find yet, , 28 - yh = fy + (m9 - md)) | 1 Vv " ; | ~ {e+ @d—moh, : by the symmetry of the formule. If both planes pass through the origin, 6= 0, & = 0, and the equations to their line of intersection become simply z a) Py it PIMC mn —mn In-In Im -Im’ OE THE STRAIGHT LINE AND PLANE. Ou (45) To find the conditions that a straight line may be perpen- dicular to a plane. Let the equation to the plane be Az + By + Cz= D, and the equations to the lne Ree BN Since the line is perpendicular to the plane it must be parallel to the normal of the plane; but the direction-cosines of the normal are proportional to A, B, C, and therefore, by Art. (32), lusty (6. TESTED are the conditions of perpendicularity. Hence the equations to a line perpendicular to the plane and passing through (a, J, y) are z-a y-B 2-7 A B C (46) To find the condition that a straight line may be parallel to a plane. Take the same equations as in the last article ; then, since the line is parallel to the plane, it must be perpendicular to the normal to the plane; wherefore, by Art. (32), AL +BM+CN=0 is the condition of parallelism. (47) To find the conditions that a given straight line may he in a gwen plane. Since every point in the line lies in the plane, the co-ordinates a, (3, y must satisfy the equation to the plane, so that one con- dition is Aa+ BB+ Cy =D. But a line which lies in a plane must be perpendicular to the normal; this gives, as the other condition, AL + BM+ CN= 0. (48) To find the length of the perpendicular drawn from a given point on a plane, the equation to which rs given. Let 2’, y', 2’ be the co-ordinates of the point, and vetd Sane SS Oa Baran ® meee Bie gs ot GL), 38 PROBLEMS RELATING TO the equation to the plane. The equations to a line perpendicu-: lar to the plane and passing through (2’, y’, 2’) are : ae Mim Mid Nd oS (2). A B C Each of these ratios is, by the Theorem I. of ile I., equal to {(@-2Y+ly- yf ene} (A? + B+ oy and also to A(x-2v)+ By - = WtGGe 2), Aa es Now if (x,y, 2) be the point where (2) meets (1), the nume- rator of the former of these is the perpendicular distance of the point from the plane. Let this be 8; then, as (z, y, 2) is a point in the plane, we have, by (1), Az + By+ Cz=D, and consequently ee D -(Az' + By’ + C2!) CE Bio If the numerator be negative we must then take the denomina- tor with the negative sign, since the value of 8 is absolute, and must therefore be considered as positive. (49) To Jind the length of the perpendicular from a given point on a given straight line. Let the co-ordinates of the given point be 2’ ays 2, and the equations to the given line Assume the Sesto to the line perpendicular to it and passing | through (2, y’, 2’) to yey Then if these lines intersect in the point (z,y, 2), r’ is the length of the perpendicular required. Now, eliminating z, y, z between the ines pairs of equations, we have a ee ee ee a a oe v-a=r-Tr, y-B=mr-mr, 2-y=nr-n'r'. THE STRAIGHT LINE AND PLANE. 39 Squaring and adding, and observing that, since the lines are perpendicular to each other, iW + mm' + nn' = 0, we have (7’-ay+(y - BY +(e -yfarsr”. Again, multiply by 7, m, m, and add, then l(z’-a)+m(iy'-B)+n@-y)=7; consequently we have re = (a! — ay +(y'- BY +(2'- yy - {He — a) + my’ - B)+n@— YF which determines 7’, the length of the perpendicular. (50) To find the perpendicular distance between two straight lines not mm the same plane. Let the equations to the lines be Pe ey Dae Hy 1 l m n (1), ery ies 1S ea Oeey, 5 Sy Fiat i Ce sins osine 4s (2) Through (1) draw a plane parallel to (2), and another through (2) parallel to (1). These planes will then be parallel to each other and at right angles to the perpendicular distance between the two lines. Their equations, therefore, will be of the forms (cx-a)A+(Y-BP)ut+E-y)v=0........ (3), (cx-a)A+YY-P)ut+e-y)v=O0........ (4), the constants A, ~, v being determined by the equations of con- ditio 1tlon, EAVASIT EGR TOV ma Oc Bare: oie « ofthe ated w/e (5), LUTTE FOREN RE RBG Bh Le POH Gp) 4X Fy aise Oe ea Wad eae ame er th cenees et F the last equation implying that we assume the constants to be the direction-cosines of the plane. To determine their actual values we proceed as follows: eliminate v between (5) and (6), when we obtain (In' —Un) X + (mn' —- m'n) p= 0; r lL y oo mn'— min in—In im’ —tm by the symmetry of the formule. 40 PROBLEMS RELATING TO Assume each of these expressions equal to w, and substitute the values of A, u, v in (7), we then obtain w {(mn' — miny + (ln - tn'f + Um - Umy} = 1. | Now, by Art. (22), the multiplier of w’ is equal to the square of the sine of the angle contained by the lines whose direction- cosines are J, m,n, 1, m', n', that is, by the given lines. Let this angle be 9, then we have wsin#=+1, and therefore mn' ~ m'n ln — In lm' — lm : aha clos pr nae er sintg. sin 0 ’ A=+t These constants being thus determined, it is easy to find the perpendicular distance required, for it is evidently equal to the © difference of the perpendiculars from the origin on the planes (3) and (4). That difference is ; (a~a)A+GrPyuty~ y)», so that if 6 be the required distance S- (mn'-mn)(a~ a) + Un tn’) (B ~ B')+(m' -Im)(y ~ 7) i sin 8 F (51) To find the shortest distance between two straight lines. Let the equations to the lines be l m n' If 6 be the distance between any two points in the lines O=(~vysty-yl+@-zy, which is to bea minimum. Now 2, y, z are functions of 7, and az’, y', 2 of 7’, and r and 7 are independent; therefore, making the differential of 6 equal to 0, we have two equations (x —- x) dx + (y-y') dy +(z-2) dz =0, (2 - 2) dz'+(y -y') dy' + (¢-2) dz’ =0. But from the equations to the lines SS eee eer. dv eo hence, eliminating the differentials by dividing each term of the THE STRAIGHT LINE AND PLANE. 41 former equations by the corresponding member of the latter, we have L(x-xv)+m(y-y)in(z-2)=0, — Ua w')tm y-y)tn (e-2)=0. Now if A, nu, v be the direction-cosines of the line 6, Z-Z2=0, y—-y =o, 2-2 =y0, and the preceding equations become IX + mu + nv = 0, UX+mut+n'v= 0. These conditions show that the least distance is perpendicular to both the given lines, and hence its length is given by the solution of the problem in the last article. (52) To find the equations to the straight line which cuts at right angles two given straight lines. Let the equations to the given lines be EO Tce a aE (1), l m n ES ye IE Sai A ray oF 1 Lay m n and assume the required equations to be Z-2Z ¥-Y 2-2 r lu Vv The values of X, uw, v are evidently the same as those found in Art. (50), and we have only to determine 72’, y’, 2'; now these being the co-ordinates of an arbitrary point in the line, we may assume that point to be the intersection of (1) and (3), so that these quantities must satisfy the equation (1), or — eS suppose ...... (4). The condition that (2) and (3) should intersect, is (2! — a!) (n'y - m'v) +(y! - B') (Uv = nr) + (2 =) (mtn—Tn) = 05 and, as 2’ —a' = z'—-a— (a — a), and similarly for the other quan- tities, this equation may be written (v' — a) (nu — mv) + (y' - B)(l'v — md) + (Z -— y) (MA -Up) = (al — a) (nip — mv) + (B'-B) Uv — wr) + (y/ = 7) (m'r — Ep) »» (5). 42 OBLIQUE CO-ORDINATES. Substitute in (5) for (z’ — a), (y' — B), (z' — y), their values derived from (4), so as to obtain an equation in 7, which gives _(@’— a) (op mb») + (Bi —B) Ov = Wd) (=i = Up) L(n'w—m'v) +m (ly — wr) + n(m'd — Ep) The value of r being thus found, those of 2’, y’, 2’ are known from equations (4), and thus the line is completely determined. If we substitute for \, u, v their values, and reduce by means of the conditions P+m+n?=1, UW +mm' +nmn' = cos 8, where @ is the angle between the lines (1) and (2), we obtain (a'—a)(Z—T cos 0) + (3' - B)(m — m' cos 8) + (y'- y)(m —7' cos 8B) sin? 0 r Oblique Co-ordinates. (53) The equations to a straight line when referred to oblique co-ordinates are in the same form as when the co-ordinates are rectangular; viz. + _ ¢ y—B z2-y¥ where however /, m, n no longer signify the direction-cosines of © the lines, but the ratios of the projections on the axes of any portion of the line to that portion, the projections being made by planes parallel to the co-ordinate planes. For if r be any portion of the line between the points (z, y, z) (a, 3, y) its projections are lr, mr, nr; but these projections are also x-a, y-B, 2-7; whence, equating these values, we obtain The equations of a straight line passing through two given — points (z,, ¥,, %,)(%»Y, 2) are the same in form as for rectangular — co-ordinates ; that is, they are ee 21F | AN eR 3 L-k% Y,-Y, @ — ® (54) The quantities /, m,n are not independent, but are con- nected by an equation of condition which may be found as OBLIQUE CO-ORDINATES. 43 follows. The distance of the point (x, y, z) from (a, (3, y) is, by Chap. 1. Art. (15), given by the formula 12-0} +(y-B)'s(2=y)'+2aly-B\(e-y)+2H(2-a)(2-7)+2e(e-a(y-P, where a, 6, c¢ are the cosines of the angles between the axes of yz, zz, xy. But from the equation to the line, we have Dia yf Pay This Fab Dh and hence eliminating # — a, y — 3, 2- y, we find 1=P +m’ +n? + 2amn + 2bln + 2clm, as the equation connecting /, m,n. When the co-ordinates are rectangular, a= 0, b=0, c= 0, and the condition is reduced to 1=P4m +n’, as it manifestly ought to be. (55) To find the angle between two lines, the equations to which are z-a y-B z2-y «e-a y-P 2z-y7 l m Oe ' m n Through the origin draw two lines parallel to the given lines, so that their equations are SCS a ——- ll — See Oe Now if 6 be the angle between the lines, 8 the distance be- tween the extremities of 7 and 7’, Oo =r -2rr cos 0+r°; but (z, y, z)(2’, y’, 2’) being the co-ordinates of the extremities of 6, we have also Ya=(e-eyriy-yYrie—zyt2aty-y') &-Z) + 2b(xe-x)(e-2)+22%c(e#-ze)y-y), ar +r? —2{ex' + yy +22 +a(ye +y'2) +b(a2' +2'z)+c(ay' + 2'y)}. Equating these two values of 6’, and eliminating z, y, z, v',y', z by means of the equations to the lines, and then dividing by 77’, we find cos 0 = Zl + mm! + mn' + a(mn’ + m'n)+ b(In' + Un) + ¢(hn' +m); by which the angle 0 is determined. 44 OBLIQUE CO-ORDINATES. If A, u, »v be the angles which the line zy _® oe ar en wre Le min makes with the axes, they may be determined by the formula just found. For let the second line coincide with the axis of z, then / = 1, m' =0, n' = 0, and 9 = X, so that cos A =/1+4+ bn + em. Sumilarly cos n= m+ an-+ el, cos v=n+am + Ol. If we multiply these equations by 7’, m’, n’ respectively, and add, we find cos 9=7' cos + m' cos w+ COS ¥ identical in form with the equation in rectangular co-ordinates, though the quantities involved have not the same meanings. The condition that the two lines may be perpendicular to each other is evidently l cos’ +m! cos +n cosy =0; or U' + mm + nn’ + a(mn' + m'n) + b(i' + Un) + e(lm + Um) =0; and the conditions that they may be parallel are L+bn+em m+an+ed n+am+bl Utbnitem mean+cd n+am'+0l° (56) The equation to the plane when referred to oblique co- ordinates must evidently be of the same form as for rectangular co-ordinates, viz. Az + By + Cz= D, since the equations to the straight lines, from which it is derived, are of the same form in both cases. If the equation be in the form z Bey de Y + S = i; p Yr p,q 7 are the intercepts of the axes, exactly as with rectangular co-ordinates. (57) To find the conditions that the straight line | OBLIQUE CO-ORDINATES. 45 may be perpendicular to the plane AD eye Catan 120) SPOR Hees a C2): If the straight line be perpendicular to the plane, it must be perpendicular to every straight line in the plane. Let the equa- tions to any one of them be BAO Y PAs 2ay oe Neal SO he 3); rE m n f ) then 7’, m’, n' must satisfy the equation FATES itras tai CoPTER ON os anaes Wats): since the line lies wholly in the plane. But the condition that the line (1) should be perpendicular to (8) is, by Art. (55), Wl + mm +nn' + a(mn' + m'n) + 6 (ln + Un) + c(lm' +Um)=0...(5). The equation (5) is to subsist for all values of 7’, m', n’ which satisfy the condition (4) ; therefore if we were to eliminate one of them as 7’, we should have an equation involving the other two ’ and m’, and as these are independent, their coefficients must separately vanish, But it is more convenient to use the method of indeterminate multipliers. Multiply then (4) by a quantity 4, and subtract from it (5); then, if we assume as the condition for determining / that the coefficient of /’ shall vanish, we have kA =1l+cm+ bn; and as m’, nm are independent, their coefficients must vanish separately : hence kB =cl+m+an, kC = bl+ amin, which are the three required conditions. ‘The quantity / is easily determined ; for if we multiply by /,m,m, and add, we find, by the condition of Art. (54), k(Al+ Bm + Cn)=1, or Lea pat CO (58) Oblique co-ordinates may be conveniently used for demonstrating various properties of the tetrahedron. Ist, The straight line joining the middle points of opposite edges all pass through one point. ‘Take O (fig. 12) one of the summits as origin, and the three contiguous edges OA, OB, OC as the axes of xz, y,z. Then if P be the middle point of AB 46 OBLIQUE CO-ORDINATES. and Q of OC, PQ is one of the lines which we have to consider. The general equations to a line passing through two points are Ee eed ae ce ded klar anton ets td Sled Let OA = 2a, OB=2b, OC = 2c; then the co-ordinates of Q(z,,¥,> 2,) are 0, 0, c, and those of P (z,, y,, 2,) are a, 6, 0; so that the equations to PQ are ; a by =e In like manner the equations to the line joining the middle points of OB and AC are | ci yb V2 alba <3 iad and those of that joining the middle points of O.A and BC are Z-A Y 2 te ec Combining the first and second equations, we find values which also satisfy the third equations; consequently all three lines pass through the point of which these are the co- © ordinates. and, If through three conterminous edges planes be drawn bisecting the opposite edges, they will intersect each other in the same straight line. The figure being the same as before, and the edges at O being taken as those through which the planes are to pass, the equation to the plane passing through ~ OC and bisecting AB is gtd ge Biel Similarly the other two equations are OBLIQUE CO-ORDINATES. 47 and it is easily seen that these are satisfied by the relations Oe i As which are therefore the equations to one straight line in which the three planes intersect. 8rd, The six planes which pass through the six edges of the tetrahedron and bisect the opposite edges, all pass through one point. Three of these planes are those which have been considered in the last problem, and it is obvious that their equations are satisfied by the co-ordinates of the point found in the first problem, viz. . j , De UIC liee T etne But that point is, by construction, symmetrical with respect to the tetrahedron, and therefore the three planes which pass through the edges terminated at A, will also pass through it, and hence that point is common to all the six planes. The point in question is the centre of gravity of the solid, CHAPTER III. TRANSFORMATION OF CO-ORDINATES. As the origin and direction of the axes to which the position of a point in space is referred are quite arbitrary, and as the simplicity of our expressions may be very much affected by the choice which we make of these, we proceed to establish formule for changing one system of co-ordinates to another. (59) To change the origin of co-ordinates, the axes remaining parallel to their original position. Let O (fig. 13) be the origin of the old axes Oz, Oy, Oz; O' the origin of the new axes O'z’, O'y’, O'z', parallel to the former. Let the co-ordinates of O' referred to the old axes be OD ay Osteo ey Let P be any point in space, and let its co-ordinates referred to the old axes be OM a7 ee Ny, Nee, and those referred to the new axes OM =z, MN =y, PN =z2, Then, as OM = 0Q0+QM=0Q+ 0M, MN=Qh+LIN= QR+MN, PNENN PPN = ORS PING we have Dien il [o et 3, | 2 ay ee ee Le as the expressions for the old co-ordinates in terms of the new. These being substituted in any function of the variables z, y, z, give a result involving 2’, y’, 2’, and therefore referred to the new co-ordinates. In the figure O' has been assumed to lie within the positive axes of the old system, and P within those of the new. But if es TRANSFORMATION OF CO-ORDINATES. 49 either chance to lie in the direction of any negative axis, the formula is easily adapted to such a case by a change of sign. Thus, if O” be the new origin, OM = MQ’ da OQ’ = MO — OR}, or L=kH -a. In like manner, if P lie towards the negative axis of y/’, ed ole and so forth. Hence, the formule (1) are true for all cases if we attach to the quantities involved their proper signs depend- ing on their positions relative to the origins. These formule hold equally for rectangular and oblique co-ordinates. (60) To pass from a rectangular system to any other, the origin remammg the same. Let Ox, Oy, Oz (fig. 14) be the old axes at right angles to each other, Oz’, Oy’, Oz' the new axes inclined to each other at anyangleye QOM=z2z, MN=y, NP =z, OM an, MIN iy NR y: Project the broken line 2’ +’ + 2’ on the axis Ox, by drawing from M’, N’, P perpendiculars to that line; the last one PM falls at the extremity of the abscissa z. Hence the line OM or z is equal to the sum of the projections of 2’, y', and 2’. Let a, 6, c be the cosines of the angles which the new axes make with the axis Ox; then, by the theory of projections, z= ax + by’ + cz’. We have here assumed that a, 6, ¢ are the cosines of the angles which the positive new axes make with the positive old axis of x, and that in the figure each of the axes makes an acute angle with Or. But if, as in fig. (15), one of the new axes, as Oz’, makes an obtuse angle with Oz, we shall have OM = On + m'n' — Mn’, or z= ax + by' — cz’, which is equivalent to the previous formula, as ¢ would in this case become negative, as it is the cosine of an angle greater than a right angle. E z) = 0, will give an equation f(a, y)=03 which is the equation to the curve of intersection of the surface — and plane. If the cutting plane be perpendicular to the plane of (7,y), the preceding expressions are reduced to t L = LACUS), i= (SIN. mee — age As this transformation of co-ordinates is generally a long and troublesome operation, it is advisable to endeavour to avoid it~ by having recourse to different methods suited to the problem — under consideration. (69) If the degree of the surface be n, the degree of the curve of intersection cannot be greater than n. : The equation of the curve of intersection is, as we have seen, found by transforming the co-ordinates till the new plane of (z', y') is parallel to the cutting plane, and then making 2’ = 0. Now, by Art. (67), the degree of the equation to the surface cannot be altered by the transformation of co-ordinates, and hence the new equation in 2’, y', z’ must be of the order ». It is clear, then, that the order of this equation cannot be increased _ by making 2’ = 0, and therefore the curve of intersection cannot — be of a degree greater than m. It may however be less, if it should happen that the vanishing of z should cause all the powers of the z‘* degree to disappear. | TRANSFORMATION OF CO-ORDINATES. 57 (70) Polar co-ordinates. In the applications of analysis to Mechanics and in the Integral Calculus, polar co-ordinates in space are sometimes found to be useful. The co-ordinates chosen are, usually, the distance of a point from a fixed point or origin, the angle which this distance makes with a fixed axis, and the angle which its projection on a plane perpendicular to the axis makes with a fixed line in the plane. To shew how we may transform from rectangular co-ordinates to these polar co- ordinates, let Ox, Oy, Oz (fig. 18) be the rectangular axes, O' the pole of the polar co-ordinates, of which the co-ordinates are OF aa AO OO One, Let P be any point, its co-ordinates being z, y,z. Then if we take the axis of z as the fixed axis for the polar co-ordinates, and the axis of x as the fixed line in the plane perpendicular to it, the polar co-ordinates of P are O'P=7r, the angle between O'P and Oz = 6, and the angle between O,N and Ox = 9¢. Hence, from the geometry of the figure, since O,N = r sin 9, z=a+rsinfcos¢d, y=PtrsnOsing, z=y+rcos 9. (12). which are the required formulz of transformation. ( 58 ) CHAPTER IV. REDUCTION OF THE GENERAL EQUATION OF THE SECOND DEGREE. (71) In a previous chapter we found that the general equa- tion of the first degree represents only one kind of surface—the plane: our next step is to investigate what kinds of surfaces are represented by the general equation of the second degree. ‘The form of this, when complete in all in its terms, may be written as Ax’+A'y’+ A'242 Byz+2Bx2+2B'xy+2 Cx+2 C'y+2C2+H=0 ... (1), where some of the coefficients are multiplied by 2, for the con- venience of future operations. Since this equation contains ten terms, it is highly desirable, before discussing its geometrical interpretation, to consider whether it may be simplified without destroying its generality. The transformation of co-ordinates gives us the means of trying this, and we proceed to show that we can always, by changing the direction of the co-ordinate axes without altering the origin, effect a very important simplification ; and that, by changing the ~ origin and not the direction, we obtain conditions by which we can determine those forms of the equation which offer distinctive peculiarities in their geometrical interpretation. (72) In the first place, we may put (1) in the shape Ug Uy EU, =O ante ete (2); where wv, is a homogeneous function of the second degree, wu, of the first degree, and w, a constant. Now if we substitute in this equation linear functions, such as we find in the expressions for changing the direction of the co-ordinates, the origin being unaltered, the different terms in (2) must alter independently one of the other, and we may therefore consider them separately. REDUCTION OF THE GENERAL EQUATION. 59 Taking then the term w, alone, we shall show that it may always by transformation of co-ordinates be deprived of the terms in- volving the products of the variables. For this purpose it may be put under the form u,=(Ac+ By + B’z)2+(B'c + A'y+ Bz)y+(B'x+ By+ A’z)z ... (8). Now the formule for changing the direction of the co-ordinate axes without altering the origin are, by Art. (60), = at, + aY,+42,, y = bx, + Uy, + bz, Z=ICl Out C 2,, the quantities a, 6, c &c. being subject to the conditions (3) and (4) of the last chapter. Substituting these values in (38), it takes the form u,=(Le, + L'y, + Lz.) (az, + vy, + a2, + + (Mz, + M'y, + Mz.) (bz, + By, + Oz) +0...... (4); + (Nz, + Ny, + Nz,) (cz, + ey, + €%,) where for shortness we have put L= Aa+ Bb+ Be, M= Ba + A'b + Be, N= Ba+ Bb+ A’, with corresponding values for the accented letters. But (4) may be put under the same form as (3), viz. Mea (Ps.+ Py, + Pz,)2,+(P2,+ Py t+Pi2y, ail UPA ad A SAT EE ee (5), where the quantities P are determined by equations of the form P= aL” + bOM™ 4 ON, the suffix of the P corresponding to the accent on the a, 6, c, and the accent on the P to that on the L, M, N. Hence P=al'+bM'+cN’, = a(Aa'+ B’b'+ Bi)+b( Ba + A'b'+ Bo)+¢(Ba' + Bb'+ A’c’) =a (Aa + Bb+ Bo) +0 (Ba+ A'b + Bo) + ¢(Ba+ Bb+ Ac) =@L+0MicNe=P.. In like manner we find ea eae Ey =P.’ 2 60 REDUCTION OF THE GENERAL EQUATION on me on RS These six quantities are the coefficients of the products of the variables in (5), and if they be made equal to zero, that equation will be reduced to u,= Px? + Ply! + Piz 271? which is thus deprived of the terms involving the products. (73) Now the three conditions Pos P30; 2 =P, = 0, ee ee (6) give three relations between the quantities a, b, c &c., and the constants in w,; and as these quantities are nine in number, and are connected, as we have seen in Arts. (60) and (61), by six equations of condition, we have on the whole nine equations of condition for determining nine quantities ; so that unless some of — these equations are derivable one from the other, the quantities a, 6, c &c. can all be determined. There remains to decide the question whether their values, and also those of P, P,, P,’ are— possible, in order to prove that the required transformation can_ be effected. For this purpose let us take the equations P,=adL+bUMicN=0 P,=adL+0M+cN=0 P=aL+6M+cN multiply the first equation by a’, the second by a’, and the last by a, and add; then, by the conditions in Art. (62), aP=L= Aa+ Bb+ Be, _ or (P-A)a-Bb- Bc=0. In like manner we find = ~~ )eceees +. (8). Bla -(P - A’))b+ Be=0 Ba+ Bb-(P - A’\c=0 Between these we can eliminate a, b, c by cross-multiplication, | and we obtain for determining P the equation (P - A)(P - ii ot hs a B* (PA) =2B BD Ones (9).a We should arrive at the same equation of ee if instead of (7) we took the equations P=0, (Pis0, Pi= als 6a oNe 1 re OF THE SECOND DEGREB. 61 and eliminated a’, b', c’; or if we took the equations P= 0; P= 0, PY Sa L' + UM’ + oN’, and eliminated a”, 5", c’. Consequently the three roots of the equation (9) are the three quantities P, P,’, P,.". (74) This remarkable cubic, which occurs in various Mathe- matical researches, is of very great importance, as will be seen from the use which we shall make of it ; and we shall distinguish it by the name of the Discriminating Cubic. But first it is neces- sary to show that its three roots are always possible, which may be done by the following method due to Cauchy. In the particular case in which B’=0, B’=0, the equation (9) is reduced to Gr (eA) (PAD SB 0 ee ee: (10), the roots of which are Beek (44) SGA 4 a all of which are possible, as the quantity under the radical is essentially positive, being the sum of two squares. Now if we put 4 1 a=\(A'+.A")+1{(A'- A"V44B'}*, B=A'4.A")-H(A'-A'ls 4 By , we can show that a and (3 are limits between the roots of the cubic. In the first place we observe that the substitution of these quantities will in virtue of (10) make the first two terms of (9) vanish, and as a— A’ and a — A” are essentially positive, since the radical in a—A’ exceeds the other term, we may repre- sent them by /’ and 4’, and as from (10) B’ = (a — A’)(a - A"), the first side of (9) becomes —~ (BV? + BYR + 2B B'hk) = - (Bh + By, which being a negative square is essentially negative. In like manner it may be shown that the substitution of 3 in the first side of (9) gives a positive result. Consequently, if we write U for the first side of (9), we have the following results: if P=, U is +, eds Uis -, fetes (3, U7 is +, P=-—o, Cis, 62 REDUCTION OF THE GENERAL EQUATION Therefore one root of the cubic lies between « and a, another between a and 3, and the third between 3 and -«, and hence the three roots are real. (75) If any one of these three real roots be substituted in the equations (8), we shall have three equations in a, 6, e, which when solved will in general give determinate and possible values for the ratios ~ , oe and these combined with the equation peeve a a oo serve to determine the values of a,b,c. These values may be found most readily in the following manner: eliminate ¢ be- tween the first and third, and then between the second and third equations of (8), when we get af(P— A)\(P- A)— BA Sb 1B (PAD SEB a {B'(P- A) + BB} =8{(P-A)(P-A)- B}, 2 Se h SS = So WwW €nce Ue a A’) (iP ae 13) yy B CH ni A) CB = An} rt: ‘Be C (PB SACP ABs by the symmetry of the formule. Let each of these ratios be assumed equal to ~: then V+04+C=1 =u{(P-A') (P-A')+(P-A)(P-A))(P-A)(P-A)- B’-B"-B"}. And, on giving to U the same meaning as before, it is obvious that the second side of this equation is equal to ao , So that Butif P,, P,, P, be the three roots of the discriminating cubic, and we substitute any one of them as P, in EUs we have dP dU aPe C5 to) Ls PE) swbene ihe: whence 1 DD See arama eee aE ea eR CE DY Gaal OF THE SECOND DEGREE. 63 so that, putting P, for P in each of the ratios which is equal to w, we find oa et 4 )CP} Ai (PH Ae HAD BR Ce, oP) Bit Py Sees eS, TN 5 Pine eee Oe (P_- P) ®,- Py A similar set of values for a’, b’, c’, a’, 6’, c’ may be found, by putting P, and P, for P, in the preceding expressions. (76) It appears then, from the preceding investigation, that since we can always find possible values for the three quantities P, and the nine quantities a, 6, c &c. from the conditions (6) Art. (73), the general equation of the second degree may always, without affecting its generality, be reduced to the form Po’ + Ply’ + Pie’ + 2Qx 4+ 2Q'y + 2Q'24+ H=0....(11), where for convenience we have dropped the suffixes of the co- efficients of the squares of the variables, and put Q=Ca+C'b+C'c, Q=Ca+C)'+ Ce, Q'=Ca'+ C'O'+ Ce’. The only restriction is that the quantities P, P’, P’ must not be all equal to zero at the same time, as the equation would then be reduced to the first degree; with this exception these quan- tities may be of any value or sign. (77) The separation into different classes of the surfaces re- presented by (11), depends on the vanishing of one or more of the coefficients of the squares of the variables, as will be seen in the following investigation. If we seek to simplify the equation still further by depriving it of the terms involving the first | power of the variables, we change the origin of co-ordinates, putting t=2,+a, Y=¥,t+B, 2z=4%, +7: The substitution of these values gives P2r?+ Py? + P’'22+2(Pa+ Q)z,+2(P'B+ Q)y,+2(P’y+ Qe, + Pot + PB + Py’ +2Qa+2Q8 4207+ H=0, and the condition that the terms of the first degree shall vanish gives the equations Pare = 0, PF B4iQ' 03) P'y4 Q'= 0.02.05. (12) 64 REDUCTION OF THE GENERAL EQUATION is These three equations will give finite and possible values of | a, (3, y, in all cases except when any one of the quantities P vanishes. If the corresponding Q be finite, then the value for a, B or ¥ is infinite: if the corresponding Q = 0, the value— is indeterminate. Hence we divide the surfaces represented by (1), and also by (11), into the following classes. I. When none of the quantities P is equal to zero, in which case the equation is reduced to Pot Py PZ (18). II. When one of the coefficients of the squares as P = 0, | while Q does not vanish, we cannot make the term involving z © vanish, but we may then determine a by the condition that the constant term shall vanish: or in this case, P'? + P’y’ + 2Qa+ 2138 + 2Q'y + F=0, an equation which must give a possible value for a, since that — quantity is involved in the first degree only. The general equa- tion is then reduced to Pap taka thr ONE en (14). The condition of P = 0 necessarily involves the condition AB’ + A'B" + A° B® -AA'A' —- 2BB'B' = 0.... (15), as this is the constant term of the discriminating cubic of which P is a root. III. When in the equation (11) P=0 and Q = 0 at the same time, the other coefficients being finite, it becomes Py’ + P'2 + 2Qy + 2Q"2+ F=0, which may be reduced to Pye Re = Tepe ee ae (16), by changing the origin of co-ordinates, IV. If we have at the same time P = 0 and P’ = 0, the equa- tion (11) becomes Pz’ + 2Qz24+ 2Q'y + 2Q’24+ H=0. If neither Q nor Q' vanish, this equation may always be re- duced, as that of Class II. to Pre + 2Q2 4+ 2Q'y=0........5. CLT. But if Q = 0 and Q'= 0, the values of a and are indetermi- nate, and that of y alone is determinate. We are then unable OF THE SECOND DEGREE. 65 to destroy the constant term, but we may get rid of that involy- ing the first power of z, so that the equation becomes dip td hfe Pee Pee (18). The equations P = 0, P’ = 0 involve in addition to (15), the relation 7 _ 4'4’+ B?~ AA’ + B?-AA'=0....(19), since two roots of the discriminating cubic in this case vanish. (78) Let us now consider the geometrical meaning of this separation: the general equation (11) being Pe’ + Py’ + P’2 + 2Qr4 2Qy + 2Q’2+ H=0, let the surface be cut by the line eA) Haat GS 29 Peal which passes through a point (a, 3, y). Substituting the values of x, y, Z, in terms of 7, we have (PP+P'm’+ P'n’) 7°+24( Pat Q) 4 P’B+ Q) mH P'y+Q*)n}r +Pa’+P'3°+P'y'+2 Qa+2 Q'B+2 Q’y+H#=0. This being a quadratic equation, gives generally two values of 7, which is the length of the portion of the line mtercepted between the point (a, 3, y) and the surface. Now the con- ditions which reduce equation (11) to Class I. make the term involving 7 disappear, and the quadratic being reduced to two terms gives two values of r which are equal, but of opposite signs. Consequently the point (a, 3, y) bisects every chord in the surface which passes through it. Such a point is called a centre of the surface, and the surfaces in Class I. are called central surfaces. It is plain, that as the equations (12) give single determinate values for (a, 3, y), there can be only one centre for such surfaces. Again, the conditions for determining Class II. give an infi- nite value for a, and finite values for (3 and y, since Q Q’ Q : ee yt? B=- 3); eS ee therefore these surfaces have their centre at an infinite distance, although they are usually said not to have a centre. 66 REDUCTION OF THE GENERAL EQUATION. In Class III. we have finite values for 3 and y, that of a being indeterminate ; hence all the points in the line, of which the equations are Q' ‘ae B= - Pp” Yaa we P' may be considered as centres, as for all these points the coefhi- cient of 7 vanishes independently of a. Surfaces of this kind are evidently, from Chap. 1. Art. (8), cylinders, as their equation involves only two of the variables, and is therefore satisfied independently of the third. The surfaces represented by Class IV. are of two very different kinds: when neither Q nor Q’ vanish, two of the co- ordinates of the centre are infinite, and the surface consequently has no centre ; but when both Q and Q' vanish, the coefficient of y will vanish if Pate Oe and therefore all points in the plane, of which that is the equa- tion, may be considered as centres. Both kinds of surfaces are cylindrical, as will be seen in the following chapter. CHAPTER V. INTERPRETATION OF THE EQUATION OF THE SECOND DEGREE. Havine reduced the general equation of the second degree to four forms, we now proceed to discuss the geometrical meaning of these equations, and the nature of the surfaces which they represent. Central Surfaces. (79) The general equation to these surfaces we found to be TER A EAT PM Sd Pap 2 9 a a aie Sek (7 9 and the different varieties of the surfaces which this equation represents depend on the relative signs of P, P’, P’, and the magnitude of H, so that we have four varieties: 1st, when H=0; 2nd, when all the quantities P, P’, P’ are positive; 3rd, when one of them is negative; 4th, when two of them are negative. All the varieties have this property in common, that they are symmetrical with respect to the origin, since the equation remains unchanged when - 2, —- y, - 2 are put for +z, +y, and +z; and this facilitates their discussion, since we may confine our attention to the absolute positive values of each variable. (80) Cones. 1st, Let H=0; then some one at least of the quantities P must be of a different sign from the others, in order that the equation Boa tele tobe 2 i=0 may represent a surface: for if all be of the same sign, the only possible values of the variables which satisfy the equation are z=0, y=0, z=0, showing that the locus of the equation is in that case a point at the origin. It is sufficient to suppose one only of the coefficients to be negative, as if two be so, we have only to change the sign of the whole equation to bring it F 2 68 INTERPRETATION OF THE EQUATION to the other case. Let the equation then be Pee ya P20 ee, (1). Since this is satisfied by z= 0, y= 0, z= 0, the surface passes through the origin. Let the straight line, of which the equa- tions are ORY wee —_ _= —_— fem ianone meet it in the point z, y, 2; then, substituting for z, y, 2 their values in terms of 7, the distance of the point from the origin, we have (PP + Pn? — P’n’)?r’ = 0. This equation can be satisfied only by PPE Pa — Poe aoe ane (Ds and then it is satisfied independently of r, so that that quantity is indeterminate. There being only one relation between /, m,n besides the general one 7’ + m’?+n’=1, there are an infinite number of straight lines, for which the condition (2) is satisfied, and as for all » is indeterminate, all these lines (which pass through the origin) lie wholly in the surface. Such surfaces are called cones, the common right cone being a particular case of them. If we put z =A in (1), it becomes Dig te Paes which is the equation to an ellipse in the plane of (z, y), the origin being at the centre, and the principal axes being parallel to the axes of z and y. Hence all sections made by planes parallel to that of (x, y) are ellipses, which become circles when P= Ff’: it is easy to see that in this case the surface is a right cone. ‘The sections parallel to the planes of (z, z) and(y, z) have for their equations Pe -P'# =- PR, Pep Pe ele, which show that the curves are hyperbolas. (81) Since an equation of the form Az’ + Aly’ + A’2’ + 2Byz + 2B'xz + 2B" xy = 0, can always, without affecting its generality, be reduced to Pz’ + Py + Pz =0 ] it appears that a homogeneous function of the second degree, pa tag OF THE SECOND DEGREE. 69 when equated to zero, represents in general a cone, the vertex of which is in the origin, unless Ist, the coefficients of the transformed equation are all of the same sign, when it represents a point. 2nd, the function can be decomposed into two possible factors of the first degree, when it represents two planes. The analytical condition that this should be the case is AB + AB? + A°B? — AA A’ - 2BBB' = 0, since two planes are a particular case of cylinders of the second degree. (82) Ellipsoid. Let H, P, P’, P’ be all positive, so that the equation is A sod Been at aed w BR eee (1). Let the straight line was, ey ee meet the surface in the point 2, y, 2, then the combination of (1) and (2) gives EL idol PAILS ee eu Thy de iad 2 RRR (2) as the equation for determining 7. Now the coefficient ofr? can never vanish, since every term in it is essentially positive, con- sequently r is never infinite, and the surface is therefore a closed surface. Hence if it be cut by any plane, the curve of intersec- tion must be an Ellipse, since, by Art. (69), the curve must be of the second degree, and the ellipse is the only closed curve of that degree: from this property the surface is called an Ellipsoid. (83) The ratios of H to P, P’, and P" are quantities which have important geometrical meanings. For let y = 0 and z= 0 in (1), which is then reduced to 1g ok Neb This determines the distances from the origin at which the axis of x is cut by the surface, which are evidently equal and on opposite sides of the origin. If, then, in fig. (19) we put OA = OA' =a, we have , A a Ta In like manner, if OB = 6, OC = c, ee ee f= —, C= —_, P ] P’ 70 INTERPRETATION OF THE EQUATION and so the equation to the surface may be put in the form a 2 2 2 ay _ a A ad BNA Fes (3). The lines a, 8, ¢ are called the axes of the surface, and the points A, B, C its vertices. It is easy to conclude from this equation that the surface does not extend beyond the points A, A’; for if it be cut by a plane z=+f, we have y 2 2 an equation which cannot be satisfied by any possible values of y and z if f>a, as the second side is then negative. The same holds for the other co-ordinate axes, so that the surface does not extend beyond B and B' along y, and beyond Cand C" along 2. (84) If two of the coefficients, as P, P’, be equal, which involves the relation a = 8, then all sections made by planes parallel to (z,y) are circles; for putting a= 6 and z=A in(8), it becomes e+? h? sd a ee on (4): The surface in this case is said to be one of revolution round the axis of z, since it may be generated by making an ellipse revolve round one of its axes. If all the quantities P, P’, P" be equal, or a= b= c, equation (2) becomes Bit 2 a re (5), which shows that every point in the surface is equally distant from the origin, or the surface is a sphere, of which the radius isa. If we change the origin of co-ordinates to an arbitrary point (a, 3, y), this equation becomes (a - a + (y - BY + (@- P= a’....(6); which is the general equation to a sphere referred to rectangular co-ordinates. Every section of a sphere by a plane is a circle ; for all plane sections parallel to the co-ordinate planes are circles, and equation (5) remains unchanged when the axes are trans- formed to another rectangular system, so that we can thus obtain all possible sections of the surface. OF THE SECOND DEGREE. vi | (85) Hyperboloid of one sheet. Let one of the coefficients, as P”, be negative, so that the general equation (@) becomes ie ieee eee Fe CL). If we seek the points where this surface is cut by the line i Sikh aS tes // amas san we find CBr Leila) |e a Lilt ee). As the coefficient of 7? may be either positive, zero, or negative, it follows that » may be either real, infinite, or impossible. Consequently the surface extends to infinity in certain directions, which are determined by the condition Pe Pane beiea=d () ee (3), and no part of it exists in the space for which Wet awed Od ed Ba Now J, m, n, being direction-cosines, lie between 0 and 1; and if we suppose P > P’, it appears that we cannot have ie He Ole es oa yi igee Therefore the space in which the surface does not extend is that bounded by a surface generated by the straight line r turning round the axis of z, forming with it angles of which the limits are determined by the preceding inequality. If we cut the surface by planes parallel to the co-ordinate planes, the equations to the sections will be found by putting constants f, g, A in turn for z, y, z in the equation (1), so that we have Pi —-P'2 = H- Pf’, RA AeA td a ae al Pes Py = A+ Ph’. The first two represent hyperbolas, and the third an ellipse ;” and as they are all possible whatever be the values of f, g, h, it appears that the surface is cut by all planes parallel to the co-ordinate axes, and consequently it is a continuous surface of one sheet. As the sections parallel to two of the co-ordinate planes are hyperbolas, it is called the Hyperboloid of one sheet. 12 INTERPRETATION OF THE EQUATION (86) The equation to the surface which limits the surface — towards the axis of 2 may be found by eliminating /, m, n_ between zy 2 peeTee and PEP ayy on a On dividing each term of the latter by the square of the corre- sponding member of the former, the result is Pa Pay IPR = ON nee (4), the required equation to the limiting surface, which is a cone of the second degree, by Art. (81). It is easily shown that this cone is an asymptote to the hyper- boloid. For, if 2’ and z be co-ordinates of points in the cone and the surface corresponding to the same values of # and y, 1 i i P® (2 -2)=(P2 +P yy -(P#+ Py - AY; or multiplying numerator and denominator of the second side by the sum of the radicals AC! HT IS ee ee (Px s Pyy + (Pe + Py Se), The difference between the co-ordinates 2 and z decreases without limit as z and y increase without limit; but 2’ is always greater than z, so that the cone lies between the axis of 2 and the surface. (87) The equation to the hyperboloid may be put in a form similar to the second one of the ellipsoid by introducing corre- sponding geometrical quantities. Let OA, OA’ (fig. 20) be the distances from the origin at which the surface is cut by the axis of z, and put each of them equal to a; let OB, OB’, each equal to b, be the corresponding quantities for the axis of y; then es as The axis of 2 never meets the surface, and we cannot assign for it a corresponding geometrical quantity; but if we assume c to be such a quantity that _ H OF THE SECOND DEGREE. 73 the equation to the surface becomes, by the substitution of these ' values, ae oy 2 —++5--—=1 s) aps) a8 6 le ee, 6.8 «=» 6 5). toe ae (5) If d = a, or the axes of the ellipse in which the surface is cut by _ the plane of (x, y) be equal, every section parallel to that plane is a circle, for its equation will be of the form Pry’ 7 h* : x‘ Bs a which is that to a circle having its centre on the axis of z, whatever be the value of 4, showing that the surface is one of revolution. It may be supposed to be generated by the revolu- tion of an hyperbola round the axis which does not meet it. If we seek the equation to the asymptotic cone to the hyper- boloid under the same form as (5), we find it to be x y° 2 a + ae nie bere eae aC (6). (88) Hyperboloid of two sheets. Let two of the coefficients of the equation (a) be negative, so that it takes the form cae eee hee fe ee GLa) then it will be easily seen, by combining this with the equations to the straight line ey 8 Lm n as in the preceding surface, that r is infinite for all values of l,m and , which satisfy the equation DED is Bag jie 2d EL eA ga he eR on ONE and consequently that the surface extends to infinity in these directions. And, as before, it may be shown that it is limited by the surface, of which the equation is Pee yom ine = 0, Or by. + P 2 — Pr = 0. This is the equation to a cone of which the axis of z is the axis, and which is asymptotic to the surface in such a way that the surface lies between it and the axis of z. If the surface be cut by planes z=f, y=g, 22h, 74 INTERPRETATION OF THE EQUATION we have, as the equations to the sections, Pp AN HP fe, Pe — P'2’ = H+ Pg’, Pe —-Py=- H+ Pl: The first of these is the equation to an ellipse, unless Pf’ < H, in which case it cannot be interpreted; therefore all sections parallel to the plane of (y, z) are ellipses, but no plane drawn at a distance along the axis of x on either side less than (Ss) meets the surface. ‘The surface therefore is discontinuous between the planes so determined. ‘The second and third equations show that all planes parallel to (z, z) and (y, z) cut the surface in hyperbolas, since they are possible, whatever values be assigned to g and f. For these reasons the surface is called the Hyperboloid of two sheets: see fig. (21.) If we assume HT 2 . ° ° e @ = 5, We see, as In the previous cases, that a is the distance on either side of the origin at which the axis of x is cut by the surface. ‘The other axes never meet the surface; but if we assume Dae oe P' 3 =e P" 9 the equation takes the form a y” 2 POR oaP and the corresponding equation to the asymptotic cone is y" 2 a B + 9 Bee ana — QO. Canad. If P’ = P", or 6 =c, the sections parallel to (y, 2) have for their equation Pevltye Gotti Vase Fog b a which shows that they are circles, and consequently that the sur- face is one of revolution. It may be supposed to be generated by the revolution of an hyperbola round its principal axis. Surfaces without a Centre. (89) The general equation to these may be put in the form Ly + Pies Ot, ne (b), OF THE SECOND DEGREE. 16 in which one of the constants may always be supposed to be positive. Moreover the sign of Q can have no influence on the nature of the surface, as the term containing it can always have its sign changed by substituting — x for x, which is equivalent to measuring the positive axis of z in a direction opposite to that before adopted: this will affect the position but not the nature of the surface. Hence there are only two forms of the equation to be considered, one when /”’ and P” are of the same sign, the other when they are of contrary signs. — (90) Elhptic paraboloid. Taking P’ and P" both positive, and Q negative, the equation is JETRO Ey ALT hel Sante (ine From the form of the equation it is evident that 2 can never become negative, and therefore the surface lies wholly on the positive side of the plane of (y, z), while it is symmetrical on oppo- site sides of the axis of x, since the equation remains unchanged when for y and z we put -y and —z. ‘The surface passes through the origin, since z = 0, y = 0, z= 0 satisfy the equation (1). If z2=0 we have Q Dy die = Qe, or y'= = @ = pe suppose. This shows that the surface is cut by the plane of (z,y) ina parabola of which p is the principal parameter. In like manner wwe see that it is cut by the plane of (z, z) in a parabola of which the principal parameter is On substituting these Q p= P' quantities in the equation, it becomes y z 1,2 2 / ote OF Ae ie ODE oases ples (2), Rene; PY + pe =p If the surface be cut by any plane 2 = f, parallel to (y, z), we nave Py + p? = wwf, which is the equation to an ellipse, whatever be the magnitude of f, so long as it is positive. Hence all sections parallel to (y, ) on the positive side of the axis of ~ are ellipses, and since f may be increased indefinitely, the surface extends to infinity in that direction. If the surface be cut by planes parallel to (@, z) and (z, y), Pag or'z=h, —- 76 INTERPRETATION OF THE EQUATION we have pz =p (pr-g), and py =p(pz-h’), | q which are equations to parabolas, of which the latera recta are the same as those of the sections made by the co-ordinate planes of (z, z) and (a, y). For these reasons the surface is called the Elliptic Paraboloid: see (fig. 22). When p=yp’, the elliptie sections parallel to (y, z) become circles, having their centres on the axis of z, so that the surface may be supposed to be generated by the revolution of a parabola round its axis. . (91) Hyperbolic paraboloid. ‘Taking P” as negative in the general form (6), we have to consider the equation Py Pe Qt . 3. In this case we can assign both positive and negative values without limit to all the variables, and coneed eae the surface extends indefinitely in all directions. If 2-0 2 @ = F se Pe! Pees z= pr suppose, showing that the plane of (x, y) cuts the surface in a parabola of which p is the principal parameter, and of which the axis is turned towards the positive axis of z. iba y=0, 2 =-—,=-p'e suppose. This shows that the plane of (z,z) cuts the surface in a parabola of which the axis is turned towards the negative axis of Z Introducing these parameters into the equation, it becomes ve Oe (2). 9 If z=0 we have Py’ - P’z’=0, which may be decomposed ae Py — P"z = 0 and P*y + Pz = | showing that the plane of (y, =) cuts the ae in two straight lines. If the surface be cut by a plane z = f, parallel to the plane of (y; Z), we have py — pz = ppT: this is the equation to an hyperbola, whatever be the value of f positive or negative; but there is a difference between thé OF THE SECOND DEGREE. 17 sections made on the positive side of the origin and those on the negative side; for in the former the principal axis of the | hyperbola is parallel to the axis of y, and in the latter to z. ‘These two kinds of hyperbolic sections are separated by the | straight lines in which the surface is cut by the plane of (y, 2). As in the previous Article, it is easily seen that all sections made by planes parallel to (x, y) and («, z) are parabolas, the former having their concavity turned towards the positive axis and the latter towards the negative axis of z Hence the surface is called the Hyperbolic Paraboloid: see (fig. 28). This surface can never become one of revolution, since the coefficients of y° and 2° can never be the same, as they are essentially of opposite signs. Surfaces having a Line of Centres. — (92) Elliptic and hyperbole cylinders. It appears from equa- ‘tion (16) of the last chapter that when the coefficients of both ‘the first and second powers of one of the variables vanish, the “equation may always be reduced to the form PiU DA as Ke dae he, tay Bi Baeeke) ye Bo A,~ 4)- 8? Pies cle L,) P,, P,, P, being the three roots of the discriminating cubic in Art. (73), or of the preceding cubic in S. These values of the direction-cosines of the chords remain determinate even when one of the roots of the cubic vanishes ; but in that case the 86 DIAMETRAL PLANES. coefficients of the variables in the equation to the diametral plane are each equal to zero, implying that the plane is removed to an infinite distance. Thus, though the three directions of the principal chords can be always assigned, one of the corresponding diametral planes may not exist. This, it is plain from Art. (97), happens in the surfaces for which AB + A'B? + A° B® - AA'A' -2BB'B' = 0. (101) The results at which we have just arrived, compared with those in Art. (73), show us that the process of reducing the general equation of the second degree so as to be deprived of the terms involving the products of the variables, is equivalent to referring the surface to three rectangular axes parallel to the three systems of chords which are perpendicular to their respective diametral planes. Accordingly the geometrical con- siderations of the properties of diametral planes have been employed by several writers for reducing the general equation : this method was suggested by M. J. Binet, Correspondance sur ? Ecole Polytechnique, vol. 11. p. 74. (102) It is to be observed that if we take a diametral plane to be one of the co-ordinate planes, as that of (7, y), and the axis of z to be parallel to its chords, the equation to the surface can con- tain none but even powers of z. For since the diametral plane bisects the chord parallel to z, the negative values of z must be equal to the positive values, and the equation to the surface must remain unchanged when we substitute - z for 2: this can- not happen if the equation contain odd powers of that variable. Conversely, when an equation contains none but even powers of a variable, we know that the plane containing the axes of the other two variables bisects all the chords parallel to the axis of the first variable, or is a diametral plane to chords parallel to that axis. From this it appears that when the equation of the second degree is in the form Rg +: Ry’ ot Oe eed, the surface being therefore central, each of the co-ordinate planes is a diametral plane of the surface, since the equation contams none but even powers of each of the three variables. ee a F | DIAMETRAL PLANES. 87 Moreover, since the co-ordinates are supposed to be rectangular, each diametral plane is a principal one, and each plane bisects the chords which are parallel to the intersection of the other two. (103) Definition. ‘Three diametral planes are said to be con- Jugate to each other when each bisects the chords which are parallel to the intersection of the other two. We have just seen that when the equation of the second degree referred to rectangular co-ordinates is in the form Wage AEM hye BSN ea Noe 2 fe the three co-ordinate planes are conjugate diametral planes in the sense just defined, and it is clear that these are the only planes which are at once conjugate and principal planes, since we found before that there are generally only three principal diametral planes in a surface of the second order. We proceed now to shew that there are an infinite number of conjugate diametral planes oblique to their chords, so that when their intersections are taken as oblique axes of co-ordinates, the equation to the surface is reduced to the preceding form. (104) Taking the equation to a central surface referred to rectangular co-ordinates in the form Idi EAT ete HEAP a ea w beans ibe Waleed ec ik let it be cut by a plane passing through the centre, Lae My N20 oie eps eile 63 (2). Now, by Art. (99), the equations to a line parallel to the chords bisected by (2), are pp, Py Pz zy Sy EE Sade Pr ese I Deas 14 N Let eat LV 2 = 0) scribe OES eA) diets hg IN ARE ore O, be the equations to two other planes passing through the centre. In order that their intersection may be parallel to the chords bisected by (2), they must both pass through (3): this gives two conditions by Art. (47), 5 Ely, UM, , NN, FETC Sey oT a Ng = (). Fre A PN oT 88 DIAMETRAL PLANES. In like manner the conditions that (4) shall bisect the chords parallel to the intersection of (2) and (5) are LL, MM, NN, Ld,, UM, NN,_ —1+4+ 1+ - +—lLs= Divi PAP HO Oe P P iP 0; and the conditions that (5) shall bisect the chords parallel to the — intersection of (2) and (4) are LL, MM, NN, , L,l,,UM, NN, PA oP eee Pp Paes Hence, in order that a system of three oblique conjugate 0, 0. diameters should exist, these six equations must hold: but they — are only three independent relations, so that if we suppose — L, M, N to be given, there are three relations for determining © cya! Ses) hyo? Biles & the four ratios +, —1, —2, —, and consequently one or other of them must be indeterminate. The corresponding plane is therefore indeterminate, and hence we see that if any plane ~ be given, there are an infinite number of pairs of planes which can be drawn so as to be conjugate with it. The intersections of these three planes two and two may be taken as axes of oblique co-ordinates, and there are thus an infinite number of oblique axes, to which when the surface is referred, its equation is reduced to the form Ag+ Aly + A’? = K. (105) In the same way as in Art. (88) we see that if a’, 0’, ¢’ be the portions of the axes intercepted between the origin and the surface, RK K\ K\ a’ ao +} ; b' ae a) : e ze 7) p (3 A; A so that the equation to the surface may be written Vad y 2 mr: + 7g Le oe =e Fe a, b,c are called three conjugate diameters, and may, like the principal diameters, be impossible, never meeting the sur- face. (106) To find the relations between oblique conjugate diameters and principal diameters of central surfaces. DIAMETRAL PLANES. 89 If we refer a central surface to any three conjugate diameters as oblique axes, its equation is of the form x 2 2 arn = a + eeee (1). We proceed to find an equation for expressing the principal diameters in terms of a’, 0’, c’. The definition of a principal diameter is that it is perpendicular to the diametral plane which bisects the chords parallel to it. Now if Les” oh svovsmercree te be the equations to any diameter, the equation to its diametral plane is, by Art. (98), & my | Lape +S < But if f, 9, 2 be the cosines of the angles between the axes of (y, 2), (a, 2), (a, y) respectively, we have, by Art. (57), as the conditions that (2) should be perpendicular to (3), k—=1+hm+gn,) k == hl +m + fn, female bMS (4), k a = gl + fm+n, where, by Art.(57), 1 PP 4»? #7 n kat BF But if we consider 2, y, z to be the co-ordinates of the extremity of a principal diameter, their values taken from (2) must satisfy (1). Substituting them, we have We are gy ar 10) 1 pa aaMoA eth where 7 is the length of a principal semi-diameter. Hence equations (4) become y & 1) 1— him gn= 0 hl - (Fe -1) ms fam 0 Be a6 90 DIAMETRAL PLANES. Eliminating /, m, n between these three equations by cross-mul- tiplication, we find y? y a 7 } Y (3 ) —-1 | —~1 ){—-1 |-f*( —-1 )-9’| 4-1 J-#’| = -1 |-279h=0, (5 1 \z 1 (= 1 f(a g (e raat tg a cubic equation in 7’, and therefore furnishing three values, which give the squares of the three principal semi-diameters. If we arrange this in terms of 7”, it becomes y® ps Pike a b” a ¢”) a ar’ {a"b” ¢! Bd h’) ae ac" (1 pe J’) te b°c? (1 —f*)} ~a°b'c? (1—f?- gf —h’ + 2fgh)=0.... (6). The roots of this equation, being the principal semi-diameters, we may call a, b,c, and then the theory of equations gives us the following relations, Ot OF Cad Bo 4 Cot ee kc ae ieee en Co ab’ +a'c’ +b’? = ab" (1 -h’)+ ae? (1-g’)+ 6°” (1-f")... .(8), abe = a'b'e (1 — f? —g? — WP + Ofgh) occ e. vee . (9), which are the required relations between the principal diameters and any three conjugate diameters. Equation (7) signifies, that the sum of the squares of any three conjugate diameters is constant, and equal to the sum of the squares of the principal axes. Equation (8) shows, that the sum of the squares of the parallelograms formed by each pair of conjugate diameters is constant, and equal to the sum of the squares of the rect- angles under each pair of the principal axes. Equation (9) shows, that the parallelopiped of which the three conterminous edges are conjugate diameters, is constant for all systems of conjugate diameters, and equal to the rectan- gular parallelopiped, of which the principal axes are diameters. In equations (7), (8), (9), we have assumed that the quantities a’, b*, c’ are all positive, or that the surface is an ellipsoid; but it is plain that it may be adapted to the other two central surfaces, by changing the sign of one or of two of the quantities a’, b*,c’. In such a case, the corresponding one or two of the quantities a”, 6”, c” will also be negative, so that the equation (9) will still subsist. DIAMETRAL PLANES. 91 (107) In the surfaces without a centre LEH IE SS eet ceo | A ae PRE (1) we said (Art. 98) that all the diametral planes are parallel to the axis, so that their mutual intersections, being parallel straight lines, cannot be taken as a system of co-ordinate axes. We may however find an infinite number of oblique axes for which the equation will be reduced to the preceding form, that is, such that two of the co-ordinate planes shall be diametral and con- jugate to the intersection of each other pair. Let AY PRE CAEE (Gee Sia, eee ae (2) be the equation to a diametral plane ; then, comparing it with Linas Sane OG SI als « (3) which is conjugate with the line [/, m, n], we have Q=XH, “Pm = NIT OP n= ANI. 4); which equations serve to determine the direction of the chords conjugate to (2). A line parallel to this we may take to be one of the new axes, as that of 2’, drawing it through any arbitrary point in the section made by (2), which we take as the origin in order to get rid of the constant term. ‘The other co-ordinate planes are to pass through this axis of z', and intersect the plane (2) in lines which are to be the axes of wv and y’, these being de- termined by the condition that the equation to the surface shall not contain odd powers of y’ or 2’, or that it shall be of the form ser Cae Jaw awe. ss. (0). Supposing this to be done, we find, on making 2’ = 0, that the equation to the section by the plane of (2’, y’) is Pyf =2Qz', which is that to a parabola, referred to an axis and the tangent at its vertex. Hence, as the axis of any parabolic section is parallel to the original axis of z, the three oblique axes which reduce the equation to the form (5) are a line parallel to the axis of the surface drawn through an arbitrary point in the surface, a tan- gent to the parabolic section made by (2), and the line determined by equations (4). As the point assumed in the section made by (2) is quite arbitrary, it is clear that there are an infinite number of systems of co-ordinates corresponding to every assumed plane. 92 SIMILARITY OF SURFACES. Similarity of Surfaces. (108) ‘Two surfaces are said to be similar and similarly place when, if we take any arbitrary point O from which radii are drawn to the one surface, we can find another point O’ such that the radu drawn parallel to the former and terminated by the other surface are always proportional to them ; so that if 15 To) 7, SOM be radi drawn from O and terminated by one surface, and 1, 7, &c. radii respectively parallel to the former drawn from O' and terminated by the other surface, | | 1 It is to be observed, that if two such points as O and O' can. be determined, there are an infinite number of such pairs of points for which the same proposition holds: for if O, be any point, the distance of which from O is p, and if along the line O'O, drawn parallel to OO,, we assume a distance ep such that p = kp’, the triangles, of which two corresponding sides are 7, p, and 7’, p', are similar, since the angles between r and ou r and p’ are equal and the sides about them proportional. Hence if r, 7’ be the radii drawn from O, and O, to the extremities of r and 7’, we shall have Die | Fiug | or the parallel radii drawn through O, and O,' are in the constant | ratio k. Such points are called centres of similarity. | (109) Zo find the conditions that two surfaces of the second order may be similar. Let their equations be Ac’+A'y’+A'2"+2 Byz+2 B're+2 Bry +2 0r+20 ‘y+2C"2+ H=0...(1), ax*+a'y"+a'2"+2by'2'420'a'z'420'x'y' +204! +2c'y'+2c'2'+4e=0 os she (2m | As the origin is arbitrary we may assume it to be the point O , relative to (1), so that the equations to a radius drawn through O | are > ~ is) aa] eke 1. © a cee ace e ha oe pex0 (3); | but if a, B, y be the co-ordinates of O' relative to (2) the equa- | tions to a radius parallel to (3) drawn through O’ are | Corsa = Bee ay ee : -_ OU ji Es Pre es Me Dk Cras OS (4). SIMILARITY OF SURFACES. 93 _ The definition of similarity gives the relation 7’ = kr, which leadsto yica+ke, y=Br+hy, 2 =y7+ kez. If then we substitute these values in (2), the resulting equation in 2, y, z must be identical with (1), and on arranging it in -powers of these variables, and comparing the coefficients, we shall obtain the required conditions. The substitution leads to | Bas aly’ + a'e + Qbyz + Wizz + 2 xy) + 2h {(aa+b'y+b'B+0)x+(aBtby+b'a+e)y+(a@y+bB4 Fa +e) | + aa’+a'3?+a'y’+ 2bBy +4 2b' ay + 26 -ab+2ca+ 2c'3+2c'y+e=0. On comparing the coefficients of the powers of the variables in this equation and in (1), we find cumera yer edin eOpoeib's!4)0° = eo — oo ——— i datby+bB+e aB+bytbate _ a'y+bB+b'ate’ _ \....(5). kC iC’ kC" aa’+a 3°+a'y°+2bBy+20'ay+2b'a3+2ca+2c B'+2c'y+e : RE ‘Since the first five equations are independent of a, B, and &, it appears that two surfaces cannot be similar, unless the coefficients of the highest powers of the variables be propor- ‘tional: but to show that the surfaces are actually similar, it is | necessary to prove that we can find real values for a, 3, y and &. For this purpose let each of the preceding ratios be put equal 1 : : ; a 3 the last four ratios give the equations h(aa + BB+ Bry) =kO-Ac...... (6), A(b'a + a'B + by) = kC'-De’...... Gay : WEAR TG arc) Te? a eee (8), | A (aa?+ a'B?+ a'y?+ 2b3-y+28'ay+20'aB+2ca+2¢' B+ 2c'y +e)= KE...(9). | But on combining (9) with (6), (7), (8), multiplied by a, B, y respectively, we find (hC+ do) at (KC' + Ac’) B+ (KC + Ae’) y= KE- N@wa«(10), which being linear in a, 9, y may be used instead of (9). These four equations lead to a quadratic equation of two terms for determining /; and if its xoots be possible, we must take only the positive one, since / is supposed to be essentially a positive ratio. This single value of & will give corresponding single 94 SIMILARITY OF SURFACES. values of a, (3, y; and hence there is only one centre of simi- larity corresponding to that originally assumed. (110) It is easy to find the geometrical meaning of the i conditions Ba at po ae Sp pe a Ie mene Mier mac E cemege enes! Ee PY A AD 1 ° AeA eee MEL rep, oy For if we suppose the surface (1) to be referred to rectangular ky co-ordinates parallel to its principal axes, we shall have i= 0, ah Ose) Ba Oe and therefore, in order that the equations (11) may hold good, ) we must have b= 02) 6) = l0get Ae 0 or the surface (2) is also referred to its principal diameters, and hence the two surfaces have their principal axes parallel. : Moreover if, in addition, (1) be referred to its centre as origin, — we should have C=0, C'=0, C'=0, and therefore the seventh, eighth, and ninth ratios of (5) give da+c=0, @Btic=0, a'y+c'=0; so that the point (a, PB, y) is, by Art. (78), the centre of the — surface (2), and therefore the principal axes of the two surfaces H are proportional, since they are corresponding radii. It is © obvious also from equations (11), that similar surfaces must t be of the same species, since, if one of the quantities A, A’, A’ vanish or be negative, the corresponding a, a’, a’ must also vanish or be negative, in order that the equations may subsist. (111) Jf two similar surfaces of the second order cut each y other, their line of intersection is a plane curve. Let the equations to the surfaces be Ax'+ A'y’+ A'2'+2 Byz+2 B'x2+2 B'ry+2Cx+2C'y+2 C'2+ E = 0...(1), | ax + ay’ + a2 + 2byz+ W'xz + 2'xy + Lex 4+ 2e'y+2c'z+e =0 .--(2); then the conditions of similarity are ANA yA) Ba BBS a UGIROT GI) FOS Te Since the two surfaces intersect, we may combine their equa- tions linearly in any way. Multiply then (2) by X, and subtract SIMILARITY OF SURFACES. 95 it from (1); then, by the conditions of similarity, the terms of the second order disappear, and we have . 2(C— dc) 7+ 2(C'- Ac’) y + 2(C"- Ac’) 2+ E— de= 0...(8). This is a relation between z, y, 2, the co-ordinates of any point of the line of intersection of the surfaces, and as it is of the first degree it represents a plane, so that the line of inter- section is a plane curve. As the combination of (1) and (2) leads to only one linear equation, we see that two similar surfaces intersect each other once only. Since all spheres are necessarily similar surfaces, it appears from this that the line of intersection of two spheres is always a plane curve, and therefore a circle. (112) If four similar surfaces intersect each other, the six planes of intersection pass all through one pownt. Let the four equations to the surfaces be u=0, u=0, Uu=0, Wa OTs then, since these may be combined two and two in six different ways, if d,,A,, A, be the ratios of similarity between the first and each of the others, by the preceding proposition the three equations U-rwU,=0, U-AM,=0, U-AU, = 0, are the equations to three planes of intersection: and if we eliminate wu between each pair of these, the equations Aw —A,uw, = 0, Ae, — A,u, = 0, A,u, — Aju, = 0, are the equations to the other three planes of intersection. Now the first three equations combined together determine the point through which the three planes pass; and since the second three are derived from the first, the values of the co-ordinates derived from combining the first three must satisfy the second three: in other words, the six planes of intersection have one common point. (113) In connexion with the preceding propositions we may introduce the following :— If two surfaces of the second degree intersect in a plane curve, their second intersection (when they have one) is also a plane curve. 96 PLANE SECTIONS. Let the equations to the surfaces be Ax’+ A’y’+ A’2*+2 Byz+2B'x2+2B'ry+2 Cr+2 C'y+2 C'2+E=0...(1), ax’ + a'y’ + a'z? + Qbyz+ 2b'xz + 2b'xy + Qex + 2c'y+2c'z+e=0...(2). Since the surfaces intersect in a plane, we may take that plane as the plane of (z, y); and therefore, making z= 0, the _ equations Az’ + A’y’+ 2B xy +2Cr+2C'y+ #H=0.... (8), ax’ + a'y’+ 2b'ry + 2cx + Wly+e=0.... (4), must be identical, since they are both the equations to the line of intersection. This involves the relations A=ia, A’=)ha, B'=)b', C=de, C'=)Ae, E=2e. To find the other intersection, multiply the second equation by d and subtract it from the first; then, in consequence of the preceding relations, we have (A"— da’) 2? + 2(B-Xd) yz+ 2(B' 2b’) wz + 2(C" - deo’) z= 0...(5). This gives a relation between the co-ordinates of the lines of intersection ; and it splits into two linear equations, z=0, and (A"— da") 2+ 2(B- db) y+ 2(B'- yo’) x4+2(C"-rAc) = 0. The former gives the plane of zy, or the plane of the first intersection ; the latter, being of the first degree, is the equation to a plane, and therefore shews that the second intersection is also plane. If the equation (5) were a complete square, or were reduced to 2’= 0, it would imply that the two lines of intersection coincide, or that the one surface circumscribes the other, touching it along the plane curve determined by the intersection of the surface with the plane z = 0. Of Plane Sections. (114) The plane sections of a surface of the second order are of course given by combining the equation to the surface with the equation to a plane, Ax + py + vz = 8, or A(e@-a)+u(y-B)+v(e-y)=0...... (1), if we suppose the plane to pass through a point (a, 8, y). The method of finding the nature of the section indicated in Art (68) PLANE SECTIONS. 97 by transforming co-ordinates is necessarily long and tedious, and it is better to avoid it by making use of the distance from a given point to the surface, as that quantity is independent of the co- ordinate axes. Let the equations to a line passing through a point (a, B, y) be Lo oy Yee Zea. ¥ l m n where 7 is the distance between (a, 3, y) and (a, y, 2); then if this line r lies in the plane (1), which we suppose to be the plane of section, /, m, m must satisfy the condition DET OG ee Pon ae eae mi tD:): Hence, instead of combining the equation to the surface with (1), we may find the nature of the section by combining it with (2) and (3), since we shall then have equations for determining all the values of 7 in the plane section. (115) To show how this may be done, let us first consider how the species of curves of the second order are discriminated in two dimensions. If the equation to these curves be Az + 2Bay + Cy? +2Dze + 2Hy + F= 9, we know that it represents an ellipse, a parabola, or a hyperbola, according as the function B’ - AC is negative, zero, or positive. Now if we substitute for 2 and y their values in terms of _r from the equation »~q y-f = = Ye y m ’ | we have (AP + 2Blm + Cm’) 7’ + &e. = 0, and we sce that the discriminating condition is equivalent to saying that the curve is an ellipse, a parabola, or a hyperbola, according as the coefficient of 7°, when equated to zero, leads to impossible, equal, or possible values of the ratio 7: m or m: 1. ‘In other words, the curve is a hyperbola when the coefficient of 7 may be split into two possible and unequal factors, a parabola when it is a complete square, and an ellipse when it cannot be divided into possible factors. This condition is equally appli- cable in three dimensions. (116) 1st. For central surfaces: let the equation to the sur- ' face be Te DOORS ge «Maree (4). 98 PLANE SECTIONS. Substituting from (2) in this, we have (PP + P'm* + P’n’) 7 + &c. = 0. Now the discriminating condition depends on the nature of the equation PP + Pm + Pr? =0, combined with Ih + mp+ nv = 0; and if between these we eliminate one of the quantities J, m, 7, as 2, we have (PP + PX) P+ (Pr? + Pw) m + 2Prylm = 0, from which we easily find the discriminating condition to be —-v(PP'/ 4 WW ei ed esd FIGR 2 2 2 or -*(5+8+5). In the ellipsoid where P, P’, P’ are all positive, this is essen- tially negative, and therefore all the sections are ellipses, as is otherwise apparent. In the cone, and the two hyperboloids, this function may be either negative, zero, or positive, and hence these surfaces may be cut by planes either in ellipses, parabolas, or hyperbolas. ‘The sections will be parabolic when PtP Pp if we consider P’ as the coefficient which is of a sign differen® from that of the other two. (117) We may show that this cutting plane is always parallel to some position of the generating line of the asymptotic cone of the hyperboloids, or of the cone itself in that surface. For if 1, m,n be the direction-cosines of the generating line, we have, by Art. (80), PP + Pin? — P20) Yor PP= P= Pw; but A, pw, v being the direction-cosines of the plane of section, an Ts v eae ees The former of these equations may for one set of values of J, m,n be satisfied by the system kP'l = P°n + P*m, PLANE SECTIONS. 99 and the latter by the corresponding system TREN r v es ] ji Multiplying together the two left-hand and also the two right- hand equations, we have 1 a Pp" iP" ~1h = ny + Mp + — Np + my, Pe p” De Pp” —~lh=nv + mp - — ny - — my. 1 1 ip? P? Adding them we find 1X + my + nv = 0, showing that the lines of which the direction-cosines are /, m, n and \, », » are perpendicular, and therefore that the plane of which the direction-cosines are \, #, v, 18 parallel to the line [Z, m, x]. Since & is arbitrary, this is true for all the values of the cosines which satisfy the equations. (118) 2nd. For surfaces without a centre: their equation is Pife peak ae — Ca, so that the equations to be combined are Pm + Pn’ =0, Dh + mp+ nv = 0. Eliminating 2 between these, we find the discriminating function to be mA? —~ PP’ -;. p When P and FP’ are of the same sign, that is, in the elliptic para- boloid, this can never be positive, and consequently the surface is never cut by a plane in a hyperbola. The section will be a parabola if \ = 0, that is, if the plane be parallel to the axis of z. In the hyperbolic paraboloid, when P and P" are of contrary signs, the function is essentially positive, except when A = 0, or all plane sections are hyperbolas except those made by planes parallel to the axis of x, which are parabolas. (119) 8rd. For cylindrical surfaces. The equation to these may be assumed to be Pe’ + Py? + 2Qzy + 2Rx + 2y + T= 0, H 2 100 PLANE SECTIONS. and the equations to be considered are PP + Pm? + 2Qim = 0, n+ mp+nv =0, from which, after eliminating m, we find, as the discriminating function, Veaarr. Tp aes consequently the section is of the same kind as the base of the cylinder of which the discriminating function is Q*— PP’, Hence a cylinder can be cut by a plane in only one kind of curve 5 excepting of course when the cutting plane is parallel to the axis of the cylinder, in which case the section consists of two straight lines. (120) The sections made by parallel planes in a surface of the second order are all similar curves. It is easy to show, by the same method as that used in the case of surfaces, that two curves of the second order are similar, when, if z, y, z be replaced by their values in terms of J, m,n and 7, so as to give an equation of the form (AP + 2Blm + Cm’) 7’ + &c. = 0, the coefficients A, B, C in the two curves are proportional. Now if the equation to the cutting plane be A(z -a)t+ p(y - B)+ P(Z—y)=0 wise ceess Ci and those to any radius oe ¢_y5 8 _2-1_, SRO © (2), we must have INR eRe vi OL ots ee eres (3). But if the equation to the surface be Pe + P+ Pie = ........5. (4), we must combine (3) with (PP + Pm? + P'n’) 7 + &e. =0...... (8). Since then a, 3, y do not appear in the coefficient of 7’, the terms in that function remain the same for all values of a, £, y, provided A, p, v remain unaltered, that is, for all parallel planes ; hence the coefficients of the terms which multiply 7” are propor- tional in different positions of the cutting plane, or the curves of section are similar. PLANE SECTIONS. 101 (121) To find the locus of the centres of sections of a surface of the second order made by a series of parallel planes. If the surface be central, let its equation be P2228 eT ae a EAR O a A OOF and if a, [3, y be the centre of any given section let the equation to the cutting plane be l(x-a)+m(y-B)+n(e-y)=0.... (2). Then as ie PCB a Rana Ae} (3), A pt Vv subject to the condition IE me nive 0) bees oo tA); are the equations to a line lying in the plane of section and passing through its centre. Hence if we substitute in the equa- tion to the surface the values of z, y, z from (8), we shall obtain a quadratic equation in r, the two roots of which are equal but of opposite signs. Therefore the coefficient of the second term must vanish, which gives the condition Panitele Die ryirs O. 6 cba. as (5); the equation (5) subsists for all values of i, », v, subject to only one condition (4): hence we have kb= Pa, k= PS. in= Py, & being an indeterminate multiplier: from which we have Rate aioe Dy _as the equations connecting a, (3, y, showing that the locus of the centres is a straight line, in fact the diameter which is conjugate to the central plane section: see Art. (99). If the surface be not central, its equation is 2 2 Zz ie instead of equation (5), we have then | Combining this with (4), we have kl=-~ 1, eee kn =; P iL 102 PLANE SECTIONS. from which we get Bm . —-+—=0, ++ Dee Py s as the equations to the locus of centres, which is therefore a line parallel to the axis, and hence a diameter. (122) To find the axes of a section of the ellipsoid 2 2 2 +t aol Rr ee eh (1) made by a plane le +My Ne = 07, oalsta eee (2), l,m, n being the direction-cosines of the plane. Since the semi-axes of the elliptical section are the greatest and least radii drawn from the centre to the curve, we must have = 0 rP=aerty +z a maximum or minimum, 2, y, z being subject to the conditions (1) and (2). Hence differentiating, the condition for a maxi- mum or minimum gives us POL YOY ce — 0 eee ae (3); with the conditions xdee ydy i zdz_ . (4 = par Tie ghee OU ene Unga )s idx + miley. nde — Ol san aye Multiply (4) by an indeterminate multiplier \, and (5) by u, and add them to (8); then, equating to zero the coefficients of — each differential, we have x +A 7+ wl=0 a Y r Bi }? +um= 0 2 2+A>+un=0. c Multiply by 2, y, 2, and add: then, in virtue of (1) and (2), r’+h=0. Hence the equations become 7” ve ; (1 - Seal 0, (1 -T)y + wm = 0, (1-5 )2+yn=05 whence L= CIRCULAR SECTIONS. 103 Multiply by 7, m,, and add: then, in virtue of (2), and dividing by u, we find aly b’m* on® fa. a quadratic equation for determining 7*. This equation may be adapted to the other central surfaces by changing the sign of one or of two of the quantities a’, 3°, c’. From this expression we can easily determine the area of the section. For the area of an ellipse is equal to the product of the semi-axes multiplied by 7: but the last term of the preceding equation (arranged in powers of 7’) is the product of the two roots, that is, of the squares of the two semi-axes. ‘Taking then the square root of this term, and multiplying it by 7, we have ae AY >—— =, oat 7" Me tse d wabe re (al + Om + c'n’y as the expression for the area of the section. Circular Sections. (123) Since all the surfaces of the second order, except the hyperbolic paraboloid and the hyperbolic and parabolic cylin- ders, give, when cut by a plane in certain directions, curves which are closed, and consequently must be ellipses, we may enquire whether under any circumstances these sections are circular. And as all parallel sections of a surface of the second order are similar curves, we have only to consider the direction of the section, choosing its position in the manner which may be most convenient. (124) Central surfaces. The equation to these is, in general, ANON Bel party 0 AC «Re a where fal Ne Wet Let the surface be cut by a plane whose equation 1s Pima TU Ue orc e + « Serv yp it being assumed to pass through the origin: we have to deter- mine whether there are any values of m and m which give 104 CIRCULAR SECTIONS. a circle as the curve of section. Let the plane (2) also cut a sphere, the equation to which is Nk) Be oe aed A cr Ena then, if the section of (1) be circular for any values of m and x, we can always assume such a value of 7 that the section of the sphere, which is always a circle, shall coincide with the section of (1). If then they coincide, their projections on the plane of (z, y) must coincide, and the corresponding equations become identical. ‘The comparison of the several terms of these will give conditions for determining m and. Substituting then for 2 im (1) and (8) its value from (2), we get (P+ Pm’) 2+ (P+ Pn’) y’ + 2P’mn zy = H, and (1 + m’*) 2+ (14+ n*) y’ + 2mn zy = 7’, as the equations to the projections. As these are to be identical, the coefficients of the several terms must be proportional, and therefore P+Pm 14m PsP? 1427 P’mn mn = ely 2 fae, “y po ON Fae = Pp? Ht “9 oe The last condition can be satisfied only by m=Q0, or n=90. For m= 0 we have from the other conditions aes and BMS ie Se he lied ayn gidemmaMl are wae) A J ihewe nl gf ec (B-a\ whence n=z(5—p| a (5) fete ae (4). For n = 0, we find similarly | P-P\ e (a -b\ m=+(5—5) oe 3) 4 Pals (5). (125) We proceed to consider how far these values are possi- ble in the different surfaces. | The Ellipsoid. In this all the quantities P are positive, and we shall suppose P< P< FP’, which is the same as a>6> ce. In this case the formula (4) is impossible; and from (5) we have CIRCULAR SECTIONS. 105 which indicates the existence of two series of circular sections, parallel to the planes, of which the equations are a (Bc) 2 - c (a’- BY 2 = 0, a(B- ey 2 +¢e(a— By 2 = 0. From the form of these it is evident that the two planes pass through the mean axis 0, and are perpendicular to the plane containing the greatest and least axes. If P= FP’, or a=), the equations to the cutting planes are reduced to 2=0, shewing that all the sections parallel to the plane of (z, y), or that containing the two equal axes, are circular, or the ellipsoid is one of revolution round the axis of z. If P’= FP’, or b=c, we have x=0, or the planes parallel to (y, 2) give circular sections, and the surface is one of revolution round the axis of z. If P=P’=FP’, or a=b=c, the expres- sions for m and » are indeterminate, or there are an infinite number of directions in which the surface may be cut by planes in circles. This indeed is obvious, as the ellipsoid then becomes a sphere. Hyperboloid of one sheet. In this one of the coefficients is negative, as P”, and then the formula (5) gives rived A S Ys me Ose ae and, in order that this may be possible, we must have }> a, or the circular sections pass through the greater of the real axes. To make the surface one of revolution, we can only have P= FP’, or a=), and there results only one series of circular sections parallel to the plane containing the two equal axes. Hyperboloid of two sheets. In this case P’ and P" are both negative, and c e a a I+ Toh ie In order that this may be possible, we must have 06>, or the * cutting plane passes through the greater of the two imaginary axes. The surface becomes one of revolution only when P’ = P", or - 6=c; and then, as m=00, the circular sections are perpendicular to the real axis of the surface. It is to be observed, that the plane which passes through the centre never meets the surface, but planes drawn parallel to it at a sufficient distance cut the surface in circles. 106 CIRCULAR SECTIONS. Conical surfaces. 'The equation to surfaces of this kind are derived from that of the hyperboloid of one sheet, by making H=0; but as this quantity does not enter into the expressions for m, the circular sections of the cone are parallel to those of the hyperboloid, of which it is the asymptote. Cylindrical surfaces. 'The elliptic cylinder may be taken as a particular case of the ellipsoid when one of the quantities, as P, vanishes: the value of m then becomes C A Deine which is possible if 6>c, or the cutting plane passes through the greater axis of the elliptic base of the cylinder. (126) Surface without a centre. For the elliptic paraboloid the formula is somewhat different. Let the equation to the m=rH+ surface be py? + pe = pp'e; and let this be cut by a plane L= met ny, which also cuts the sphere e+y +2 = re. The equations to the projections on the plane of (y, z) are py + pe — mpp'2 — npp’y = (l+n)y?+(1+ m’) 2 + 2mn yz — 2mrz -— Ary = 0. In order that these may coincide, we must have m=0 or n= 0. If m= 0, then Rey ere and n=a(? =e P P If ~ = 0, then 1 mak, and mai (PoP), ~P P If p'>p, the first is possible and the second impossible ; so that the equation to the cutting plane is put (p-p)y=9, giving two sets of circular sections parallel to planes which pass through the axis of z. If p>p’, the second formula is possible, and the equation to the cutting plane becomes px ¥(p-pyz=9, CIRCULAR SECTIONS. 107 showing that there are two series of circular sections parallel to planes passing through the axis of y. If p =p’, the equation to the cutting plane is z=0, or the circular sections are all parallel to the plane of (y, z), and the surface is one of revolution round the axis of z. (127) The following proposition is worthy of note :—Any two circular sections belonging to different series lie on the surface of the same sphere. If the equation to the surface be Dat ote Pe td 0, SEA (1), that to the plane of any circular section of one system is UP a patae 3 ye (2), and that to one belonging to the other system is (ee P ye (CE Pye oe nee (3). On multiplying these together, we have (P'- P’)2-(P'- P)2-(D+D,)(P'- Py 2-(D-D)(P- Py =z el) 1) eee. (4), an equation of the second order representing the two cutting planes. Since these planes intersect the surface (1), any equa- tion derived from combining linearly (1) and (4) is the equation to a surface which passes through the intersections of (1) and (4), that is, the circular sections. But on subtracting (4) from (1), we have P(2+y'+2)+(D+D,) (P'-P'Yz+(D-D,) (P'-P)«- DD,-H=0, which is the equation to a sphere; hence the two circular sections are on the same sphere. If the equation to the surface be py +p2—- pyre=0...... a a a Gop those of two planes of circular sections belonging to different Peau Are St aNAK Dt py ycid = 0. K0.a i. MEBs pe +(p'-p)yy-4,=0. oe oe OED Ng whence, as before, pa'—(p'—p)y’-(d+ d,)p'« -(d—d,)(p'-pyy + dd,= 0... .(8). 108 SURFACES OF REVOLUTION. On adding (5) and (8), we find i i p(vry'+2)—-{(d+d,)p’ + pp'} x -(d-d,)(p'-p)’y + dd, =0, the equation to a sphere on which lie the two circular sections. (128) From what has preceded, it appears that all surfaces of the second order, except the hyperbolic paraboloid, and hyperbolic and parabolic cylinders, may be generated by the motion of a circle of variable radius which moves so as to be always parallel to one plane. When the surface is of revolution, the plane of the circle is perpendicular to the line of the centres of the sections, but in other cases it is oblique, as may be easily seen ; for, by Art. (121), the equation to the line of centres of parallel sections is the diameter conjugate to them, and this can never be perpendicular to their planes, unless it be a principal con- jugate diameter, which is the case only in surfaces of revolu- tion. Conditions that the Equation of the Second Degree shall represent Surfaces of Revolution. (129) When asurface is one of revolution, all the sections made by planes perpendicular to the axis are circles, of which the cen- tres are on the axis ; and as any line which, passing through the centre of a circle, bisects a line which does not pass through the centre, is also perpendicular to it, any plane passing through the axis and bisecting a system of parallel chords must be also perpendicular to them. In other words, it is a principal plane to a system of chords perpendicular to the axis. Hence, in surfaces of revolution, there are an infinite number of principal planes, the conjugate chords of which are parallel to one plane ; and, conversely, if we investigate the condition that a surface of the second order may admit of an infinite number of principal planes, the chords conjugate to which are parallel to one plane, we shall obtain the condition that it may be a surface of revolu- tion. It is necessary, however, to add that the principal planes are at a finite and determinate distance, since in surfaces of revolution they all intersect in the axis. SURFACES OF REVOLUTION. 109 (130) The direction-cosines of a system of chords conjugate to a principal diameter are, by Art. (100), given by the equations ~~ 4)%,-4)-F 2 _P-4AG,-4)-# UP Py) (Pe PP) ; CPP) CPs 2) ‘ yp, ata A Ned lad sath sith Pertenece | 1) C2) Es) Clerea 2s) P,, P,, P, being the three roots of the discriminating cubic. In order that there may be an infinite number of such systems of chords, these expressions must become indeterminate, which they will be if both the numerators and denominators of each vanish. ‘The denominators can be made to vanish only by making P, = P, or P, = P,, that is, by making two roots of the discriminating cubic equal. ‘The three numerators equated to zero give (P,- 4) (P,-A4)-B=0, (P,-A)(P,-4)-B’ =9, Pietra Oi nw. Li oats «Clete ai a Paar” © B These equations are not inconsistent with the preceding con- dition, for it will be seen that they satisfy the equation aU Wee (8), which is the condition that the discriminating cubic U = 0 shall have equal roots. The equations (2) combined with U=0, or Beep 4) (P44) (P.-A)- BYP 4)- BY(P,-A - 2 BBB = lead to B(P, - A)= B’ (P, - A’)= B? (P,- 4) =- BBB’; whence ee eee ne (4). B B B These equations give two relations between the coefficients of the general equation of the second degree, which must be satisfied in order that the surface may be of revolution. But we must add the condition that the principal planes correspond- ing to the chords determined by /, m, » are at a finite distance ; this, by Art. (100), is expressed by saying that P, shall not vanish, or that the two equal roots of the cubic shall not be zero. 110 SURFACES OF REVOLUTION. Hence, the condition for a surface being one of revolution may be written in the form BB } : 4~ a = A a =A — ar < (131) Since, by the first of equations (8), of Art. (73), the direction-cosines of the chords are subject to the condition (P,-A)l- Bm- Bn=0, the preceding equations (4) change it into BB+ BB mv BBi ole aaaate (6); and if 7a 7-- Lim be the equations to a line passing through the origin and parallel to the chords, the elimination of 7, m, between these and equation (6) gives BB'x+ BB'y+ BBz=0, or = ~ en + = = "05 Cr) as the equation to a plane parallel to all the chords, and there- fore perpendicular to the axis of the surface. To find the equation to this line we have to consider that it is the. inter- section of all the planes which bisect all chords parallel to (7). Now the equation to the plane which bisects all chords parallel to a line [7, m, m] is, by Art. (96), (Al+B'm+B'n)2+(B1+A'm+ Bn) y+ (Bl+ Bm+ An) z + Gi Cm+ On = 00 2(8)5 and when it is also perpendicular to the chord, we have, by Art. (100), Al+ B'm+ Bin= Sl, B+ A'm+ Bn=Sm, Bl+ Bm+ A'n=Sn; but, as we have seen, Sis given by the same cubic as P, and consequently, instead of S we may put any one root as Bes by doing so, equation (8) becomes (Px+Cyl+(Pyt+ C)m+ (Petr O)n=0... ol on: The axis of revolution is the line of intersection of all the planes given by this equation, when J, m, n vary, subject to the condition BBE BB mt BBuS 0). ae (6). As there is only one relation between 1, m, n, two of them must be indeterminate, so that if we eliminate one between (6) and SURFACES OF REVOLUTION. 111 (9), the coefficients of the others must vanish separately, whence we get the equations Pz+C iG aot CG eee Boe BB’ or, putting for P, its values from (4), and multiplying by BB'B’, we have BC , BG f. » ‘Be Ge Bape)? (+ ape)? Gee which are the required equations to the axis, showing it to be perpendicular to the plane (7)- (132) If the general equation of the second degree is defi- cient in two of the terms involving the products of the variables, that is, if two of the quantities B vanish, the conditions (5) become indeterminate and therefore nugatory. Let the two which vanish be B and B’; then if between the two equations B B ihe date hey aaa : A'-—A B A’ = A’ 7 Be SAG bik) we eliminate the indeterminate ratio = , we have C18 = 1tA)) CAG AS ee tl 2) as the condition that the surface may be of revolution; but we must add to it that A” shall not vanish, since when B= 0 and B= 0, A” is the value of P.. The equations to the axis of revolution (10) become, in this Raw, Tab case, after eliminating from them the ratio 7 by means of (11), xL + Serie apes ait 0 (138) Vitae aa y vie Wisse s elere . If, im addition, B’ = 0, or all the terms involving the products of the variables be wanting, the condition (12) becomes A aR OLE Ae A 9 8, 0 es (14); and from symmetry we may add A= A’. Any one of these conditions being satisfied will make the surface be of revolution. Taking the first, the equations to the axis of revolution are ae Oe Ome A te Gi 0 (15), with similar expressions in the other cases. 112 RECTILINEAR GENERATING LINES. Rectilinear Generating Lines. (133) We have seen before, that cones may be generated by the motion of a straight line which passes constantly through a fixed point ; and cylinders by that of a straight line which moves so as to be always parallel to a given position. This suggests the enquiry, whether any of the other surfaces of the second order may be generated in a similar manner. It is evident, d prior, that this is impossible for, the ellipsoid, since it is a closed curve ; for the hyperboloid of two sheets, since it is not a continuous surface; and for the elliptic paraboloid, since it is bounded in one direction. We may therefore confine our attention to the hyperboloid of one sheet, and the hyperbolic paraboloid. (134) The equation to the former is 2 2 2 Dy haps ale ae es See our (1), which may be written in the form* oe 2 y” ai wos =1- Wee aaa (2) Now this equation may be satisfied by either of the following systems of linear equations : i being an arbitrary constant. Each of these equations is the equation to a plane, and therefore each system represents a straight line. Since then the equation (1) or (2) may be satis- fied by the equations to two straight lines for each value of 4, there are two straight lines which lie wholly in the surface repre- sented by (1). As 4 admits of an indefinite number of values, and to each value of & there corresponds a position of the line in each system, we may, by assigning a proper series of values to 4, cause the line represented by either (A) or (B) to trace out the surface (1). Hence there are two ways in which the hyperboloid of one * This method is due to Bobillier, Correspondance Mathématique et Physique de Bruwelles, vol. 1v. RECTILINEAR GENERATORS. 113 sheet may be generated by the motion of a straight line, the one corresponding to equations (A), the other to (B). (135) It is easy to find the condition to which must be subject the direction-cosines of the generator: for if its equations be and we substitute for z, y, z their values in J, m, » and 7, the resulting equation in 7 must be indeterminate, since the line lies wholly in the surface, and therefore the coefficient of each term must vanish separately: that of r° gives us the equation a % m i n? ae Oia hence the generators are all parallel to the generators of the asymptotic cone (Art. 87). From this it appears, that the gene- rator is never in three positions parallel to the same plane; for if this were the case, the three corresponding generators of the cone would lie in one plane, which is clearly impossible, as a cone of the second degree cannot be cut by a plane in more than two straight lines. (136) There is an important difference between the mode of generation of this surface and that of cones and cylinders. For in the latter, the generating lines either pass through one point or are parallel, and consequently any two lie always in one plane: but in the hyperboloid, if we consider one system of generators as (A), and take any two individuals of the system, as Beef) UCC H( ‘), 114 RECTILINEAR GENERATORS. we find that the condition of their intersection given in equation (5) of Art. (30) leads to the equation (k — ky = 0, which cannot be satisfied so long as & and #’ are different : in other words no two lines of the system (A) ever intersect. The same may be shewn of the system (B), and consequently no two generators of the hyperboloid of one sheet are ever in the same plane. Surfaces generated in this manner have been termed by French writers, who first studied their properties, “ surfaces gauches”: perhaps the nearest equivalent expression in English is “skew surfaces,” and that term we shall use for the future. Surfaces which can be generated by the motion of a straight line are called ruled surfaces, and are divided into the two classes of skew surfaces, of which the hyperboloid of one sheet is the type ; and developable surfaces, of which the cone may be taken as the type. The reason of the term developable, and the nature of the distinction between these two classes of surfaces, will be explained in the chapter on Tangent Planes to Surfaces. (1387) On the other hand it will be readily seen, that every line of the system (A) meets every line of the system (B), since the condition of their intersection leads to an identical equation which is satisfied whatever be the values assigned to /& in each equation. This leads to a very simple geometrical mode of regulating the motion of the generating line: for if we take any three lines of the system (B), the motion of a straight line will be completely regulated by constraining it to intersect these three lines. ‘This will appear more clearly from the following con- siderations: if B,, B,, B, be any three lines of the system (B), and if at any point in (B,) we draw two planes, one passing through (B,) and the other through (B,), their line of intersec- tion rests both on (B,) and (B,), and of course also on (B,), as the two planes pass through the same point in that line. For each point in (B,) there is only one such line of intersection, and con- sequently a line, which is made to pass through any point in (B,), is completely determined by being constrained to rest on (B,) and (B,), so that the motion of a line is completely regulated by being constrained to pass constantly through three given straight RECTILINEAR GENERATORS. 115 lines. It is to be observed that the three directors (B,), (B,), (B,) must not be all parallel to one plane, since they are taken out of a system of generating lines of the hyperboloid, no three of which, by Art. (135), can be parallel to the same plane. ‘This serves to distinguish this surface from the hyperbolic paraboloid, as we shall see presently. We may of course equally well take any three lines of the system (A) as directors, and constrain the generator to rest always on them. Hence the hyperboloid of one sheet admits of two modes of generation by the motion of a straight line, the directors in the one mode being some of the generators in the other. (138) Having thus shown that the hyperboloid of one sheet may be generated by the motion of a straight line which rests on three rectilinear directors which do not intersect, and are not all parallel to the same plane, it remains to prove that it is the only surface so generated. For this purpose it is convenient to choose our co-ordinate axes as symmetrically as possible with reference to the three directors, and that is to draw them parallel to these lines, which is always possible as the three directors are supposed to be no two in the same plane. The following construction gives us the means of doing this. Let B, B,, B, (fig. 24) be the three directors. Through B draw a plane BCD parallel to B,; through B, a plane BEF also parallel to B,. ‘These planes must evidently intersect in a line A, parallel to B,. In like manner through B and B, draw planes parallel to B,, and through B, and B, planes parallel to B. These six planes form a parallelopiped, at the centre of which, O, we shall place our origin, the axes of z, y and z being parallel to B, B,and B,. Let the sides of three contiguous edges of the parallelopiped be 2a, 23, 2y, parallel to x, y, z respectively. Then the equations to the three directors are CB) (y=+B, (B) fz=+7, iaigetoake tn ee “c=--a, y=- [. If now the equations to the generating line be fiat iN th Mls Pir 2 1 m Fimo 116 RECTILINEAR GENERATORS. the conditions that it shall pass through (B), (B,), (B,) respec-@ tively are y-B_ zoey | m n eet aot Fite iat zw—-a y+ cue ee On multiplying these three equations together, /, m, are elimi- _ nated, and we have (c-a)(y-B)(@-y=@+a) (y+ B) (+ y), which may be reduced to ayz + Pxz + yay + aBy = 0, the equation to the locus. Since this equation is not altered when the signs of the variables are changed, the surface has a centre and is therefore the hyperboloid of one sheet, that being the only central surface of the second order which admits of a rectilinear generator. The form of the equation will be seen to be analogous to that of the hyperbola referred to its asymptotes; of which the rea- son is obvious, for the axes, passing through the centre and being parallel to three lines which he wholly in the surface, are in fact three positions of the generating line of the asymptotic cone. (139) The equation to the hyperbolic paraboloid is 2 2 ¥ _ ea = XL, ?P and this may be satisfied by either of the two systems of linear _ equations 2 1 Z_= ' f+ age beateiee (A), Pp p Pp p- ee fs Ue Ss el i ar eae, Pei tet Bite eecececee (B); Pigvel PoP and consequently, as in the case of the hyperboloid, for every value of & there are two straight lines which lie wholly in the SR ie snes. jane RECTILINEAR GENERATORS. LG surface. Hence, by assigning all possible values to 4, we can obtain from either system a consecutive series of positions of a straight line which lies wholly in the paraboloid: or there are two modes by which this surface may be generated by the motion of a straight line. From the left hand equation of the two systems it Agee that ine generator # ee parallel to a fixed plane py - pz = 0, or p"y + pe =, according as it belongs to (A) or (B). This serves to distinguish the surface from the hyperboloid, of which we saw (Art. 135) that no three generators are parallel to the same plane. (140) It is easy to shew, as in Art. (136), that no two lines of the same system ever intersect, so that the hyperbolic paraboloid is a skew surface, but that every line of the one system inter- sects all the lines of the other. Hence the motion of the gene- rator in one system will be completely regulated if we constrain it to rest constantly on three of the lines of the other system considered as directors: these lines are not arbitrary, but must be taken parallel to one plane, since all the lines in each system (A) and (B) are parallel to one plane. We may also consider the surface as generated by the motion of a straight line which rests constantly on ¢wo rectilinear directors, while it remains parallel to one plane. These conditions will regulate completely the motion of the generator ; for if the two director-lines be cut by a plane parallel to the director-plane, the two points of inter- section will determine the position of the generator, and for every parallel plane there is only one such position. (141) Let us now shew that a line subject to the geometrical conditions of resting on two given straight lines, while it remains parallel to a fixed plane, will trace out the hyperbolic paraboloid. For convenience we shall use oblique co-ordinates, taking the fixed plane as that of (z, y), the axis of y as passing through the points where the director-lines meet that plane, the origin bisect- ing the line joining the points, the plane of (x, z) parallel to the director-lines, and the axis of z equally inclined to them. ‘The equations to the directors will then be Jean; Reise she. S309 ah oil) Pree NSO Re AZ IUNS ..e. ai jain C2), 118 RECTILINEAR GENERATORS. Since the generating line is parallel to the plane of (2, y) its equations are t-a y-B BES Wrery th Five ore lu The conditions that it shall pass through (1) and (2) give Wi Ph aig ale ln a ul rv a whence eliminating A and pu, we have ayy = hx, and as y = 2, the final equation is ayz = hx. This is of the second degree, and as it involves x and not 2”, it is a surface without a centre ; and as it cannot be the elliptic para- boloid, or parabolic cylinder, it must be the hyperbolic paraboloid. (142) There is a very simple method of constructing practi- cally the hyperbolic paraboloid, which we may here notice. Since through every position of the generating line we may draw a plane parallel to the director-plane, and since parallel planes cut any two lines proportionally, it follows that the generators cut the director-lines proportionally. Consequently, — if we take any two finite straight lines not in the same plane, and divide them into the same number of equal parts, lines or _ threads joining the points of division will form a portion of a hyperbolic paraboloid. fos Vas) CHAPTER VII. OF CURVES IN SPACE. (143) In the first chapter it was shown that a curve con- sidered as existing in space is represented by two equations between the three co-ordinates; these equations being the equations to any two surfaces which, by their intersection, determine the given curve. Also in Art. (44) it was shown that the straight line is the only locus given by the intersection of two planes, that is, of two surfaces of the first degree: in the present chapter we shall briefly consider the nature of curves of a higher order, confining our attention chiefly to those deter- mined by the intersection of the surfaces of the second degree ; but we shall first premise some general remarks on curves in space. ‘These are naturally divided into the two classes—those which lie wholly in one plane, or plane curves, and those which do not lie wholly in one plane, or curves of double curvature, as they are called. The former may always be considered as determined by the intersection of some surface with a plane, and their properties are most easily studied by considering them as existing in two dimensions only; so that it is unnecessary to treat of them here. (144) It is to be remarked, however, that the most general equations to a plane curve in space are not—the general equation to a surface of the same degree, and the equation to a plane: these would involve too many arbitrary constants. or, as was shown in Art (9), a cylindrical surface may always be supposed to pass through the intersection of any two surfaces: and if we assume the generating line of the cylinder to be parallel to one of the axes, the equation to this cylinder, combined with the equation to a plane, will determine the plane curve in question. Now this cylinder is of the same degree as the given plane 120 CURVES IN SPACE. curve, for it is easy to see that all plane sections of a cylinder are of the same degree. Hence, the equations to a plane curve of any degree in space are perfectly general if we combine the equation to a cylindrical surface of the same order, parallel to one of the co-ordinate axes, with the equation toa plane. Thus, in order to obtain the general equation to a plane curve of the second degree, it is not necessary to take the general equation to surfaces of that order containing nine arbitrary constants, and the equation to a plane containing three constants, making twelve in all; but it is sufficient to combine the equation to the plane with that to a cylindrical surface of the second order, which, if it be parallel to one of the co-ordinate axes, contains five con- stants only: the total number therefore of disposable constants in the equations to a plane curve of the second degree in space is eight only. (145) It is not difficult to find an analytical condition by which to distinguish between plane curves and those of double curvature. For, as has been said, a plane curve may be con- sidered as represented by the general equation to a plane G2 +, OY. 4102 =. nisisnas sfersed 1)s combined with the equation to some cylinder which we may write in the form Sf (26 yy eS Cees. i eee These two equations leave one of the variables independent, of which the other two may be considered as functions. If therefore we differentiate equation (1), considering two of the variables as functions of the third, we may, by means of the resulting equations, eliminate the constants which determine the particular plane, and so obtain a relation between z, y, and z which is common to all plane curves. For the sake of sym- metry, we may, in differentiating, consider each variable as independent, and introduce the condition of dependency after- wards. After three differentiations we have adz + bdy + cdz = 0, ad*z + bd’y + cd’z = 0, ad*x + bd*y + cd®*z = 0; eliminating a, b, ¢ by cross-multiplication, we find d°a(dy d*z— dz dy) + dy (dz d*x - dx d*z)+d°z (dx d’y — dyd*x)=0 CURVES IN SPACE. 121 as the required condition. If xz be considered as the indepen- dent variable of which y and z are functions, d*x = 0, d*x = 0, and the preceding relation becomes dai dv dx’ dx (146) The intersection of two surfaces of the second degree is in general a line of which the projection on any plane is a curve of the fourth degree. To this proposition, however, there are exceptions which we shall consider. When two surfaces of the second order have a common princi- pal plane, the line of their intersection is projected on this plane in a curve of the second degree. If we assume the plane of (x, z) to be the principal plane common to the two surfaces, their equations cannot contain odd powers of y, and will there- fore be of the form | Az’ + Aly’? + A’e’ + 2B’rz + 2Cr + 2024+ H=0, aw +ay’?+ a2) + Wez+ 2x+A2le+e=0; if we multiply the first by a’, and the second by 4’, and sub- tract, we have (dad -— A'a) 2’ + (A'ad - A’) 2’ + 2 (Ba - AD) xz +2(Ca' —- A’c)x+2(C'a - A’c') z+ Ka - A’'e= 0, which does not contain y, and is therefore the equation to the projection of the line of intersection on the plane of (#, z) or the principal plane, and is of the second degree. This proposition evidently includes the intersection of similar surfaces, which by Art. (111) is a plane curve. (147) The projection of the intersection of two surfaces of the second degree may be a curve of the third degree, as is seen in the following remarkable proposition, due to M. Quetelet.* All plane curves of the third degree are the pro- Jjections of the curves of intersection of surfaces of the second degree. The general equation of the third degree in two variables is ax’ + bay’ + b.a’y + ay + cx? +dzy+cy’ +ex+ey+ f= 0...(1). * Correspondance Mathématique et Physique de Bruxelles. 122 CURVES IN. SPACE. Now if we change the direction of the co-ordinates by the for- mule r= lx, — my); Y = MZ, + ly,, the coefficients of the terms of the third degree in the trans- formed equation will involve 7 and m in the third degree, and therefore any one of them equated to zero must give a possible value for the ratio 7: m; that is, it is always possible to trans- form the co-ordinates so as to deprive the equation (1) of one of the terms of the third degree. Let this be the term involving y°; then the equation will be in the form x (ax’ + by’ + b.xy) + cx? + day+ cy’ +ex+ey+f=0.... (2). This evidently may be considered as the result of the elimination of z between OLA DY: Aeley lees, Ae eee (3), and cx’ + day+uze+eoy+err+eyr+f=0.... (4). But (38) and (4) are the equations to two surfaces of the second degree, and (2) is the equation to the projection on (z,y) of their curve of intersection ; consequently all the curves repre- sented by (2), and hence by (1), or all plane curves of the third degree, may be considered as the projections of the intersection of two surfaces of the second degree. We may remark that the equation (3) may, by changing the co-ordinates z and y in their © own plane, without altering z, be put in the form Az’ + By’ =z, from which we see that it represents one of the paraboloids, the axis of the surface being perpendicular to the plane of projection. (148) Moreover the curve of the fourth degree, in which is projected the intersection of two surfaces of the second degree, may sometimes be split into two equations of lower degrees. Thus if, as in the theorem of Art. (113), the two inter- sections are plane curves, the curve of the fourth degree may be divided into two curves of the second degree. In such a case one of the surfaces of the second degree may be replaced by a system of two planes, which by Arts. (92) and (94) may be con- sidered as a particular case of surfaces of the second order. CURVES IN SPACE. 128 (149) If the equation to a surface of the second order be given, it is easy to assign the equation to the surface which shall intersect it in two given planes. For if COE ROR Gotta swhale sails CL) be the equation to the surface, and Oe Oat me See euetiOn ee ot): 2) be the equations to the given planes, the equation Te OTRAS carer OA CDE \ being any constant, is the equation to the surface required. This is easily seen on combining equations (1) and (3) by sub- traction, for we then get yy = 0... 0... ec cece ee ee (ADS which is satisfied either by u,=0, orby »,=0; that is, the planes represented by these equations pass through the intersections of the surfaces represented by (1) and (3). If the surfaces, instead of intersecting in two plane curves, touch each other along one plane curve, the equations to the two planes must become the same, or wv, = v,. Hence the equation URL TNE SAA ecg ea ER is the equation to a surface of the second order which is in- scribed in, or circumscribed about, the surface of which the equation is u,= 0, the equation to the plane of contact being u,= 0. Again, if v, = 0 be the equation to a plane in which the surface (1) is touched by some other surface of the second order, the equation to the latter is u,+ wv, = 0 pane yt ine ei Ole Now if we suppose the surfaces (5) and (6) to intersect, we have, on combining their equations by subtraction, NEE nO Soares ists es Nes onema la CT), which is satisfied by | 1 Nw, ~ Pig =0, or Neu, Hee UO re (8), and these, being linear equations, represent two planes. Hence it appears that if two surfaces of the second order be inscribed in, or circumscribed about, another surface of that order, their lines of intersection, when they cut each other, are plane curves. 124 CURVES IN SPACE. (150) Again, if two ruled surfaces of the second order inter- sect along a generating line common to both, the curve of intersection will be projected on any plane in a straight line which is of the first degree, and another curve which must be in general one of the third degree. Thus, for example, if the cone of which the equation is 2-m (“’+y’°)=0 be cut by the cylinder of which the equation is z* +m (x + y*) — 2mxz + maz — ma (« + y) = 0, one line of intersection is the straight line of which the equations aS y=0, z2-mz=0, and which is projected on the plane of (z, y) along the axis of x, while the other line of intersection is projected on the plane of (x, y) in a curve of the third order of which the equation is 2y-a)\(e+y)+e@x=0. (151) After plane curves the most interesting class consists of those which can be drawn on the surface of a sphere. These are of course determined by the intersection of a sphere with some other surface depending on the nature of the curve in question ; the equation to the surface being deduced from the definition of the curve. The best method of studying the pro- perties of such curves is however, not by referring them to three co-ordinates in space, but to two curvilinear co-ordinates on the surface of the sphere ; a method closely resembling the co-ordinate geometry of plane curves. It would be out of place here to explain this method, and it will be sufficient to refer the reader to the original memoirs on the subject, which are by Gudermann in Crelle’s Journal, Band. vi. and x111.; Davies in the Edinburgh Transactions, vol. x11.; and Graves in an Ap- pendix to a Translation of two Memoirs by M. Chasles : this last work in particular I would recommend, as the method there pursued is the most symmetrical and elegant. In the present place I shall content myself with shewing how, from the defini- tions of some of these curves, we may deduce the equations to the surfaces which intersect the sphere. CURVES IN SPACE. 125 (152) To find the equations to the equable spherical sprral. This curve is defined in the following manner :—If a meridian PRP’ (fig. 25) on a sphere revolve uniformly about an axis PF’, which is a diameter of the sphere, while a point JZ moves uniformly along the meridian from P to P’, so as to describe an arc on the meridian equal to the angle through which the meri- dian has revolved, the locus of M is the spiral in question. Taking the axis PP’ as that of z, PAP’, the initial position of the plane of the meridian, as the plane of (2, z), the equation to the sphere ig C+y+ ver Gy G46. 0 648, 6.2 8 Bs Gy. 667 8) 66 Now let POM= 0, AON = 4, then, by the definition of the curve, 9= 9. But we have generally, by Art. (70), zx=rcos@cos¢, y=rcos Osin ¢; and in this case z=rcos'0, y=rcos @sin 0. Hence 2 +y'=7 cos’ 0 (cos’ 0 + sin’ 8) = ra. Therefore the surface defined by the equation CS ial OM) ees Aan 3) by its intersection with the sphere determines the spiral in question. It is easy to see that this second surface is a right circular cylinder perpendicular to the plane of (z,y), the diameter of its base being the radius of the sphere. If we wish to represent the curve by two equations involving each two variables only, we may subtract the second equation from the first. We have then PtP ATD ce FONE aie Si des OTE (3), which is the equation to a parabolic cylinder perpendicular to the plane of zz. Hence the curve may be considered as deter- mined by the intersection of a right circular and a right parabolic cylinder at right angles to each other. Again, if we eliminate z between (1) and (3) we obtain, as the equation to a cylindrical surface perpendicular to the plane of iu, 2), pe (ot eae SOR earner ra Cit < (4), which is of the fourth order. Hence the equable spherical spiral may be represented by any one of the following systems 126 CURVES IN SPACE. of equations to cylinders: he) pA wey 2=r(r-—-2,) “+y=rez, gZ=ar(2—-y’), ry—2)=2, Zar (2 —- ¥'), or by the combination of the equation to the sphere with any one of them. (153) To find the equations to a spherical ellipse. The spherical ellipse is a curve traced on the surface of a sphere such that the sum of the distances of any point from two fixed points is constant. Let S, H (fig. 26) be the two fixed points on the surface of the sphere, C the middle point between them. If P be any point in the spherical ellipse, SP, HP arcs of great circles, then the definition gives us , SP + HP =a constant = 2a suppose. Through P draw PM, an arc of a great circle perpendicular to SH, and let SH=2y, CM=¢4, PM=9. Then, in the night angled triangle SPM, we have by Napier’s rules cos SP = cos (y — ¢) cos 0. In like manner from the triangle HPI cos HP = cos (y + ¢) cos 8. Now cos SP + cos HP = 2 cos od HP) COs eee cos HP — cos SP = 2 sin nee sin eas ; and SP + HP = 2a; therefore, after reduction, we find Pia 27 CE BO COS a sin y sin @ cos 6. nema! (Pagers: abs tadede sin} (SP — HP) = squaring and adding, we find sin a 2 2 cos Biligy ye [<7 cos’ g + sin’ °| COS el. sin 2 a Now if we take OA as the axis of z, OC as that of y, O being the centre of the sphere, and call r the radius of the sphere, we have, by Art (70), x=rcosgcos 8, y=rsin > cos 0; CURVES IN SPACE. 127 so that the preceding apa is equivalent to Vy = Sis) een cere (1). This is the equation to a right elliptical cylinder perpendicular cos” aes , ny cos’ a ne a to the plane of (z, y), and, being combined with the equation to the sphere TR Tp Er aa peer cas Pol Ca it determines the spherical ellipse. If we subtract (1), multiplied by sin’ a, from (2), we have (1 —tan’a cos*y)'2* + cos’y y° + 2 = 7" cos’a .... (3), which is the equation to an ellipsoid 2 2 2 FoR ATR 4 Ee mo B aE “3 = 1 eoeeer ee oe © © oO (4), 2 2 2 cos a y’ cos’ a if e=rcs'a =r a= cosy’ “ * T—tan® a cos” Y Hence the spherical ellipse may be considered as the inter- section of a sphere with a concentric ellipsoid. Its equations may also be exhibited in a symmetrical form analogous to those of the straight line. For if we eliminate 2° between (2) and (4), we have me Ay ay a Pa # (aa) tare) aot But if f, g be two corresponding values of x and y, Tet \ eee Leh NG Ts) tles)-2 7» and hence by subtraction and obvious reduction Bee ED YR Tee PI BH! | (8 Ge Go ABS B (a — ) c’) C(b =a. ? by the symmetry of the formule. (154) After curves given by the intersection of surfaces of the second degree, there are scarcely any of interest except the heliz, the equations to which we proceed to find. The helix is a curve traced on a right circular cylinder, in such a way that the co-ordinate parallel to the generating line of the cylinder is proportional to the arc of the circular base intercepted between the foot of the ordinate and a fixed point. 128 CURVES IN SPACE. Taking the centre of the circular base of the cylinder as origin, the axis of the cylinder as the axis of z, and making the axis of z pass through the fixed point in the circular base, and calling s the intercepted portion of the circular arc, the definition of the helix gives us the relation z=ks, k being the coefficient of proportionality. But if a be the radius of the circular base of the cylinder, Ss a s a a Z . therefore Z=aCO87 , y=asin v0 (ae these two equations taken together are the equations to the curve; but it may also be expressed by either of these com- bined with the equation to the cylinder pe SEY pas T hy kha n being any integer, it appears that the same values of x and y correspond to an infinite number of values of z, or the ordinate z meets the curve in an infinite number of points. These points are separated from each other by an interval 27ka; and if we call this h, we have h : zZ Zz Oh ey ad ee Zz Since cos — = cos | 2m~7 + — |, andsin —=sin| 2m7 + — |, ka ka ka ora” and the equations to the curve may be put in the form ZL = a COS 2a = asin ae : 7)? fo a sGr These equations show that the helix is projected on the planes of (x, z) and of (y, z) as the curve of sines, while it is obvious that it is projected on the plane of (z, y) as a circle. ( 129) CHAPTER VIII. OF TANGENTS AND NORMALS TO SURFACES. (155) Definition. If through any point in a surface a straight line be drawn, meeting the surface again in at least one other _ point ; and if, as the second point moves up to the first along any _ given curve traced on the surface, there be a limiting position of the cutting line, the line in that position is called a Tangent Line to the surface. Since an infinite number of curves passing through one point may be traced on the surface, there are an _ infinite number of tangent lines which can be drawn at any given point. We therefore cannot determine the equations to any one line, but we may find the condition to which must be subject the constants in the equations to all lines which are tangents at the given point. (156) Let the equation to the surface be HACE AY ea OVEN ad dae ik alt atte (1), and the equations to any line passing through a point (2, y, 2) gi RY ea ] Br 7 NE ERE ae (2), z,y',z being the current co-ordinates of the line. Then if %,,Y,, 2%, be the co-ordinates of the point nearest to (#, y, 2), in which the line meets the surface again, they must satisfy the equation to the surface (1) as well as those to the line (2), so that we have JC LD aE ee ul Aim bebe (3), and ee ere ey INT 2 = Zt I. win we (4), r being the length of the chord between the point (2, y, 2), and (z,,¥,,%,). Substituting the values of z,, y,, 2, from (4) in (3), and expanding by Taylor’s theorem, we find F(a, y, 2) + (7 _ m a n =| gor Re =U dee oo K 130 TANGENTS TO SURFACES. where R is a function of z, y, 2 and positive powers of 7. But as (x, y, 2) is a point in the surface, the first term of (5) vanishes — and the equation becomes dF dF dF cars — amt Do Tous vale os 6). 7 (0 + m Fan T+ Br) (6) This is satisfied either by 7 = 0, or by [indicate yas 97h. i Rts beats yee: da dy dz The former of these merely gives the point (2, y, z); the second is an equation for determining 1, and therefore one of the other points in which the line (2) meets the surface. But if, as we a P| ‘ assumed, 7 correspond to the point nearest to (@, y, 2), and if we suppose this point to move up to (a, y, 2), 7 diminishes without limit, and by our definition the straight line becomes ultimately a tangent. But unless R become infinite (which case we do not here consider), the limit of the equation (6') is dF dF dF {—- +m —+%" —= Tet gg th gg 7 Ones tee (7), the required condition which must be satisfied in order that the lines represented by equations (2) shall be tangents to the sur- face (1). This gives one relation between /, m, , which, jomed with 7? + m? + n? = 1, leaves one of these quantities independent ; we have, therefore, drawn through one point (2, y, 2) a series of lines passing through one point and subject to one geometrical condition, and which must consequently constitute a surface of some kind, the nature of which we proceed to find. (157) To find the locus of the tangent lines which can be drawn to a surface at one pownt. | The equations to any one tangent line are 1, m, n being connected by the equation of condition dF dE a o Gt i, — — ee a | are temas edi 7s 0 (7) Now any particular tangent line is determined by the quantities 1, m, n, and as the locus includes all the lines it must be TANGENTS TO SURFACES. 131 independent of J, m, 2; so that if we eliminate these quantities between (7) and (8), we shall obtain a relation between 2’, y/’, 2! and x, y, z, which, being true for all the lines, is the equation to their locus. ‘The elimination is easily effected by multiplying each term of (7) by the corresponding member of (8), when we find Cho, ais), LE ee, f: aa Dat pte Tepe Cian ct eat This being a linear equation in 2’, y’, 2’ (which are the current co-ordinates of the tangent lines, and therefore also of their locus) shows that the locus of the tangent lines is in general a plane, which is called the tangent plane to the surface. I say that the locus is 7 general a plane, because it has been assumed in equation (7) that the differential coefficients ae , saat as dx dy dz do not all vanish at the point of contact. If this were the case both equation (7) and equation (9) would be nugatory, and it would be necessary to have recourse again to equation (5), but this exceptive case (which can occur only for isolated points and lines on surfaces) we shall treat of in another chapter. Some writers assume at once the existence of the tangent plane, but from what has preceded it is clear that this is a proposition requiring proof, and that in fact it is not always true. (158) ‘There is not so close analogy between tangent planes to surfaces and tangents to curves in two dimensions as the student might at first be disposed to imagine. It does not by any means always happen that the tangent plane touches the surface in one point only ; it may touch it along a line, and it may cut it along one or more lines, which may even pass through the point of contact. ‘These three kinds of tangent planes may be seen in the surface produced by the revolution of a circle round an axis in its own plane, but which does not meet the curve; such is the surface which bounds the ring of an anchor. For a plane perpendicular to the axis, which touches the surface at all, touches it in a circle; and all planes which touch the surface at points outside of this circle meet it in the point of contact alone, while those which touch the surface inside of the circle cut it in acurve. In order to determine in any given surface what is the K 2 132 TANGENTS FO SURFACES. nature of the tangent plane, we must combine the equation to the tangent plane dF dF @-) FU - Zt e-9Z-0 with the equations F(z,y,2)=0; F(#,y',2)=9, since, when the plane meets the surface, the point (2’, y’, 2’) is in the surface. If these lead to the conclusion = 2, y’ =¥Y,; g= 8, the tangent plane meets the surface at the point of contact only, otherwise it either touches the surface along a line or cuts it along one or two lines; the former is distinguished from the latter by two factors becoming equal. (159) Generally the curve of intersection of a surface with its tangent plane has a double point at the point of contact.* (‘This includes the case of contact at one point only.) For if we have the equations EN ae ig e j= 10, Fe -2)+ FY -)+ Ze -D=0, which are the equations to the surface and the tangent plane at the point z, y, z, the direction of the tangent line of the curve of their intersection will be determined by combining their differentials with respect to z’, y’, 2’, or dk dk dF abel Det gale dz! = 0, dic! dy“! * de dF, dF, dF fa sch Pap aale ay) * ae From these we may determine, in general, the ratios of dz: dy': dz’, which determine the direction of the tangent lines. But if z =z, y'=y, 2’ =z, the two equations become identical, and consequently the ratios indeterminate: therefore to determine them we must employ in addition the equation dF a ak a, 12 12 12 dz" sae dy” dy t dz” dz a’F a’F d’F 2 ‘dz dx 'dy' = 0. + PE ani Gada ‘dz’ + tains * This remark is due to Mr. A. Cayley, Fellow of Trinity College. TANGENTS TO SURFACES. 133 The combination of these equations will give the ratios dz’: dy': dz' by means of a quadratic equation, implying that there are two directions of the tangent line at the point ; 2. e. that the point is double. (160) The preceding remarks will enable us to explain the difference between the two kinds of ruled surfaces of which we spoke in Art. (136). A ruled surface is one which may be generated by the motion of a straight line which moves subject to certain conditions, and the two kinds are—that in which the successive generating lines do not intersect, or skew surfaces, as we have termed them, and that in which the successive gene- rators do intersect, and therefore lie in the same plane. In both kinds of surfaces the generating lines must lie in the tangent plane: for if we consider the equation (5) as expanded in terms of 7, the condition that r shall be a portion of the generating line implies that the value of r derived from (5) shall be in- determinate, and therefore that the coefficient of every term in (5) shall separately vanish. Hence, this includes the condition of tangency (7), and therefore the generating line is one of the tangent lines, and so lies in the tangent plane. Now the tan- gent plane at any point P of a generator is determined by the conditions of passing through this line and any other tangent line at the point P; in the same way the tangent plane at any other point P’ in the same generator passes through that line and some other tangent line at the point P’. Hence, the two tangent planes pass through one common line (the generator PP’), and they would be coincident if the tangents at P and 1g were in the same plane. But this cannot be the case in skew surfaces: for since a curve may for a small space be supposed to be coincident with its tangent, we may consider the elements of the tangent lines at P and P’ to lie in the surface, so that the generator, in moving from the position PP’ to the consecutive one, rests on these tangent lines; and if they were in one plane, the two positions of the generator would be also in the same plane, which is inconsistent with the definition of skew surfaces. Hence in these the tangent planes at each point in a generating line are distinct, so that in skew surfaces a tangent plane touches the 134 TANGENTS TO SURFACES. surface in one point only, but cuts it along the generating line. On the other hand, in ruled surfaces in which the consecutive positions of the generator intersect and are therefore in the same plane, the tangent planes at each point of a generator coincide, so that a tangent plane touches the surface along a generating line. From this we may easily deduce the characteristic pro- perty of these surfaces; for if G,, G,, G,, G,, &c. be consecu- tive positions of the generator, the surface may be considered as the limit of the planes which pass through G,G,, G,G,, G,G,, &c. Now we can turn each of these elemental plane areas round the line which is common to it and the succeeding one as a hinge until they both lie in the same plane—as GG, round G, till G,G, and G,G, are in the same plane, and so on in succession—until all the elemental plane areas lie in the same plane, and therefore the surface which is the limit of these planes may be in the same manner unfolded into a plane surface without introducing discontinuity; so that if we suppose the surface to consist of a thin flexible and inextensible substance, it could be unfolded into a plane without tearing or rumpling. On this account these surfaces are called developable surfaces : they will be more particularly treated of hereafter in the chapter on envelops. (161) Definition. The normal to a surface at any point is a line perpendicular to the tangent plane at that point. To find the equations to the normal. Let 2’, y', z' be the current co-ordinates of the normal, z, y, 2 those of the point in the surface through which it is drawn; then, since the normal is perpendicular to the tangent plane of which (9) is the equation, and as it passes through (z, y, z) its equations are w-x2 y-y 2-2 TTT TES. Va nace (10). de dy dz (162) If the equation to the surface be put in the form z= f(x,y) it is desirable to express thie differentials of F' in terms of those of z: in this case F(a, y, 2) = f(a, y) -2= 0. TANGENTS TO SURFACES. 135 - Therefore aF d dz sd oad dF dx = GIy »¥) = dx’ dy aa »y) = fe so that the equation to the tangent plane eas ; OZ: az angie iy Se sith 7 Mn) Sul haseraglte (11), and the equations to the normal dx (163) On comparing the equation to the tangent plane (9) with the equation to the plane in Art. (39) we see that the direction-cosines of the tangent plane, and therefore of the normal, are Raed (ain 36) y-y+F @-2)=0.. (12) dF dF dex dy Boa) ae) (SG) ay dz dy dz dz dy dz dF de OA AAT! $°. (13). { dk \ ‘ dk \* f dF \’\3 da dy dz )} The perpendicular from the origin on the tangent plane is, by Art. (48), Red Peis dF dF dees dy dz ( (Fy (164) If the equations to the surface be in the form U=C, where w is a homogeneous function of ” dimensions, the equation to the tangent plane is much simplified. For, by a well-known property of such functions, Pe ee is co 700 5 dx dy dz so that the equation to the tangent plane becomes ae ie es (15); 136 TANGENTS TO SURFACES. and the length of the perpendicular from the origin is cee (1 du\* du\ du\?\ (ae) +) +) (165) Curve of contact. If, instead of finding the locus of all the tangent lines which touch a surface at one point, we subject them to any geometrical condition, their poimts of con- tact will trace out on the surface a line which is called the curve of contact. In the case in which all the tangent lines are constrained to pass through some point not in the surface, the curve of contact is easily determined. For since the tangent line is contained in the tangent plane at any point, it is sufficient to make the latter pass through the given point; we thus find the equation to a surface which, by its intersection with the given surface, deter- mines the curve of contact. Let the equation to the surface be Ee, yea) = 0, and that to the tangent plane ! dF 1 dF , 0 : @ -2) T+ -G+@-2) FG =0 If this pass through a point a, 3, y, we must substitute a, (3, y for z, y’, 2’, and we have dF dF dF Beater rs Re 2 ox aM Diep Moc) This, considered as a function of z, y, z, represents a surface which, by its intersection with F(z, y, z)= 0, determines the poimts of contact of lines passing through a, 3, y, that is, the curve of contact. (166) It is easy to shew that in this case the curve of contact always lies on a surface of a degree less by unity than the degree of the given surface. Let the equation to the given surface be put in the form M=& tus t+ui,t+ Ke. +4, =, n-1 n-2 where the different terms are homogeneous functions of the TANGENTS TO SURFACES. Tad degree indicated by the suffixes. Then the equation (17) becomes (a - 2) eee ees + &e.| +(B-y) {oe de cre + &e.| dx ad dy dy dy [ati By 5 Ms +(y - 2) Paget + + &e.), - - du +B du dt - du, he du, ie du, , stent dz dy dz dz dz d. du, du, du, | (du, du,, du, ji WG ge ip aaa a er aE Drags esear But, by the well-known property of homogeneous functions, ar du, - ue +2 ee TU a Mls - din +2 Wess =(n-1)u__; ies dy dz eC add dy dz car &c &e. therefore, observing also that U+u,,+U,. + &. =e, the preceding equation is reduced to a a +2 i +¥ a =ne-Uu,,—2u,,— 3u,,—- &c....(18). And as uw is of m dimensions, its differential coefficients must be each of m — 1 dimensions ; therefore the curve of contact lies on a surface of m — 1 dimensions in 2, y, 2, or of a degree less by unity than the given surface. Cor. Hence we see that in surfaces of the second degree the curve of contact is a plane curve. It is evident that the curve of contact, which has just been determined, is the apparent outline of the surface to an eye placed at the point through which all the tangent lines pass; or it is the boundary between light and shade if the surface be illuminated by rays issuing from that point. (167) When the point is removed to an infinite distance, the formule for the curve of contact fail, but we easily obtain more simple equations. For in that case all the tangent lines are 138 TANGENTS TO SURFACES. parallel, and therefore the direction-cosines 1, m, m are constant ; so that the condition which they must satisfy, l at m gh +7 gis 0 dx dy dz a is a relation between 2, y, 2 and constants which, combined with the equation to the surface, determines the curve of contact. It is obvious that this equation is of a degree less by unity than that of the given surface. (168) We proceed to illustrate the preceding theory by applying it to the surfaces of the second order. The general equation to these is Ax’+ A'y’+A's?+2 Byz+2Bxz+2 B'ry+2 Cx+2 C'y+2C'24+E=0...(1); hence, by equation (9) of Art. (157), the equation to the tangent plane at the point (z, y, 2) is (Ac + Bz+ By +C) (#-2)+(A'y+ Bz+ Bae+ C)Y'-y) +(A"2+ By + Ba+ C’) (2-2) =0....(2), z',y', 2 being the current co-ordinates of the plane. On re- ducing this by (1), we have (Ac+B'2+B'y+O)2's+(A'y+ Bet Bat C) y'+(A’e+ By Bas CO) 2 +Ca+C'y+C'2z+ H=0....(8). If we apply to (1) the method of finding the centre of the surface used in Art. (78), we shall find, calling the co-ordinates of the centre a, 3, y, the conditions that the terms involving the first powers of the variables shall vanish to be Aa+ B'B+By+C=0, A'B+By+B'at+C'=0, A’y+BB+B'at+C'=0. By means of these, eliminating C, C’, C" from (2), the equation to the tangent plane becomes { A(e—a)+B'(2-y)+B'y-B)} (@- 2) + {4(y-B) + B@-7) +B(e-a)}(y'-y) + {A'@-7)+B(y-B) +B (@-a) }(2-2)=0...(4). Now if J, m, be the direction-cosines of the diameter passing through the point (z, y, 2), we have Cen Yr ee ay [ieee an ae ee te which, combined with (4), give the equation (Al+ B'n + B'm) (a' - z)+(A'm+ Bn+ BD (y'-y) +(A'n+ Bm+ Bl (2 -2)=0... .(5). Se TANGENTS T0 SURFACES. 139 (169) On comparing equation (5) with the first equation of Art. (98), it will be seen that the coefficients of the variables in the two are the same, and consequently that the planes are parallel. Hence the tangent plane at any point of a surface of the second order is parallel to the plane diametral to the diameter passing through the point of contact. And from this it follows that the six planes drawn at the extremities of three conjugate diameters and parallel to their diametral planes are tangents to the surface, so that the parallelopiped formed by them circumscribes it. The volume of this parallelopiped is eight times that of the parallelopiped of which three conjugate diameters are conterminous edges, and therefore by Art. (106) is constant. ‘The following are more particular examples. (170) The equation to the ellipsoid is 2 2 a be a +o spires and therefore, by sora the ae to the tangent plane is ae sent To determine whether i ne meets the surface in more points than one, we have to combine the preceding equations with 12 12 2” Subtracting the second equation multiplied by 2 from the sum of the first and third, we have (-2! Y-y , @-2 =0. a b° Gs This can be satisfied only by © rn a= 2, and therefore the tangent plane meets the surface in one point only, which is the point of contact. The equations to the normal at the point z, y,z are, by equation (10) of Art. (161), @(e-2) BYy'-y)_¢@-2) x Yy @ 140 TANGENTS TO SURFACES. If p be the length of the perpendicular from the centre, on the tangent plane, we have by equation (16) of Art. (164) 1 xy y? 2\1 -= Te Sree . DMO hay To find the locus of the intersection of the tangent plane with the perpendiculars on it from the centre. The equation to the tangent plane is BLAMYY ) Ake The equations to a line through the centre, perpendicular to it, and therefore parallel to the normal, are PAA] At the point of intersection 2’, y', 2’ are the same in (a) and (6), while z, y, 2 satisfy the equation to the ellipsoid. Equation (b) may be put in the form el = =(az” + By? + ee) he A a ink c by the Theorem I. of Art (22). Multiplying each term of (a) by the corresponding member of (¢), and squaring, we have (2? + y? 42°F = aa” + By” + ez", which is the required equation. This surface is the Surface of Elasticity in the Wave Theory of Light. If we call 7, m, m the direction-cosines of the normal, from which, by multiplying by a, 4, c, squaring and adding, we find B= al + bm + en’, so that the equation to the tangent plane may be put in the form Ix + my + nz =(al+ Bm? + cn’). (171) The equation to the hyperboloid of one sheet is Dk Yt dor 2 eA Em | Si ree Ta . TANGENTS TO SURFACES. 141 and therefore the ie to the tangent plane at (z, y, x) is LW Yo 22 TAR Tae To find where this meets ate alee we combine the equations NES io SI Lae hac gt BR C bi? C with the equation to the eS tae when we have Bomegy (co y” ree Ey Zz eae zz’ A Weed Cer ste mired Wat Nt ae) | 00 sr Rel We in rr a ea e)- (Gr F) “(ate ) (ep which may be reduced to BY YR Pee, ( ab 5 (==), This may be split into the two linear equations ry’ — ye — ERIE game — Yh iy sane, ab C ab c either of which, combined with the equation to the tangent plane, gives the equations to a straight line. Consequently the tangent plane cuts the surface in two straight lines, which are in fact the generating lines passing through the point of contact. (172) For the cone 2 y’ 2 a ie 2 we find the equation to the tangent plane to be and the relation for determining the points where the tangent plane meets the surface, will be found to be th ab which, since it consists of two equal factors, shews that the tangent plane touches the surface along the line determined by the equations ey’ = wa’ oyy’ az! aie ole ae Te ’ I a’ b? 2 ey that is, the generating line passing through the point of contact. 142 TANGENTS TO SURFACES. If the vertex be the point of contact, or if z= 0, y= 0, z2=0, the equation to the tangent plane becomes nugatory, since each term vanishes by itself. In fact there are an infinite num- ber of tangent planes at that point: for since the tangent plane at any point touches the surface along a generating line, and all the generating lines pass through the vertex, the tangent plane at every point of the surface passes through that point. The vertex of a cone is the simplest case of the kind of singular point to which allusion is made in Art. (157). (173) Let the equation to the hyperbolic paraboloid be in the form zy = az; then the equation to the tangent plane is vy + y= az + 2), which meets the surface in two generators determined by its intersection with the planes Ee apg EST The equations to the normal are Eo Dee eee BRR a Ee To find the locus of the intersection of the tangent plane with the perpendicular on it from the origin, we have to com- bine the equation to the tangent plane with TED ei a Each of these ratios is by Art. (22) equal to a” bY? Hie ae ot eee Tee ay aoe vy + yx — az dz i xy in virtue of the equations to the tangent plane and to the sur- face: hence, we find 3 a? + yy? ae 2” av 4+ Jee ue 2? gz” ay y” vs 2? ea mena ars area" *: PT ye SEA I een ee ee ae y Z On substituting these values of z, y, z in the equation to the surface, and dividing by z* + y” + 2", we find as the required equation to the locus 2 (2? + y" +2") 4 az'y' = 0. TANGENTS TO SURFACES. 143 (174) In a central surface of the second order to find the curve of contact when the tangent lines pass all through one point. Let the equation to the surface be Ay’ + Aly + Ae =1; then by equation (15) of Art. (164) the equation to the tangent plane is Ara + A’yy' + A’z2' = 1, xz, y', 2 being the current co-ordinates of the plane. If the co-ordinates of the fixed point be a, (3, y, the condition that the tangent plane shall pass through it gives Aaz + A'By + A’yz=1, the equation to a plane which by its intersection with the surface determines the curve of contact. Since the equations to the line joining the centre of the surface with the point a, B; Y> are it appears that the curve of contact is parallel to the plane, which is diametral to chords parallel to this line, and con- sequently, by Art. (121), this line passes through the centre of the curve of contact. When the tangent lines are all parallel the equation of Art.(167) gives Ajy a A'my + A’nz = 0, the equation to a plane passing through the centre, which by its intersection with the surface determines the curve of contact. It is obvious that this is the diametral plane to chords parallel to the tangent lines. (175) To find the conditions that two central surfaces of the second order shall intersect everywhere at right angles. If the surfaces cut each other at right angles, the tangent planes at every point of their line of intersection must be perpendicular. Now let the equations to the surfaces be eg ee ey 2 SAS Fall St BAN ie 144 TANGENTS TO SURFACES. then the se to their lenge ig i> at a see (a; se) yy LW. ; ee Be nes =a BC 3 ace hs from which we see that their Rea are proportional to are Ls et). Wak Coven : tase mae 72 ar a vate Clb CPD eC. eae Oey and since the planes are to be perpendicular, we have x ye 2 But on subtracting the second equation of (1) from the first, we have neal 1 oa aL 1 Bae 1 aes, Pry at eee +-—+j/=0...... 4). x (- ve Ty G z) + 2 (; | ( ) Equations (3) and (4) are two relations between z, y, z the co-ordinates of any point of intersection of the surfaces; and as they are of the same form they must be identical, with the exception of a factor. Multiplying then (3) by A, and equating the coefficients of the variables in the two equations, we have e—d=) —~B=c-—y=X; whence a? - B=a°- 3’, @-C=Ha-y, F-C=p-y. These latter equations show that the principal sections of the surfaces have the same foci: such surfaces are called confocal, and possess many remarkable properties. Since equation (3) cannot be true generally, unless one at least of the terms be negative, it appears that one of the surfaces must be a hyper- boloid. ( 145 ) CHAPTER IX. OF TANGENTS TO CURVES, NORMAL AND OSCULATING PLANES. (176) A curve in general is determined by the intersection of two surfaces, and at every point of the line of intersection two tangent planes can be drawn, one to each surface; the inter- section of these two planes is a tangent line to the curve. To find its equations let ON 1 Os 0 be the equations to the two surfaces; the equations to their tangent planes are ‘a eae @-a)2sy- NE+e-)L=0. As these are ae at the same point, the co-ordinates z, y, z are the same in both; and when they intersect, z’, y’', 2’ are the same. Hence, eliminating the speintele: 2-2, y-y, ¢7-z ‘in succession, we find a ea i du dv dudv dudv dudv dudv du dv" *” dy dz dz dy dz dz dz dz dx dy dy dz as the equations to the line which is a tangent at the point (z, y, 2) to the curve determined by the intersection of the surfaces Aa CRO From the equations = seiner 146 TANGENTS TO CURVES, » we find ; dx > dy me dz (2) ; du des divdb dude dude . duude ts dusdcn a ee eed ed dy dz dz dy dz dx dz dz dx dy dy dz so that the equations to the tangent line may be put under the form “ go y¥ -y 252 (3), eo, 8 © 6 @ €)6 0/72 18) 6.52 35 ' dx dy 7 —_ = ee i — == — <_ ae eeee 4 >) or ae Slap (Zim By Sy oe (z' — z) (4) a form which is convenient when two of the variables are given” as functions of the third. (177) If one of the equations, as « = 0, do not involve one of. the variables as z, it represents the projection of the curve on the plane of zy, and the aan | @ 2) Hay-yT=0 represents both the projection of the cya on the plane of (z, y) and the tangent at the corresponding point of the projection of the curve on that plane. Hence, generally, the projection on any plane of the tangent to a curve coincides with the tangent at the corresponding point of the projection of the curve on that plane. (178) Normal plane. A normal to a curve being defined as” a straight line drawn perpendicular to a tangent arora the point of contact, it is clear that there are an infinite number of such lines all lying in a plane perpendicular to the tangent line, which is called the normal plane. The equation to this plane is evidently . du dv du dv CON. du dv du dv o1ax dz dx dx dz YY 4 dy dz dz dy du dv du dv ( dy dy a)@-2- Ost tb ye or (a' — x) dx + (y' — y) dy + (2 - z)dz=0.... (6). The differentials in this equation are to be got rid of By means PRA Iain ant proc NORMAL AND OSCULATING PLANES. 147 of the equations to the curve, from which we can find the ratios dz: dy: dz. (179) If we put ee Ae eae), (et wy dy dz dz dy dz dz daz dz ae -Ou ay An Le the direction-cosines of the tangent and of the normal plane are, from equations (1) and (5), x(F EEE) 1/du dv _ du dv\ 1/(dudv_ du de A\dy dz dzdy]’ A\ dz dx dx dz)’ A\d« dy dy ie) or, from equations (3) and (6), ae dz d dz Ba eee g Sp ce ee Seenetan icc. casa? 70 AT (de® + dy? + dz)’ (da? + dy?+ dz)’ (de? + dy'+ dz) But by the direct application of geometrical conceptions, these may be expressed in simpler forms. A curve, whether plane or of double curvature, may be considered as the limit of a polygon, the sides of which are diminished in magnitude while they are increased in number indefinitely. Now if As be the length of a side of the polygon, the difference of the co-ordinates of the extremities of which are Az, Ay, Az, the direction-cosines of the side of the polygon are Az Ay Az As’ As’ As’ But as the sides of the polygon are diminished in magnitude while their number is increased indefinitely, the side coincides ultimately with the tangent to the curve, and the limits of these ratios are consequently the direction-cosines of the tangent; and as the length of the side is ultimately the increment of the arc s of the curve, the limits of these ratios or the direction-cosines of the tangent are, in the language of the differential calculus, expressed by dz dy dz ds” ds°. ds” HS OLIN ac Pate H dz? (dy (de _ |. ence (=) (2) +(3) ; or ds* = dx’ + dy’ + dz’. (180) Osculating plane. If we consider a curve as the limit L2 148 TANGENTS TO CURVES, of a polygon, the sides of the latter are not necessarily all in one plane, but any two conterminous sides must be in the same plane, since they intersect. ‘This plane passes through three of the angular points of the polygon; and as the polygon approaches to the curve as its limit, this plane tends to assume ultimately a determinate position, in which it is called the osculating plane. To find its equation we must express the condition that it passes through three consecutive points of the curve; that is, through three points of which the co-ordinates are Ly Ys, % a+dz, yr+dy, z+ dz, a+2dxe+@a, y+2dy+@y, 2+ 2dz+ dz. | ; | Now, if in any equation F(z, y, z)=0 we substitute these — values of the co-ordinates, it becomes in succession, F(a, y, z)+ dF (, y, z)= 9, F(a, y, 2) + 2dF (a, y, 2) + @F(@, y, 2) = 03 and when these are taken simultaneously they are equivalent to — F(z, y,2)=0, dF(«#,y,2)=0, @F@,y, 2) = 9. The equation to a plane passing through a point (2, y, 2) may _ be assumed to be A(¢ —2)+ Bly'-y)+ C@-2)=0...... (Aj; and if it be the osculating plane, that is, if it pass through three — consecutive points of the curve, we must have, by what has just been said, Adz + Bdy + Cdz=0 0 .cccsecseee (2), Ad’z + Bd’y + Cd’z2=0......0 200, (3), Eliminating A, B, C in turn between (2) and (3), we have _ 4g ee Ee eee (4); dyd*z — dzd*y ~ dzd*x — dad*z dxd’y - dyd*x i so that by eliminating A, B, C between (1) and (4), the equation - to the osculating plane is (dyd’z — dzd’y) (a - x) + (dzd’x — dxd’z) (y' — y) + (dad’y — dyd’x) (2 —z)=0....(5) It is obvious that this is perpendicular to the normal plane, and therefore passes through the tangent. In applying equation (5), if we keep it in the symmetrical form, we must consider z, y, 2 as functions of some other | NORMAL AND OSCULATING PLANES. 149 variable ; but if we consider any one of them to be independent, of which the other two are functions given by the equations to the curve, the formula necessarily becomes unsymmetrical. Thus, if z be taken as the independent variable, so that d’z = 0, equation (5) becomes eyea 2 dz d*y\ ., Ow d’y — i _ _- —. _ - —— — =A cal tr ky (2 ek Be) e-9- By -+ Be-5-0.© The following are applications of the preceding formule to particular examples. (181) The equations of the equable spherical spiral are, by Art. (152), a? 4 yY+e=a4r, x+y’ - Iz =0, or Z+2re=47, 2 +y’- 2rze=0. From these we find dy r-x bd Fenge BEIT ene eee so that the equations to the tangent become, after obvious re- ductions, yy 4(e—r)a'=r2, 22 +ra = 4-92; and the equation to the normal plane is r—-x Y, This, after reduction, may be put in the form hae, Naas of 28 Y x y) hele eye)? in which we see that it is satisfied by the equations (z-xz)+(y¥-y) r (aa ey = 0) ¢-o! Zane ak Za and therefore it contains the radius of the sphere drawn to the point (z, ¥, z). The equation to the osculating plane is, taking z as the independent variable, a {ey—r(y—2))+ yy? + 22 = 492’ + rez (y’ - 2”). (182) ‘The equations to the spherical ellipse are (Art. 153) rad y° 2 2 BLA SE Gane ik 2 2 . acl, e+y+z=r, 150 TANGENTS TO CURVES, af ef aft u y- Gf : 2h’ x (cid 8) b? (a ~ ¢) e (& — a)- The equations to the tangent are ala—a) yyy) 22-2) ~3-713 IN 8 LE LT eee CCE NO ae) Cia - cat IPR (nae bee Gu Ce) web (ae Bi(ae te) eb ae The equation to the normal plane is 2 2_ gy }? 2 eae, Ve ees oe Gaile sr (a aha a’) ‘ and therefore it passes always through the centre of the sphere. The equation to the osculating plane is somewhat complex, but if we put dike = 0) ="Asy ab" hae oh Be ad) and Bz’ — Cy’ = Bh’ - Cy’ = L, Cx’ — A?’ = Cf’ - Al’ = M, Ay’ — Bz’ = Ag’ — Bf’ = it takes the very symmetrical form L 3 ! M 3 ' N iS ’ SI (x — B+ ay (y BD) ps (Z - 2) =0. (183) The equations to the helix being in the form =a C08 ~, y=asin = h h? the equations to the tangent are ey =a, a-u2 y-Yy z-2 -Y on h If 0 be the angle which this line makes with the axis of z, we find tan 0 =“: @ is therefore constant, and the tangent is inclined h to the plane of (x, y) always at the same angle. The tangent meets the plane of (x, y) in points forming a curve, the equation to which is easily found. For, making 2’ = 0 in the equations to the tangent, we have A(z -x)-yz=0, h(y'-y)+2zz2=0, np NORMAL AND OSCULATING PLANES. 15! between which and the equations to the curve we have to elimi- nate z, y,z. The last equations may be put in the form ha =he+yz, hy = hy — xz: whence, by squaring and adding, h* (2? + y”) = ha’ + ae’, he ‘ and therefore 2= "5 (Gere yea), (a ae y” a ay (2? + y" — a’) Z2=acos ———*“—-*+, y=asin ———“——. a . a But multiplying the equations by z and y and adding, we have we + yy =a; substituting in this for z and y, their preceding values, we have 1 at (2 4 Cie Lat a’) (x” ut y” ia a) a a zx cos +y' sin = 0; which is the equation to the involute of the circle. The equation to the normal plane is ay —yz +h(z'-2z)=0. The equation to the osculating plane will be most readily found by making z the independent variable, and therefore d’z = 0; this gives us hwy -y'z)+ @ (2 - 2)=0. In both of these equations, if we make z’ = 0, y'= 0, we find ‘=z, that is, both planes cut the axis of z at the same point, which is the corresponding co-ordinate of the point in the curve: their line of intersection is therefore parallel to the plane of (z, y). It is easy to see that both planes are inclined at a constant angle to the plane of (z, y), the direction-cosine of the normal plane being h ih (2° ms y 4 hey" (a a. hey! -and that of the osculating plane being 2 fa’ ti }? Cs as y?y? (a 4 hy? the complement of the former, as is otherwise obvious. 5) (2RIS28) Line of Greatest Slope. (184) The line of greatest slope on a surface, starting from any point, is such that its tangent always makes a greater angle with a given plane than any other tangent line to the surface drawn through the same point. Since all the tangent lines at any point of a surface lie in the tangent plane, that one which is perpendicular to the intersection of the tangent plane with the given plane, makes the greatest angle with the plane. Hence the line of greatest slope on any surface has its tangent at every point perpendicular to the intersection of the tangent plane and the given plane. Take the plane of (2, y) as the given plane, and let the equation to the surface be VE GOROE CAV EOD bi che putenedate’s ie ots Gi so that the equation to the tangent plane at the point (%, y, 2) is dF ,, dF, eae, Bede ir gg dil di ahs ae ia aia the intersection of which with the plane of (2, y) is given by the equations z=0 EINES (ans ED aes EP (2) eda dy dx dy Te Maida tee et The equations to the tangent to the required curve, at the point (x, y, 2), are in the form “de dy = rete e es G5 and if the lines (2) and (3) are perpendicular, we have dF dF eer — — ALH=0 i crrcrnnces 4). oP dy oi dz (4) This differential equation, which expresses the general pro- _ perty of the line of greatest slope, combined with the equation (1), will serve to determine it; and if between (1) and (4) we — eliminate z, we shall obtain a differential equation between — z and y, which on integration gives the projection of the line — of greatest slope on the plane of (z, y). (185) Let the given surface be the ellipsoid 2 2 2 2 oe te 9 3 —+%4 Gano: Tae LINE OF GREATEST SLOPE. 153 and the given plane, as before, the plane of (x, y); then equation (4) becomes “ dy = Jy dz = 0, } or a i 108 dy = 0; a 7 in which the variable z does not appear. The integral is a® = Cy”. The constant C is to be determined by the condition of the line of greatest slope passing through any given point: if the co-ordi- nates of this be a, 3, the preceding equation becomes a\@ mY B (c) (8): The intersection of this cylinder with the ellipsoid will give in general a curve of double curvature: but if the point (a, (3) lie in one of the principal planes of (y, z) or (x, 2) the line of greatest slope is plane; for if it lie in the plane of, (y; '2); a = 0, and therefore z = 0, showing that the section of the ellip- soid by the plane of (y, z), is the line of greatest slope. (186) Let the given surface be that represented by the gene- ral equation @ z= 3 Zs the nature of which will be explained in the following chapter. In this case equation (4) becomes adxz + ydy = 0, the integral of whichis .2, 2 2. Cube Cis so that the line of greatest slope in these surfaces is projected on the plane of (2, y) in a circle, the centre of which is at the origin; that is, the line is determined by the intersection of the surface with a right circular cylinder, of which the axis is the axis of z. Gol bas) CHAPTER X. OF THE GENERATION OF SURFACES BY THE MOTION OF CURVES. (187) In investigating the equation to the plane (Art. 33), and those to the hyperboloid of one sheet, and hyperbolic para- boloid (Arts. 188 and 141), we considered these surfaces as traced out by the motion of a straight line constrained to move in a certain manner. An extension of this method gives us the means of conveniently classifying and discussing surfaces of which the equations are of a degree higher than the second. The general theory of the process for finding the equations to surfaces defined as generated by the motion of a curve may be explained in the following manner. (188) A line is represented by two equations to surfaces involving 2, y, 2 and constants; we may suppose one of the constants in each equation to be arbitrary, and to admit of an indefinite number of values, corresponding to which the surface assumes different forms or positions; for the form and position of a surface depend on the values assigned to the constants in its equation. If we assume the equations to be solved with respect to these arbitrary constants, or parameters as they are styled, they are in the form Fay ys yey F(a, y,- 2) = 6.) OA, or, as We may write them, WIC DO ee earn ey Now, on assigning different values to ¢ and c,, the line deter- mined by these two equations will change in form or in position, or in both, in consequence of the change in the surfaces. If ¢ and c, be independent, the line may be made to occupy all points within the space for which the equations (1') are satisfied GENERATION OF SURFACES BY THE MOTION OF CURVES. 155 by possible values of the variables, so that it will trace out a solid locus. But if we assume a relation to exist between c and e,, which may be expressed by the equation eG (Capen OTR. (Caos at 62) then for each value of ¢ the curve will assume only one deter- minate form or position, and in passing through all its forms and positions it will trace out a surface. To determine the equation to this surface it is clear that we must obtain a relation between z, y, and z, independent of the quantity c, which determines by its successive values the successive forms and positions of the curve. ‘This is easily done by eliminating ¢ between the equa- tions $item OLA CC) ay ibe todo beset (3), the result of which is evidently RL EOD Buc hae de ER OE (4) (189) Hence, when the parameters in the equation to the generator are given explicitly, it is very easy to find the equation to the surface, as when the relation between the parameters is given, the elimination is at once effected. Thus, for example, in finding the equation to the plane, since we suppose the gene- rator to move parallel to itself, the quantities a@ and 6 of the equations Z=az+p, y=bze+q, of Art. (26) are constant, and p and g are the variable para- meters, so that if they be written on one side of the equations, these then become y — gz = p, y — bz = q. Now the relation between p and gq is to be found by the con- dition that the generator shall pass always through the director, of which the equations may be written z=lz+h, y=mzr+hk, the condition for which is, by equation (5) of Art. (30), p-h q-k Ge be ibm? and this equation is the form of (2) appropriate to the present case. Hence, eliminating p and g by means of the equations Z-az=p, y-bz=4q, etal w-az-h y-be-k Oa Ol wb mi’ ) 156 GENERATION OF SURFACES as the equation to the plane, which is evidently of the first degree in 2, y, z. (190) In what precedes we have assumed that there is only one variable parameter involved in the two equations to the generator, but the subject may be considered more generally. For we may suppose the equations to the generator to contain parameters c,, ¢,....¢,, so that they may be written T (Es Y 25 Crs Cae» » 0.) = Oy | flat) s 23,015 Cae Coen Then in order that the motion of the generator may be com- pletely regulated, so that it shall trace out a surface, the » parameters must be connected by ” — 1 relations, so as to leave one independent quantity only. If these relations be (Cosa ieC.) 105) We Ci te sO ee Wee sin ee Cvs Came the equation to the surface will be found by eliminating the x parameters between these » — 1 equations and the two equations of the generator, making m + 1 in all. (191) It usually happens in practice that the relation between the constants is not given directly, but is to be deduced from the geometrical condition that the generator shall pass always through some given director-line, as was the case in the pre- ceding example. It is easy to see that such a geometrical condition always corresponds to one relation between the pa- rameters. For if the equations to the director be Lee ye) 5051 eh Ae /,12) 10, in order to express that the generator passes constantly through it, we must make xz, y, z the same in these equations and in the equations to the generator FILSY NERC ee C0 (ERY) eG Cen Between these four equations we may eliminate the three quan- tities z, y, z, so that we shall obtain a result involving ¢,, c,...¢ only, and which may be written as P(e) G «. -6,) = 0, which is one relation between the parameters. 2) = O...eee (2), the equations to the generator in any position are scan eh) aod paki hs ee x,y’, 2 being the current co-ordinates of the generator. Now, since two successive generators intersect, the points 2, y, 2, @,,Y,, %, must be so taken that the tangents to the curves at those points are in the same plane, see Art. (160); and the equations of the tangents to (1) and (2) being aris aM hake Wo uaa iste sents le old Megs dT oe z' - i - ae a i tere peo Gye the condition for these lying in the same plane is, by Art. (30), (dydz,—dy,dz)(«-x,)+(dzdz,- dadz,)(y-y,)+(dady,—dz,dy) (2-2,)=0 me oh? &t sOaW 4S. Yoh MOSWION SRLS. HG): The differentials in this equation may be eliminated by means of the differentials of (1) and (2), and then (6) is a relation between the quantities z, y, &c., which, combined with the six equations (1), (2) and (3), gives seven equations, between which the six quantities z, y, &c., may be eliminated, and a relation obtained between 2’, y’, 2’, which is the equation to the develop- able surface. (196) Since the successive generators of a developable surface intersect, they will by their intersection determine a curve to which they are all tangents. This curve has been called by French writers the “‘ Aréte de rebroussement ” of the develop- able surface, and the term has been translated into the English phrase “edge of regression”: perhaps, however, the name “cuspidal edge” expresses better the meaning of the French words. This curve is a remarkable line on the surface, as it is a prominent edge which offers a salient angle in all plane sections 160 CYLINDRICAL SURFACES. except those which pass through a generator. It appears from the nature of the curve that the surface falls away from it in two sheets, so that it is an extreme boundary to the surface: it is evidently a curve of double curvature, since if it were a plane curve its tangents would lie all in one plane, and the developable surface would then be reduced to a plane. ‘The student may perhaps obtain a more distinct idea of the nature of this line by the inspection of fig. (27). (197) Every developable surface has a cuspidal edge peculiar to itself; im the case of cones it is reduced to a point, and in cylinders this point is removed to an infinite distance. Hence, when the equations to any curve of double curvature are given, we may find the equation to the developable surface of which it is the edge; for if its equations be EXC Y 2) = 0, CE ee ea and those to its tangent at a point z, y, 2, “w-x“2 y-y Z-2 dz ih jay Wane we may between these four equations eliminate z, y, z, and obtain a relation between 2’, y’', z', which is the equation to the developable surface. The method of finding the equations to the edge when that to the surface is given, will be found in the Chapter on Singular Points and Lines in Surfaces. There is a third mode of considering the generation of de- velopable surfaces, which we shall explain in the next chapter. Cylindrical Surfaces. (198) Cylindrical surfaces are those which are generated by the motion of a straight line which always remains parallel to a given position. ‘To find the general equation to such surfaces, let the equations to the generator be m n then /, m,n, being the direction-cosines, are constant, while x, y, 2 vary from one position of the generator to another. The preceding equations may be written in the form ly' —mz' =ly-mz, lz -nz' =lz-nz.......(2), CYLINDRICAL SURFACES. 161 in each of which the second side may be looked on as a single variable parameter, so that they are given explicitly as c and C, in Art. (188). But in order that the generator may move so as to trace out a surface, some relation between these parameters must exist; and we may express this by writing the equation be ING) oso. st che Ss (3). Eliminating a, 8, y between (2) and (3), we have bea Tt = bh (ly Ing ea ee (2) as the general equation to cylindrical surfaces, the individual surface being determined by the nature of the function ¢. (199) If we assume that the generator is determined by the more general equations le+my+nz=8, la+imyinz=8,.... (5), we easily find the general equation to cylindrical surfaces to be Lo+ my +nz = (le+ my +nz).......... (6) so that if we have given an equation in which one linear func- tion of z, y, zis made equal to any function of another linear function of these variables, that equation represents a cylindrical surface. | (200) To find the equation toa cylinder of which the director is a plane curve determined by the equations At+ py +228, F(z, y,2)=0.....:(7; let the equations to the generator be wae: eae oe between these four equations we have to eliminate z, y, and z. Each of the ratios (8) is equal to A@ -atuly -y)t+r(e' — 2) _ Aw't+ wy'+ v2 - 8 IX + mp + ny IN + mu + nv in virtue of the first of equations (7). Hence if we put IN + mu+ ny = k, we find Z=2'- : (Az + wy’ + vz' — 8), y = y' — 5 (Aa! + wy + v2! ~ 8), U n / Uy f i z=2—-— —(Az' + + yz — 0); ci i an ); M 162 CYLINDRICAL SURFACES. and these values being substituted in the equation F(z, y, z)=0, give a relation between 2’, y’, z' and constants, which is the equa- — tion to the cylinder. ! (201) Let the director be the ellipse yond y ZO, Faatepaie o In this case A = 0, w= 0, = 1, 8 =0, & = n, whence na’ — Iz ny’ - mz! t= 5 = ——-, n n so that the equation to the cylinder is (na! — lz’) i (ny'-mz'P Sarena nae BN Abe asl poerksy (REDE (202) Let the cylinder circumscribe the ellipsoid Ag + Ai A ee ee (10); then by Art (167) the director is determined by equation (10) combined with Aig Amy + A Ne = (ci), | the equations to the generator being cleats Be A 560, But on combining equations (11) and (12), we have in virtue k of (10) Aga Ady +i aime Te yy. loys | and we may use (11) and (13) for the elimination of z, y, 2 instead of (10) and (11). From (12) we have z=2-lr, y=y'-mr, 2=2-nr; which substituted in (11) and (12) gives Alz' + A'my' + A’nz' - (AP + A'm’? + A’n’) r = 0... (14), Az” + A’y? + A’2”—1-(Ale' + A'my' + A’nz') r= 0... (15). Eliminating 7 between (14) and (15), we obtain (Al+A'm’'+A'n’)\ Az"+A'y"+A'2"-1)=(Ala's A'my'+A'nz'y ...(16), as the equation to the circumscribing cylinder. If p be the semidiameter of the ellipsoid which is parallel to the generators — of the cylinder, its direction-cosines are J, m, n, and hence from — (10) we find CONICAL SURFACES. 163 which, substituted in (16), makes the equation to the cylinder Ax? + Aly? + A’2?—1 = p (Ala' + A'my' + A'nz'y: in this may be recognized the form (5) of (Art. 149) for the equation to a surface of the second order which circumscribes another surface of the same order. Conical Surfaces. (203) Conical surfaces are generated by the motion of a straight line which passes constantly through a fixed point. To find their general equation, let a, b, c be the co-ordinates of the fixed point; then the equations to the generator may be written as PAY ie 2 9% 1 j pam wee cbse ea ly whence fidinid hes a ims Net Nay are ad Hehe (2) z-C 7 ~-—-C nr Now a, 6, c are constant, while 7, m,m are the variable para- meters, since they vary with the position of the generator: hence : it Saki, in equations (2) the ratios — , — may be considered as parameters nm n given explicitly, and in order that the generator may in its motion trace out a surface, the one of these must be some function of the other, which is expressed by writing : =o (=) DOE eae (3), whence eliminating /, m,n by means of (2), we have ool ard EEE (4) 24 = Ibs CoE as the general equation to conical surfaces. If the fixed point be taken as origin a = 0, b = 0, c= 0, and the equation (4) becomes z= 9(2)... fg ea Pe are 75 This may be written as in which form we see that it is equivalent to saying that it is a M 2 164 CONICAL SURFACES. homogeneous function of z, y, z equated to zero: so that we have thus extended to all conical surfaces the remark made in Art. (81) on cones of the second degree. (204) Let the cone be that of which the director is the plane curve determined by the equations le tmy +2: = 90, se Aa, Yj 2) = Oe eee If the co-ordinates of the vertex be a, 5, c, the equations to the generator may be written as =e O_o since it passes through the point (a, 6, c), and also through a point z, y, z of the director. ach of these ratios is equal to E(z'-a)+m(y'-b)+n(z2-c) I(2'-a)+m(y'-b)+n(e'-0) L(a-—a)+m(y-—b)+n(z-c)~ 8 — (la + mb + ne) 3 in virtue of the first of equations (6). Hence if we put la + mb + ne = d, we find CA) le’ + my' + nz’ —d’ , 2 (d-9)(y-5) Ix’ + my’ + nz! — d’ _(d- 8) (2-0) Ixc' + my’ + nz! - d’ De — ye Z=C- which values substituted in F(z, y, z) = 0, give a relation between z’, y', 2 which is the required equation. If the director lie in the plane of (z, y), so that 7= 0, m=0,n=1, 8=0,d=e, the resulting equation is r(¢ ied bz Eee) Ay pe. 7, If the director be the circle of which the equations are z=0, C+ yf = 9", then the equation to an oblique cone with a circular base is (az! — cz’) + (b2' - cy'f = 9 (z'-— cy... (9). (205) Let the cone circumscribe the ellipsoid AE A Uf AA et ae (10); i Me CONICAL SURFACES. 165 then by equation (17) of Art. (165) the curve of contact is given by the combination of (10) with Adres Hye C2 =e oo aie. CLL), a, b, c being the co-ordinates of the vertex of the cone. The equations to the generator may be written Gere Dee Ganz eo oe ee ee eee (12). On subtracting (10) from (11), and multiplying each term by the corresponding member of (12), we have ADEE CA YU eA aie es ale ane (18), and the system (11) and (13) may be used instead of (10) and (11) for the elimination of x, y, z. Multiply the numerator and denominator of each member of (12) by Aa, A’b, A’e respec- tively, and add, then by Theorem 1. of Art. (22) each member of (12) is equal to Aa (zt -2)+Abiy-y)+Ac(e'-2)_ Aar'+ A'by'+ A’cz'-1 Aa(a-2z)+Ab(b-y)+A’c(e-z) Ad’+ A+ A’P-1?’ in virtue of (10) and (12). Again, doing the same with Az’, A’'y', A’z’, we have each ratio equal to Ag (xv -2)+ Ay y-y)+A2Z (2-2) Az? + Ay? + A’2?-1 Ax’ (a-2)+ Ay (b-y)+ AZ (e-2) Aaz'+ A'by' + A’cz'-1 and hence equating these ratios, we have (Aa’+ A'l’+ A’c’-1)(Ax?+ A’'y”?+ A’2”—-1)=(Aaz' + A'by'+ A’cz'-1F eee CLAY, as the equation to the cone. ee Let the distance of (a, 5, c) from the origin be 7, and its direc- : : a b tion-cosines therefore ~,-,£, and let p be the portion of 7 1" Hae? intercepted between the surface and the origin, or the diameter to the plane of contact, then at the extremity of p z=a®, eee zi OL, r r r which being substituted in (10) give Fee ts Ad’+Ab’+ A’? =, p so that the equation to the circumscribing cone may be written (7? — p’) (Az? +A'y” + A’z” — 1) = p? (Aaz' + A'by' + A’cz' - 1Y. Ca r66r4 Conordal Surfaces. (206) Conoidal surfaces are generated by the motion of a straight line which passes through a fixed axis and remains always perpendicular to it. © Let the equations to the axis be —— = +— = —__ ............... (1) y m n then, since the generator is perpendicular to this axis, it lies in the plane LE MY ENZO eae 6 eet ees and also in some plane passing through the axis, the equation to which may be written n(y—6b)-m(z-¢)=6, {n(a-a)-I(z-0)}....(8), and 6 and 0, being the arbitrary parameters, there must be some relation between them, as A : whence we have ; ‘ a i n(y-b)—-m(z-c¢ lo my + nen g {ARR , as the general equation to conoidal surfaces. If we take the axis of 2 as the axis of the surface, we have 7 = 0, m= 0, n= 1; a=0,6=0,¢= 0, so that in this case the equation becomes -+(2) showing that the ordinate parallel to the axis of the conoid is equal to a homogeneous function of 0 dimensions in 2 and y. This is the class of surfaces treated of in Art. (186). (207) Let the director be the plane curve determined by the equations le+my+nz=6, F(a, y, 2z)=0, and take the axis of z as the axis of the surface; then the equations to the generator are oo wae x, y', 2 being the current co-ordinates of the generator: from these we have ey lermy lermy le +my’ ye oe hones BY Wide wimnyy De Ore ne 6 — nz CONOIDAL SURFACES. 167 , O- nz , O- ne whence t=2% ———, y=y —— #2=2, lx + my la + my which yalues, substituted in F(z, y, z) = 0, give , O— nz , Omen Fz ER BORE abi » # |=0 la + my la’ + my as the equation to the conoid. (208) Let the director be the circle of which the equations are rien eae face ates then the equation to the conoid is a $2° =a, org 2 ar by", This surface is called the cono-cuneus of Wallis, that mathema- tician having been the first who conceived it and investigated its properties. It is easy to see that planes z'=+c parallel to (2’, y’) cut it in two straight lines so long as ¢ is less than a, but when cis greater than this value the planes do not meet it, so that the surface is limited within the space included between the planes Ze=+d, z2=- a. (209) To find the equation to the tangent plane and the nature of the contact in the cono-cuneus. Dropping the accents we may write the equation to the surface as (?-—2)2- By’ =0; then, z’, y’, 2’ being the current co-ordinates of the tangent plane, its equation 1s y?z2' _ (q? — 2°) va’ + Byy! = 22°. To find the lines in which this meets the surface, we must com- bine these equations with (a2 a" — by" ='0, The second equation gives Byy' = x’ (2 —22') + (a - 2) ax; from which, on eliminating y and y’ by means of the first and third equations, we have (a - 2) (a — 2)" w2=a2(2-2)+(P-2)2; whence, by squaring both sides and omitting the terms which destroy each other, we find (@— 2) @ = 2”) z= 92" (ze — ZS + az (A - 2) (ze - 2) 2. 168 CONOIDAL SURFACES. This may be put in the form (z— 3°) [(@ — 2) {x (2 + 2’) - Qaen'l - 24 (2 - 2] = 0, which it is easy to see may be split into the two factors 2-2 =0, (a — 2) {x” (2 + 2°) - Qazx'} — x2" (2 - 2') = 0. The former, combined with the equation to the tangent plane, gives a straight line which is in fact the generator, the other gives a curve of the third order. Hence the tangent plane cuts the surface in two lines, one of the first, the other of the third order. (210) Let the director of the conoid be the helix of which the equations are Z=aCOSnZ, y=asin nz. The equations to the generator are ae, ee See ae “y-yx=0, or 2 sinnz-y' cos nz=0; and the equation to the conoid is, therefore, z sin nz’ — y' cos nz’ = This remarkable surface, which may be called the “skew screw surface,” to distinguish it from another of which we shall speak presently, is that which forms the under surface of a spiral staircase, and is consequently one which is frequently presented to the eye, and is also easily constructed. To find the equation to the tangent plane; the equation to the surface being whence x sin nz — y cos nz = 0, that to the tangent plane is sin 2 (v' ~ x) - cos nz (y' — y) +2 (a cos nz + y sin nz) (z'-2z)=0; De LO may be reduced to cos NZ sin nz yu - xy +n(a+y’)(z'-2z)=0. but since (211) Let the conoid circumscribe the sphere OS iain Yi) ab iV) eetiects nso ee extol the equations to the generator being u=zy-Yr=0, v=z2'-z=0, arene SKEW SURFACES. 169 the equation of Art. (198) becomes 2 of dF da 4 dy dy Pie so that in this case the director consists of (1) combined with w(x-a)+y(y—-B)=0............(2). The equations to the generator a = y > 2£=2 a ‘ Bre 2 2 a) ty Y-B)_ 2 @- a) +y' YB) ty «#(«@-a)+y(y- fp) 0 in virtue of (2), consequently xz (w-a)t+y' (y- B)=0; ie — ws CNL I al Dad CE : 12 129 Y x" yy” ety : U D ’ a “-a whence pany Oe, Vapeo es us tase z+ y and, on substituting in (1), we have (Ba — ay’) = (4? + y”) {r? - (2' - yF} as the required equation. Skew Surfaces having more than one Director. (212) In the families of surfaces which we have been considering there is only one director, so that the requisite eliminations are sufficiently simple; we shall now give some examples of the investigation of the equations of ruled surfaces which have more than one director. The equations to the generator, when put in the form ety 0) Seay Lee een, appear to contain six parameters, but as they can always be reduced to the form z=az+p, y=bze+4q, we see that there are really only four independent parameters. The condition that the generator shall pass through a director gives one relation between the constants, and if there be three directors we have thus three equations involving the constants 170 SKEW SURFACES. which, combined with the two equations to the generator, give us five equations, between which the four parameters may be eliminated, and an equation obtained between z, y, z, which is the equation to the surface. (213) Find the equation to the surface generated by the motion of a straight line which passes through the circumference of a circle, and also through two straight lines at right angles to each other and parallel to the plane of the circle, their shortest distance passing through its centre. ‘Taking the centre of the circle as origin and the axis of x coinciding with the shortest distance, the equations to the directors are (1) e=0, Yist.2°= a, 2) 250, = 0, (8) y= 0 The equations to the generating line are Ti tM pte ay l m n As it intersects (2), we have t-b_2z pes flay x l n L- As it intersects (3) Z+b y i eae PC Re oe eee As it meets (1), a, B, y may be supposed to satisfy these equa- tions, and as a = 0, the equations to the generator give Y z—b eI AY ae relay em. uly.” ad Sind ete b Aeon so that Y 2 a (a+by (@_-by BF is the required equation. (214) Find the equation to a surface defined in the follow- ing manner. On the opposite sides AB, CD (fig. 28) of an — oblique parallelogram are described two semicircles having their planes perpendicular to that of the parallelogram ; the surface is traced out by a straight line which rests on these semicircles and on a straight line MON passing through the centre of the — pn az SKEW SURFACES. stag! parallelogram and perpendicular to the plane of the circles. Take the plane of the parallelogram as the plane of (z, y), and the straight line WOW as the axis of y: then if O be the origin, the equations to the three directors may be written as Be. eaeare dO Gleb ss pacitele « oo CL), iar 0, (es 8a ee SL SI (2), y = b, feb) athe eT ce Peta, oer etre (3). We may at once express the condition that the generator passes through the director (1) by writing its equations iid coy Oy me eat Wal op meee eae CO The conditions that it shall rest on (2) and (8) give the equations fabs) + ayy) (Ol) Sai datilet. ots (5), fa(6—-B)+a¥+7(6-PY=r"........ (6); from which, by subtraction, we have PL COd ghd ts ONG Ve On cesrares, Ses orotate ii) This equation may be satisfied by 3 = 0, but that would corre- spond to the generator passing always through the origin, in which case the surface would be a cone; we take, therefore, the other solution rm yd EN ae (8) ; which, being substituted in (5), reduces it to (B= 8 ligutsnb 7% 2a") 8. a lavyd. 1900) Between the four equations (4), (8), (9) we may eliminate the three parameters a, (3, y, and we obtain as the final equation, which is that of the surface, {ary + b (x + 2*)\? = Br’a’ + 0b? (7? - a’) 2. (215) When there are only two directors, some other geome- trical relation must be assigned in order that the motion of the generator may be completely regulated. The condition most usually taken is that the generator shall remain parallel to a fixed plane ; in this case we may find a general equation for the class of surfaces so generated. ‘Taking the plane of (x, y) as parallel to the fixed plane, the equations to the generator are oe eG, tp ee toy: 172 SKEW SURFACES. Now the conditions of the generator resting on two directors give two relations between the parameters Pla, B,y)=9, (a, B, y) = 93 from which we may find values for two of them in terms of the oe B=9(@), y=4,() Eliminating then a, 8, y between these equations and those to the generator, we have ¥Y = xh (z) + ?, (z) as the general equation to ruled surfaces, generated by the motion of a straight line which is always parallel to the plane of (2, y). (216) As an example of such surfaces take the following. A sphere touches a circle of equal radius in the centre; the surface is traced out by a straight line which, always parallel to a plane perpendicular to that of the circle, touches the sphere and passes through the circumference of the circle. Take the plane of the circle as that of (z, y), its centre being the origin, and the plane of (2, z) parallel to the fixed plane ; the equations to the generator are» — a) Y= Ge uy eee eae those to the directors are GO ESE Mz Oo ge et (2), “+y’+2-2a2=0, BPy+z2-—ae=0...... (3), the last equation being the condition that the generator shall touch the sphere. The condition of the generator passing through (2) gives the equation Wii 7 = Bich s/s wile le Ree le the condition that it shall pass through (3) gives B (a'B + 2ay — By’) = 0. This may be satisfied by 8 = 0, which would lead to the equa- tion of a cylinder perpendicular to the plane of (z, y), a solution which evidently satisfies the geometrical conditions. The other factor gives ZBit2ay.— Sy 0. 4.10 ds Se (5). Between (1), (4) and (5) we may eliminate a, B, y, and we obtain as the equation to the surface xy’ = (a? — x”) (x* — 2a2). Sele sri) Developable Surfaces. (217) In Art. (197) we showed that any developable surface may be considered as generated by the motion of a straight line which is always a tangent to the cuspidal edge. Let the equations to the cuspidal edge be t= (2), y=W(2); then, if 2’, y', 2’ be the current co-ordinates of the tangent, its equations may be written x -$(@)=9 @)@-2), y -¥@)=V@( -2); between these two equations we may eliminate z, and we then obtain an equation between 2’, y’, z', which is the equation to the surface. (218) Thus if the cuspidal edge be the helix Z=acosnz, y=asin nz, the equations to the tangent are x — @ COS nz = — na sin nz (z' — 2), y —a@ sin nz = na cos nz (z' - 2). To eliminate z, multiply by cos nz, sin nz and add, then z cos nz+y' sin nz =a; Squaring both equations and adding, we have a +4? — @ = nia? (z' 2); 5: (2 at y” ke a’) na whence z2=2 3 and therefore , 12 12 2\2 12 12 2\2 ae, H ae ry a . i = x + a a z' cos {ne + tra +y sin {nz + ed, == a; a a which is the equation to the surface. (219) This surface is called the “ developable screw surface,” to distinguish it from that of which we treated in Art. (210). It is obvious from the form of the equation that the surface falls back from the cuspidal edge ; for as the equation becomes im- possible when e+ y? we deduce then from (2), the equations Se ee ag Y, —- 3 seo 0 f from which we find mee y,-B: 2-7 by Of ve (az, + By, + y%) Se = 2 ye b) 2 ? LL, + YY, + %%, = 71, 5 ; =—+2= Are SEES Ly Y, oat r; r 1 and therefore ay + By, + y2,=7, (7, tM). 00 eee O). This being the equation to two planes, shows that there are two circles which are curves of contact corresponding, one to an internal, the other to an external sheet. The equation to a tan- gent plane to the sphere is opie LY Stee 7 nak hia chy Cerne 2,» Y,> 2, beg connected by the equation NON a eee as GY iat ieee ye EE Or (c); and by equation (a). We have now to find the locus of the ultimate intersections of the planes determined by equation (0). Differentiating (4), (c) and (a), considering ~z,, y,, 2,, aS vari- able, we have PATE IUOUN e 20e tt Oe eae a iclezig (2) OF Freres onl) ORIG) PSone (ys d (0) = p(c') + (@’) gives, on equating the coefficients of each differential, AE = px, + a, hy = wy, + B; AZ = 2, + Y- Multiplying these by z,, y,, z, respectively, and adding, we find, in virtue of (a) and (c), «dz, eT y,dy, Ss z,dz, adz, + Bdy, + ydz, Il Are = en +7, (1, +12) Substituting the value of u derived from this equation in the first of the preceding, it becomes Ri (Geer areca re (Te 15) 2,3 190 TUBULAR SURFACES. and therefore, by the symmetry of the formule, wx, Ue Us _ ee . es os (7, a ”,) x, iG ee rt 1) Y Gr ats (", £ ",) a If we multiply numerator and denominator of each of these ratios by z, y, z respectively and add, each of them is by Art. (22) equal to eryre—r? r, (ax a By 37 72) = 2 at PUR Again, if we do the same with a, (3, y, each is equal to ax + By + yz-7,(7, im), CE EY OX a BO nt ae? 2 Equating these two nee we find fertypre—r blast p+y—-(7,+7,) }={ae+ By t+ yz-r,(7,47,) 23 as the equation to the Wot et surface, which touches the two spheres. It is evident that it represents two cones, the vertex of one being between the two spheres, and of the other without. (241) Tubular Surfaces. A tubular surface is the envelop of a sphere of constant radius, the centre of which moves along some line called the axis, either plane or of double curvature. Let the co-ordinates of the centre of the sphere be a, 2, y, then its equation may be written (x-al+(y-BY+@-yyer"...... (ale since the centre lies always in some given line, the two equa- tions to this line give two relations between a, (, y, so that as r is constant, there is only one independent parameter. By the general theory, the equation to the envelop will be found by eliminating a, 6, y between (1) and its differential (c-—a)da+(y — B)d6 + (@-y)dy=0...... (2): combined with the two relations between a, #3, y. The characteristic is determined by the combination of (1) and (2): and as (2) is evidently the equation to a plane, which passes through the point a, B, y and is perpendicular to the axis, the characteristic is a great circle of the sphere; and the tubular surface may be supposed to be generated by the motion of a circle of constant radius, the centre of which moves along a curve to which its plane is always normal. TUBULAR SURFACES. 191 (242) Let the axis of the tubular surface be the straight line Gene ny 7 Tae ae Te es tote (1) 5 then, since (2—al +(y—P) + (@- yl =7r’........(2), (x —a)da+(y-f)dB+(@-y)dy=0........ (3). But (1) gives us da_ dp dy. ea antina” whence (3) becomes l(x-a)+m(y-PB)+n(@-y)=0...... (4). Again from (1) we have ee RE REI OT ee iat bm nm Pam +n? from (4), and therefore a=I(le+my+nz), B=m(et+my+nz), y=n(let+ my +nz). Substituting these values in (2), we have, as the equation to the envelop, {(1 — Px — (my + nz)? + {a — m’)y — m(le + nz) + {C1 - n*)z-— n(le + my)P =7", which is that to a right circular cylinder having the line (1) as its axis. (243) Let the axis of the tubular surface be the circle deter- mined by the equations Reais IEEE [ode CUMS er lpe vin as «8 0 (layne then the equation to the sphere may be written as (Braye (Geo) it SI als pee (2). Since the characteristic is a great circle of the sphere (2), it appears that the envelop in this case is the same surface as that deduced in Art. (223), and it is unnecessary to repeat the analysis, but we may show how to find the edge of the envelop. The result of the elimination of da and dj3 between the differentials of (1) and (2) is Bx - AY = Overseer er cvees (3), the equation to a plane which, combined with (2), gives the characteristic. To determine the edge, we differentiate (3) with 192 ENVELOPS TO SURFACES respect to a and 3, paying regard to (1), and after eliminating these quantities, we find Vom fk y” na 0, which, combined with the equation to the envelop, gives 2 2 2 Sup —c. If r > ¢ this is possible, and combined with the preceding equation determines two points on the axis of z, which represent — the edge of the envelop. If, =— +=, = 0. e-—a@ hve Differentiating with respect to J, m, n, v, we have zal + ydm + 2d = dv... cece cece GL); tdi. mdm — nda = 2 ves. oceen (2); ldl mdm ndn ay (it iia, aa, 2 a fiews& m ne bi = vdv wna 4 CED + a ali oe .tS); —A(1) - # (2) - (3) gives, on equating the coefficients of each differential, y NE = wl + Gave seen seen ee A) hy = uM + ss eevee ee ee eor ees (5), VL) A = “so © @ 0 @ 6 2 6 OG oc @ € 6 8 6 Fd LA (6), hao {eo toe Cm ms a 1(4) +m (5) +” (6) gives, by the conditions, Ae Lek GRE St oa AGF | @(4) + y(5) + 2(6) gives, putting 7’ = 2° +y° + 2’, la my nz Ny? = ww + + LU ae ate eS _— °C 4 le gay nz Se 1p ee ee 9 whence A (7° - v’) pea Poe (9) (4) + (5) + (6) gives ; 2.2 2 peek ee ee ‘ C (v” = ay (v” — b*y (v ~ ey ; whence ’(7” — v”) = : De C7 Je AUG 8 \ovats aes ocean's (10), 1 1 and therefore d= Pre Ta and p= Taye 194 ENVELOPS TO SURFACES. Substituting these values in (4), we have Pe ee ee o(r-v) \ro-’ va}? x ol whence = ri weve 4 oh om Similar] RAINS z On and J See Multiply by z, y, z, and add, then by (9) and (10), A ak Se dD) poe ep oa : This is the equation to the surface of a wave of light propa gated through a crystalline medium. See Fresnel, Mémoires d P Institut, vol. vit. p. 186; Ampére, Annales de Chimie et d Physique, vol. xxxtx. p. 113; and Smith, Cambridge Trans actions, vol. vi. p. 85. Griigs +") CHAPTER XII. ON PARTIAL DIFFERENTIAL EQUATIONS TO FAMILIES OF SURFACES. (246) We have seen in the preceding Chapters that families of surfaces might be expressed by means of equations involving arbitrary functions, on the form of which depends any particular individual surface of the family. We might eliminate by dif- ferentiation the arbitrary functions from these equations, and thus obtain partial differential equations to the families of surfaces : but it is equally possible to obtain these directly from the equations to the generator, as we proceed to shew. Let the equations to the generator be Mie, Y, 2, 2, b, c..2) = 0," f(a, 4,2) a, by 6:1.) 05.0); in which a, 6, c,...are m parameters, connected by - 1 equa- tions of condition ; so that there is only one really independent, of which the others may be considered as functions. Now, to begin, let there be only two parameters, @ and 4, of which b may be taken as a function of a: then, if we differentiate the two equations (1), first with regard to z, and next with regard to Y, considering z, a, and 6 as functions of these variables, we obtain four additional equations, while we introduce three new quanti- ties, viz. db da da We have therefore, on the whole, five quantities, ake db da da da ? dz 5) dy 5) which may be eliminated between the six equations consisting of (1) and their four differentials. It is obvious that the result of the elimination must be a partial differential equation of the first order, since we proceed only to one differentiation. 0 2 196 PARTIAL DIFFERENTIAL EQUATIONS. »* If there be three parameters, a, b, c, on proceeding to the — second differentials, we obtain twelve equations, but we have then to eliminate twelve quantities, viz. a da’ da’ d@° da’ da*4dy") dz dadg sar i which is in general impossible; we must therefore proceed to the third differentiation when we find twenty equations between — which we have to eliminate eighteen quantities, and the result — gives two differential equations of the third order. It is easy to find, in general, the order of differentiation to which we must proceed in order to eliminate m parameters. Let 2 be the required order of differentiation ; then the number of quantities in the series ” de” dy) Ga aacdy may yh roe. Wie is 1(m +1) (2 + 2), while the successive differentials of the m—1_ parameters 6, c... with respect to a, together with the quantities themselves, give (m-—1) (m+ 1) functions; so that the total number of quantities to be eliminated is T(n+1) (+2) + (m-1) (w+ 1). On the other hand, the number of equations, including the original ones, together with their differentials up to the »™ order inclusive, is (2 + 1) (7 + 2). In order, then, that elimination may be possible, we must have (+41) (m+2)>3(m-+1) (m+ 2) +(m-—1) (m+ 1), or 3i(n+2)>m-1, from which the lowest value of 7 is n= 2m — 3. (247) If the equations to the generator be given in the explicit — form th 0, a0 och (0) Mie eee ene (1) the partial differential equation to the family of surfaces is easily — found. For, supposing the functional equation to be i AY 9) 8 eran. 2 eee ee (2), ‘ we have Fee Foyt GeO t CYLINDRICAL SURFACES. 197 Now, if the curve (1) lie on the surface (2), the values of the differentials dz, dy, dz, derived from (1), must satisfy equation (3). But if for shortness we put du dv du dv du dv du dv if du dv du do mca a2 dy” dz dx dxedz ~~ de dy) dy dx "” dx dy _ dz we have from (1) piel Gia Reo ol eee a (4). Eliminating dz, dy, dz, between (3) and (4), we find dF dF dF Pi ast — =0 Be Ea Oe ppg as the required differential equation. dz dz If tp=—,q=—,wW aeEt PR Cg ih ean k dF dF dF dF ——s —— = Ht as a a dy dzt~”? and therefore the partial differential equation may be written also under the form Pp + Qq=R. Cylindrical Surfaces. (248) The equations to the generator are, in this case, i m 1 me l,m, n, being constant: hence dz dy dz PED gl ae” these equations, combined with tT Ute eo, l ee +m ge +n Ae dx dy dz as the partial differential equation to cylindrical surfaces. This equation may be applied to find the conditions that the general equation of the second degree may represent a cylinder. The form of the general equation is Az’ + By’ + C242A'yz+2B2x4+2C'xy+2A'e+2By+2C2+F=0, give us 0, 198 CONICAL SURFACES. and we deduce from the preceding equation 1(Ax+B'2+ C'y+A')+m(By+C'x+A'2+B')1n(Cz+A'y+ Bas C’)=0. Now, so long as the coefficients of z, y, z, in the latter of these two equations, are supposed to be finite, it is evident that it cannot hold good for all the values of the three variables which satisfy the former: we must have, then, since the coexistence of the two equations for all such values of the variables is required by the nature of the case, lA+mC'+nB =0, mB +nA'+IC' =0, nC +IB'+mA’' = 0, LA’ +mB'+nC = 0. These are four relations between only two independent quan- tities (since the variables are, in fact, any two of the ratios — l:m:n); and therefore, in order that they may coexist, there must be two equations of condition between the constants. ‘These are easily found by eliminating J, m,, between the first three, and between the last and the first two, and the results are AA” + BB” + CC” -—- ABC -2A'BC' =0, A'( AC —- BB) + B (BC - AA') + C" (AB - C") = 0. Il Conical Surfaces. (249) ‘The equations to the generator may be written where /, m, m, are the parameters, and a, [, Y, constant. From these we have Lom nn’ and, dividing each member of the latter equations by the corresponding one of the former, dx dy dz a —_— = ea y—P 2-7. by means of which equations, eliminating the differentials from dF dF dF ae dx ae dy dy 32 oh, dz= 0, i : ; } , a ‘ CONOIDAL SURFACES. 199 dF LE LE we find eee) | “Bz — y) a = 9s as the differential equation to Fis surfaces. If we make a= 0, 3 =0, y = 0, that is, if we suppose the vertex of the cone to be at the origin, the equation becomes dk dk dF Dee oe Af eth BEL NO, dx dy dz shewing that # is a homogeneous function of z, y, z. Conoidal Surfaces. (250) If the axis of the surface be the line l m n t-a y-( OLB OR Aer hPL _— noma +r (z—-c)} dx - (da + pdc) = 0, {pq + 8 (z-¢)} dy - (da + pde) = 0, {1+ 9°+t(z-c)} dy —-(db+ qde) =0, {pg + 8 (z—)} dx - (db + qdc) = 0. From these four equations we easily find {lige+r(z-oc)} {it¢g+ie-—e)} —~{pqt+s(z-c)P=0. But, on combining (1) and (8), we have +p +gj(e- f=, ae “+p age which value, subsuneed a in the preceding equation, gives, after obvious reduction, (lip? +9 ay. +p (1+p” +9). {7 (1+9’) - 2pqs SF t(1+p’)} +(rt-s°) p'= 0, as the required differential equation. whence wo = DEVELOPABLE SURFACES. 203 The partial differential equations of developable and tubular surfaces may be investigated also symmetrically in the following manner. Developable Surfaces. (255) Let z, y, 2, be the co-ordinates of any point of a develop- able surface ; a, (3, y, the variable parameters. Then ahs Otte ty Zee deen eriene tet ens G1))s Dade Uap thy aie we cera eres (2); a, 3, y, being subject to two equations, ¢ (a, B, y) = 0} Sicko del she eis es ase 3), x (a, B, y) = 0) Se the functions and x being any whatever. _ From (1) and (2) there is also adz + Bdy + ydz=0. tle ee eta CET: Suppose F’= 0 to be the equation to the sevlofabils surface ; and put dF LE y. dF W. dx dy dz I aoe ene oer, dx” ys pana aa : dydz ° dedx ~° dxdy then we shall also have Cd Vay Widz = OFF on hak fiat (5). By the aid of an indeterminate multiplier, A, we shall get from (4) and (5), observing that, by virtue of (1), (2), (3), 2 and y may be regarded as independent variables, Pie! Now the only equations connecting a, 3, y, , y, 2, with da, dp, dy, are (2), and the differentials of (3); all which three equations are satisfied identically by putting da=0, dB=0, dy=0, 204 TUBULAR SURFACES. without subjecting to any limitation the absolute or relative values of 2, y, 2, a, 3, y. Differentiating, then, the equations (6) on this hypothesis, we get Nee a Urea > Wie sa (udx + w'dy + v'dz), 2 = a — (ody + w'dz + w'dz), 2 = ME - . (wdz + vdx + u'dy). Eliminating dy and dz by cross-multiplication, we get dx / 7 37 Ae! | / t | O(ew-u") + V (w'e'- ww’) + W(w''-v')} = Rdz...(7), where R= uvw — uu? — vv? — ww” + 2u'v'w’, Observing that R is a symmetrical function of w, v, w, uv’, v', w’, it is evident that we shall have also a | Viwu-v)+ We'w'- uu')+ U(u'r'- ww')} = Rdy...(8), ze 1 W (uv -w”) + U(w'u'-w') + V(o'w'-u') |= Rdz...(9). Multiplying the equations (7), (8), (9), by U, V, W, respectively, adding, and attending to (5), we get OU? (vw — u”) +V (wu — v”) + W? (uv — w”) +2VWo'w' - uu') + 2WU(w'' — wv!) +20 V (u'r - ww’) = 0, as the symmetrical form of the partial differential equation of developable surfaces. Tubular Surfaces. (256) Let p be the radius of each sphere; a, 3, y, the co- ordinates of the centre of any one of the spheres; then, z, Y, 2 being the co-ordinates of any point of the envelop, we shall have (c-a) +(y—B) +(-yy=p’........ (1), (x-a)da+(y - 9) dB+(ze-y¥) y=). pee 12) The quantities a, 3, y, are subject to two equations, J (a, B, y) = 0) NOME 0 | eee (3). TUBULAR SURFACES. 205 From (1) and (2), we get (c-a)dx+(y-)dy+(e-y)dz=0......(4). Suppose F’= 0 to be the equation to the tubular surface ; then we shall have also (gy + Vdy + Wdz= 0... 0000000065). Now a, 2, y; Z y, Z, being connected by the equations (1), (2), (3), it is evident that a, 3, y, z, may be regarded as functions of two independent variables, x and y: we have then, from (4) and (5), by the aid of an indeterminate multiplier A, AU+2z-a=0 Ar +y-B-ol Ere uN han es (6). AW+2-y=0 Now the only equations connecting a, DB, y, 2, y, 2, with da, d, dy, are (2), and the differentials of (3); but all these three equations are satisfied identically by putting dae 04d n= 0. dy =03 without subjecting to any limitation the absolute or relative values of x, y, 2, a, (3, y: differentiating, then, the equations (6) on this hypothesis, we get = es or OT ea and therefore, performing the differentiations, (1 + Aw) dx + Aw'dy + Av'dz = ~— Udy, (1 + Av) dy + Aw'dz + Aw'dx = - Vdd, (1 + Aw) dz + Av'dx + Au'dy = —- Wad. Eliminating dy and dz from these three equations, we get =e U0+A{U(o+ w) - Vo' - Wo} + {U(ow — wu”) + ul (Vo' + Ww') - Vow - Woe'}...(7), where R=(1+du) (14dv) (14dw)-0? (w?407 400”) — M(ue"-+00"4-ww"?- 2u'v'w’), a symmetrical function of uw, v, w, w,v,w. We must have, therefore, also - ae V+r0{V(w + u)- Wu' - Uw} + 1{V (wu - v?) + 0' (Ww' + Uu') - Wu - Uww'}...(8), 206 TUBULAR SURFACES. - ae W+r{ Wu + v) - Uo' - Vu} + { W (uv — w?) + w! (Uu' + Vo') - Uov' - Vuu'}...(9). Multiplying the equations (7), (8), (9), by U, V, W, respectively, adding, and paying attention to (5), we get 0= 0+ V*+W* +d {u(V?+W") +0( W740") 4 w(U*+V") — 2u VW - v’WU - 2w' UV} +’ { 0? (ow — u?) + V? (wu - v”) + W" (uv — w?) + 2VWe'w' — uu’) + 2WU (w'u' — vv!) + 2UV (ule — ww’) =0: but, from (1) and (6), \? = gue eR hence we obtain, for the symmetrical form of the differential equation to tubular surfaces, (0+ V4 W*)? + p (U4 V2 4+W) fu (V4 W*) +0(W*+ U’) + w(U*+V") — 2u' VW - 20' WU - 2! UV } +p’ 1 U* (ow — u?) 4 V? (wu - v0?) 4 W? (uv — w’”) +2 VW (v'w' — uu’) + 2WU(w'e' — ve) + 2UV(u'r'- ww )b=0.* (257) ‘The transformation of the partial differential equations from the symmetrical to the unsymmetrical form is readily effected. Suppose, in fact, the equation F'= 0 to be reduced to the form Fe 7 f(a, yY)=0: then it is easily seen that, P»4,7, 8, t, denoting the partial dif- ferential coefficients of z with respect to x and y, according to the usual notation, U=-p, V=—-q, W =1, “u=-T, v=-t, w= 0, u=0, v=0, W=-— 5S. If we substitute these values of the partial differential coefficients of F' in the partial differential equation to the surface, we shall * The symmetrical investigations of the differential equations of Developable ; and Tubular Surfaces, given in the text, have been extracted from the Cambridge — Mathematical Journal, for November, 1844. TUBULAR SURFACES. 207 vt—s’= 0, and the equation to tubular surfaces (l+p'+9°Pt p(1+ p+ g) {r (1+ 9) - 2p98 +¢t(l+p’)} +p’ Gt—s*)=0, the equations to these surfaces obtained in Arts. (253) and (254). ( 208 ) CHAPTER XIII. ON SINGULAR POINTS AND LINES OF SURFACES. (258) In the Chapter on Tangencies it was stated that, under certain circumstances, the equation to the tangent plane becomes nugatory in consequence of the vanishing of all the terms. We now proceed to consider the nature of the points where this occurs. It is to be observed that, since the vanishing of the three dif- ferential coefficients ong lane , Involves either one or two dz dy’ dz relations between x, y, z, besides the equation to the surface, the singularity can occur only at isolated points or along isolated lines, and not throughout any extent of surface. In this and succeeding investigations we shall have frequent occasion to use the differential coefficients of the first and second order of a function of three variables: we shall therefore, for shortness, use the following notation. If the equation to the surface, cleared of radicals and fractions, be expressed by the equation FP (2,°Y, 2) =. 8 to ee then we shall put dF dF dF oe = is dy ead V, We = W, UE | uU ae D) Te w dx’ Oo Chane dz” : TEE pape Ee fre BoE en | dydz dzdz ; dady ‘ 4 Now at any point (2, y, 2), let the variables receive the incre- - SINGULAR POINTS AND LINES OF SURFACES. 209 ments dz, dy, dz, then, by T'aylor’s 'Theorem, the equation (1) becomes F(a, y, 2) + Udxe + Vdy+ Wdz + k(udz’ + vdy’ + wdz*) + udydz + vdzdz + w'dady SCRE) ON re gist voy a. ea RAP Ie The conditions for a singular point are Ee eal = Oma Mas Onde eros mento) 5 which, together with (1), reduce equation (2) to (uw) dx’ + (v) dy’ + (w) dz’ + 2{(u') dydz + (v') dedx + (w') dxdy} H&E, = 077, 204), the bracketed letters indicating the values which they take when we substitute for x, y, z, their values at the point in question. If we suppose dz, dy, dz, to be the limits of the increments of the variables, the equation (4) will, at the limit, be reduced to the terms involving the lowest powers of these quantities ; that is, to (u)da*+(v)dy*+ (w)dz’+ 2 {(u') dydz+(v')dzedz + (w') dxdy\ = 0...(5). If all the quantities (), (v), (w), (wv), (v'), (w'), be zero, we must retain the terms of the development which involve partial differential coefficients of F of the third order, neglecting the remaining terms of the series, and so on successively, until we arrive at an order of partial differential coefficients of which, at any rate, all do not vanish. In the examples which we shall adduce, however, we shall not have occasion to proceed beyond second differential coefficients. Equation (5) gives a relation subsisting between the incre- ments dz, dy, dz, in the surface at the singular point. These are the same for the surface and for a straight line touching it at. the point; and therefore equation (5) gives a relation between the increments dz, dy, dz, on the tangent lines at the singular points: or, since the very same relation must hold good also for the co-ordinates of all points of these lines, we may substitute Z, y, 2, for dz, dy, dz, in (5), and we find (0) @ + (0) y? + (w)2+2(w')yz + 2(o') za + 2(w')ay=0....(6) as the equation to the locus of the tangent lines at the singular point, which is taken as the origin of co-ordinates. ‘This equa- Pp 210 SINGULAR POINTS AND LINES OF SURFACES. tion, except for particular values of the coefficients, is that of a cone of the second degree. It may happen that this equation may be decomposed into two factors of the first degree, and then it will represent two planes. The condition that this may be the case is (24) (&) (w) — (a) WY - (&) (YF = (we) (wf + 2(w') (0') (w') = 0. If it had been necessary to proceed to third differential coeff- cients, we should have found generally the equation to a cone of the third degree, and so on: exceptions arising from the same cause as in the instance of the equation of second dif- ferential coefficients. From the nature of the singular points which we have been — investigating, it is evident that more than one tangent plane will belong to them; an infinite number in the case of the tangent cone, which may be regarded as the locus of the ulti- mate intersections of the tangent planes at the point. It appears therefore, as might have been anticipated, that a plurality of tangent planes is indicated by the indeterminate forms assumed by the direction-cosines of tangency at the point. If the three equations (8) are satisfied by assigning certain relations between the variables, then the curve formed by the intersection of the surface (1) with that indicated by the relation between the variables which satisfies (3), is a locus of singular points; that is to say, it is a line in which two or more sheets — of the surface intersect, at each point of which line the surface will of course have two or more tangent planes. If the equa- tions (1) and (3) are satisfied simultaneously by assigning certain definite values to x, y, z, and not when they receive values differing slightly from these, the point will be single and not one of a series of singular points, and will have a tangent cone. If for possible values of two of the variables on one side of the singular point we find impossible values of the third variable, that point is a cusp. Ifthe same occur at every point of the singular line, it is called an edge of regression (aréte de rebrousse- ment): such, for examples, are the curves which are the loci of © the ultimate intersections of the generating lines of developable surfaces. SINGULAR POINTS AND LINES OF SURFACES. 211 Ex. 1. Find the nature of the point at the origin in the surface (x’ 4 yf = 2"? = az’ + by? ~ oz. Here, putting 2+ y+ 2=?r, UO = 22 (2r" = a"), V = 2y (27° — 8’), W = 22(27 + c’), = 2 (27° - a’) + 82’, v = 2(27" — 0’) + 8y’, w= 2 (27° +c’) + 82’, u = 8yz, v = 82r, w' = Bay. Now when z= 0, y= 0, 2=0, U, V, W, all vanish, while =~20, v= —-20'", w=2e, v=0, v0 =0, w'=0, so that the equation to the tangent cone at the origin is aa + by? — eae Ex. 2. The equation to Fresnel’s wave-surface in biaxal crystals is (7? +y' + 2) (ax + By’ + 2’)- (B40) 2-8 (+0) y¥ -C(@+0)2+a@b'e'=0; find whether it has singular points, and determine their nature. Here U=22{@'(r’-B-0)+@04+ By ic2}, =2y {0 (r-C-a@)+ax + by’ + &2*}, W = 22{(rP-a@- 8) +027 +b y+ &2}, where page +4? + 2" Now if we put y = 0, 7” = 0’, and assume accordingly a — 0b; b? — c’\s -20(S—5), e-na(5—5) ? we shall satisfy the equation to the surface, and also make U, V, and W vanish: hence, as the double signs of z and z may be combined in four different ways, there are four singular points on the surface. To obtain the equation to the tangent cone we must find the values of uw, v, w, vu’, v, wv’, at the singular points. These are readily seen to be = pt - ¢ 2 a 2 w= 8a". = 4, v=- 2(a' - 8) (Bh -*), aa oe cre a> ania w= 0. u=0, v =4ac{(a-d)(b-e Dy} = P2 ore SINGULAR POINTS AND LINES OF SURFACES. : Substituting these values, and dividing the whole by a’ —b’)(0-e sane CALE = 2). a—e we find, as the equation to the cone, 2 2 2 2 f a’ — Cc v2 PH as Aa7e Yost a— B {(@ ix 6”) (8 - cy} . ry The existence of these singular points in the wave-surface was first pointed out by Sir W. Hamilton. Ex. 3. Let the equation to the surface be 2(et+y'+2)+ ax’ + by =0; then U=22(z+a), V=2y(2+b), We=a2'+y'+ 82’. } At the origin, where z= 0, y = 0, z = 0, these three quantities vanish ; therefore there is a singular point at the origin: also u=2(z+a), v=2(¢+6), w= 6z, U = 2y, v = 22, w=0, (u)= 2a, (v)= 26, (w)=0, w')=0, ()=0, (w')=0. The equation to the locus of the tangent lines becomes, then, ax’ + by’ =0, which, a and } being supposed to be both positive, can only — represent the axis of z. The cone in this case degenerates into a straight line; and, as z can never be positive, since that j renders z and y impossible, it appears that the point under — consideration is a cusp. The surface surrounds the negative — axis of z, which it touches at the origin, so that its form resem- bles the shape of the flower of the convolvulus. If a and 6 be of contrary signs, the equation to the locus of the tangent lines is ax’ — by = 0, which represents two planes perpendicular to the plane of 2, y. Ex. 4. Let the surface be the cono-cuneus of Wallis, the equation to which is ay’ - 2 (? — 2) =0. Here U=- x(c’- 2), V=2a’y, We 22°. These all vanish when z = 0, y = 0, independently of the value RE GIS RIM, Fh SINGULAR POINTS AND LINES OF SURFACES. 913 of z; hence the axis of z is a locus of singular points or a singular line. u=— 2(c° —'2"), v u=0, v= 422, w =0. ll 2 2 20°, Ws 22, The equation to the tangent lines becomes, in this case, ay” —(¢ — 2) xz” = 0, where 2’, y’, are accentuated to distinguish them from z, the undetermined co-ordinate of the point of contact. The pre- ceding equation is equivalent to those of two planes perpen- dicular to the plane of zy, diy 2 (One, f. de = 03 ay’ -(? - 22 =0. By assigning different values to z, we obtain different equations corresponding to successive points taken along the axis of z. Ex. 5. The equation to the héligorde dévelopable is . f2rz (v+y'- a) [2 GA Ge ee get) nae een ye pre rasa ae Qnz (a@+y’-a’) Putting oa = §, it may be ascertained that a Ties g 2% cos 0 —- y sin @) | i a(e?+y’ -a’y V = cos Ge Seco ieee a(2+y?-ay 27 ; eters (x cos 9 — y sin 8). But, as may easily be shewn, zcos $- ysinO=(2?+y'- a’); therefore, if we assume 27rz 272 ee EN Ree iia at the preceding expressions will vanish, and therefore the line determined by these equations, and the equation to the surface, is a locus of singular points. This line is the intersection of the surface by the cylinder a a y” pe np z=asin 214 SINGULAR POINTS AND LINES OF SURFACES. and is evidently the generating helix. Since in the equation to the surface z* + y’ can never be less than a’, it appears that no part of the surface lies within the helix, which is ther efore truly an edge of regression. On proceeding to the second differential coefficients, and sub- stituting in them the critical values of z and y, we find, retaining only the terms which become infinite from involving (z*+y*- a’)! in the denominator, . Qqr2z Q7rz A ees 2irZ (uw) = — 2 sin —* cos ales (v) = 2sin * cos ag (w) = 2 h h Bee PUES 27rz (w') = sin® =~ ~ cost SE; h h so that the equation to the locus of the tangent lines is 1! Cy oa (y" — 2”) ay + vy (a? - y’) + Qa 7% 2 (vz +y'y) = 0, where the accentuated letters are the current co-ordinates of the tangents, and the unaccentuated the undetermined co-ordinates of the point of contact. This equation may be decomposed into two factors 2 / / a / x-xXY+2%7r—-2z=0 Y Y h ? ge+y'y=0, which are the equations to two planes. (259) *'There is a species of singular lines on certain surfaces of an entirely different character from the singular lines of which we have treated above. It occasionally happens that a single tangent plane will touch a surface not only at one point, but in a series of points forming a curve line. We shall investigate the analytical condition for the existence of such singular lines in surfaces. The equation to the tangent-plane at any point z, Y, 2, 18 Uz + Vy' + We = Ux+ Vy + We. * Singular lines of this species were first discussed generally by Mr. Greatheed, of Trinity College, in the Cambridge Mathematical Journal, vol. 11. p. 22. ~ a ‘acento a SINGULAR POINTS AND LINES OF SURFACES. 215 Let the right-hand member of this equation be represented by P: then, if the plane touches the surface in a curve line, V ’ 4 Sp? “pp? Temain constant while the co-ordinates vary in agreement with the condition Fey PT Met. and to another condition, which, together with that, determines the curve. OVA ALS Then, as Pp’ P’ A , are all constant, Ee ——_—_—S—_Ee i —- i hence, effecting the differentiations, and denoting each member of this multiple equation by dQ, we get dU =udz + w'dy + v'dz = UdQ, dV =vdy + wdz + w'dz = VdQ, dW =wdz+vdz«+u'dy = WdaQ. Eliminating dy and dz by cross-multiplication, Rdz = { U(ww — u”) + V (w'e' - ww') + W (w'e' — vv')} dQ, where # = uvw — (uu” + vv" + ww”) + Qu'v'w', a symmetrical function of uw, v, w, u', v', w'. Similarly we must have Rdy ={V (wu - v?)+ W(e'w' — uw’) + U(u'e' — ww')} dQ, Rdz = { W(uw - w”) + U(w'w' — wv’) + ;Voo'w' — uu')} dQ. But, from the equation to the surface, we have also the condition Udz + Vdy + Wdz = 0. Hence, multiplying the previous equations by U, V, W, respec- tively, and adding, the first side of the equation disappears by the last condition, and we have U* (vw — wu?) + ;V? (wu - 0”) + W* (uv - wv” +2VWe'w' - uu’) + 2WU(w'd - ov) + 2UV (W'' - ww’) = 0; which equation, combined with F(z, y, z) = 0, determines the curve of contact; and this condition must subsist, in order that the surface may be touched by the tangent plane in a curve. 216 SINGULAR POINTS AND LINES OF SURFACES. This condition may be expressed more briefly in terms of the partial differential coefficients of z taken with respect to x andy. Conceive the equation to the surface to be put under the form f(a, y)-2=0: then U2 9, = 9,. Wo A, i ee ee u=0, v=0, wes: substituting these values in the equation, we reduce it to r¢é-—s= 0. This is the condition which subsists for every point of develop- — able surfaces, as might have been anticipated, since in the case of this class of surfaces the tangent plane at every point touches them along a straight line. Instances of singular lines of the | species which we have been considering occur on the wave- surface, being circles the planes of which are at right angles to the wave-axes. See the Cambridge Mathematical Journal, vol. I. p. 838. CHAPTER XIV. ON THE CURVATURE OF CURVES IN SPACE. (260) Let any number of points P, P’, P’, P”, ... (fig. 29) be taken in a curve AB in space. Join PP’, P’P’, P’P”,... by straight lines: these lines will be chords of the curve, and when the number of the points is increased without limit, will ulti- mately assume ratios of equality with the corresponding elements of the arc of the curve. Produce PP’, P’P’, indefinitely to points 7, 7’; PT, PT’, will ultimately be tangents to consecu- tive points of the curve: and it is evident that the amount of the curvature of the curve in the vicinity of the point P may be properly measured by the ratio of the angle 7'P’T" to the chord or elemental arc PP’, that is, by the rate at which the tangent changes its direction in passing from any point of a curve to a consecutive one. (261) In the osculating plane PP’P’ draw two normals £0, K’'0O, from the middle points A, A’, of the chords PP’, P'P’: these normals will include an angle HO’ equal to the angle TP’ T’: call this angle de; the point O, in which the two normals intersect, will be the centre of a circle passing through the three points P, P’, P’. This circle is called the osculating circle to the curve at the point P, in consequence of having two consecu- tive elements PP’, P’P’, in common with the curve. Each of the distances OP, OP’, OP’, is a radius of this circle. Sup- posing the elements PP’, P'P’, to have been taken equal, then the bent line KPK’ will ultimately be an elemental arc ds, both of the curve and of the osculating circle: and, since it subtends at O an angle dz, we shall have in the limit, p being the ultimate value of OP, 218 ON THE CURVATURE OF CURVES IN SPACE. This result shews that the curvature of a curve, as indicated by the ratio 2 , varies from point to point of the curve inversely as s the radius of the osculating circle: for this reason, this line is called the radius of curvature of the curve at the point. The angle de is called the angle of contingence. The radius of the osculating circle is sometimes called the radius of absolute cur- vature to distinguish it from the radius of spherical curvature, which is the radius of a sphere passing through four consecutive points of the curve. (262) If we suppose the osculating plane PP’ P’ to revolve through a certain angle about the chord PP’, we shall bring it into the same plane with the consecutive osculating plane P’P'P", the three elements PP’, P’P’, P’P"”, being thus brought into a single plane. If the plane PP’P’P” be then turned through a certain angle about P’P’’, we shall have four elements of the curve in a single plane ; and so on indefinitely. Thus, a curve in space may be in this way reduced to a plane curve. Con- versely, by opposite movements, we may change a plane curve into a curve not contained in any one plane. The ratio of the angle between two consecutive osculating planes, which we will call d@, to the length of the elemental arc ds, may be taken as the measure of the torsion of a curve in space at any point. This torsion of curves in space may be regarded as a species of cur- vature, and, as it is of an entirely different nature from the curvature which we have considered above, curves of this class have been called curves of double curvature. The ratio = is s never infinite in continuous curves, and is always zero in curves lying in one plane. (263) To calculate the angle of contingenceat any point of a curve in space. Let x, y, z, be the co-ordinates of any point P (fig. 29) of the curve, ds being the elemental arc PP’: let As Marais ee es ds and let w’', v', w', denote analogous quantities at the consecutive ON THE CURVATURE OF CURVES IN SPACE. 219 point P’. Then, de representing the angle between the tangents at P and P’, we have, putting for brevity de=«,, COS &, = UU + vv + ww’. But w', v', w’, are the values assumed by w, v, w, when 2, y, become z+ dz, y + dy, z+ dz: hence, by 'Taylor’s theorem, @5 : 1 u=ut+du+idu+—durt.... 2.3 ! 1 72 1 3 MME A RO tee eh Se tae ete l phe wt Ue ey Pe ea bt and therefore, attending to the relations U+evrw=1, udu + vdv + wdw = 0, ud*u + vd*v + wd’w = — (du’ + dv’ + dw’), the two last of which result from the first by differentiation, we see that cos «, = 1 — 3 (dw’ + dv’ + dw’) er ~ (ud + vd*v + wd*w)+.. 2 eg oe : but cose, = 1 Gites 2 hence, by taking infinitesimals of the second order on the two sides of the equation, In comparison with which those of the third and higher orders vanish, we get e” = du’ + dv’ + du’, or, restoring the values of w, v, w, (hi, da dy\’ dz \ sore — Me RA te OS 2). de = h d 7, d Fait d Si (2) Performing the differentiations indicated in the equation (2), we have, Ye dy, dz, ds, being all supposed to be variable, de = —, =; {(dsd*e - dad*sy + (dsd’y — dyd’sy + (dsd*z — dzd’s)’}4. P.. squaring the binomials in this expression, we shall obtain, for the coefficient of ds’, (d?x} + (d°y) +(d’zy; =. 220 ON THE CURVATURE OF CURVES IN SPACE. for the coefficient of (d’*s)’, dx’ + dy’ + dz’ = ds’; and, for the coefficient of ds d’s, — 2 (dx d*x + dy d’y + dz d*z) = - 2ds d’s: hence we see that ihn = {(d*z) + (d°yy + (d°2} - (d’s¥ 4... (3). From (3) we get de* ds* = {(d’x) + (d’yy + (d’z)} ds® — ds’ (d’s)’; and therefore, since ds’ = dz’ + dy’ + dz’, and ds d*s = dx d’x + dy d’y + dz d’z, we have de'ds* = {(d’x) + (d’yy + (d’zY} (dz’ + dy’ + dz’) — (dxd’*x + dy d’y + dzd*zy = ne d’z-dz d’yy + (dz d’x-dx d’z) + (dx d’y — dy d’xy, ds = —. <3 Udy d*z-dz d*y)' + (dz d’x- dz d°z)+(dz d’y- dy dx) \4 " - (4). It may be remarked, that the three binomials ade the radical are the same as the coefficients of z'- z, y'-y, 2'- 2, in the equation to the osculating plane, Art. (180). From the different expressions (2), (3), (4), which have been obtained for de, we may get a variety of formule for the radius of curvature at any point of the curve, by virtue of equation (1). Thus, taking the expression for de given in (3), we have ds* O* {aay + (ayy + (Czy - sy} (264) This method of obtaining the radius of curvature serves for the determination of its magnitude, but gives us no informa-_ tion respecting its position. We shall proceed, therefore, to — develop another method of investigation, which will determine — at once both the length of the radius of curvature and the co-ordinates of the centre of the osculating circle. Hl The normals AO, £'O (fig. 29) are ie intersections of the osculating plane PP’P" with two consecutive normal planes‘ ON THE CURVATURE OF CURVES IN SPACR. 291 hence the determination of the centre of curvature O is coin- cident with the determination of the point of intersection of these three planes. The equation to the osculating plane at the point P, the co-ordinates of which are z, y, z, will be (Art. 180) A (vw -—2)+By'-y)+C(z'-2=0.... (5); where A = dy d*z — dzd’y, B = dzd’x — dz d’z, C= dzd’y — dy d’x. The equation to the normal plane at the same point is (Art. 178) (a — x) dx +(y' -y) dy + (2-2) dz=0....(6). The equation to the normal plane at the consecutive point z+dz, y+ dy, z+dz, will be (x — x) (dx + d’x) + (y' — y) (dy + d’y) + (z' — 2) (dz + d’z) — (dz’ + dy’ + dz)=0; and therefore, at the point in Which the two consecutive normal planes intersect, (v'— x) d’x+(y'-y) d’y+(z -2z) d’z-ds*=0....(7). Multiplying (5), (6), (7), by 1, \, X’, respectively, X and 2’ being indeterminate multipliers, adding together the resulting equations, and equating to zero the coefficients of y —y and z —z in the final equation, we have (7 —2)(d+rAdx+ 2 dz) = 'ds®.... (8), B+rdy+N@y=0 ...... (9), OF dz PN id*2 = 00000" (10). From (9) and (10) we have Bdz — Cdy = (dy d*z — dz d’y) ='A, and Cd*y — Bd’z = (dy d’z — dzd’y) =A; and therefore, from (8), (z — 2) { A’+ dx( Cd*y- Bd’z)+ d’x (Bdz — Cdy)} = ds® (Bdz- Cay), or (z' — x) (A’ + B’+ C”) = ds’ (Bdz — Cady). 999 ON THE CURVATURE OF CURVES IN SPACE. Similarly we have (y' —y) (4° + B’+ C’)= ds’ (Cdz — Adz), (2' — z) (A? + B’+ C’) = ds’ (Ady - Badr). Hence, observing that p=(@- al +(y- yf + @-2y, we have p (A?+ B+ C’)= ds® {(Bdz - Cdy) + (Cdz — Adz) + (Ady- Bdzy\4. — Developing the squares of the three binomials under the | radical, the terms multiplied by A’, B’, C’, will evidently be A’ (dy? + d#) = A’ (ds* - dz’), B’ (dz + dz’) = B’ (ds’ - dy’), C? (dx? + dy’) = C’ (ds’ - dz’): thus the radical becomes {(4?+ B’+ C*) ds’ - (Adz + Bdy + Cdzy}}. But, restoring to A, B, C, their values, we see that Adz + Bdy + Cdz is identically zero: hence the radical becomes (A? + B’+ C)4 ds; and for the radius of curvature we get ds® O” (44 BPs OF! f ds® ~ {(dy d*z- dz d’y) + (dz d’x — dx d*z¥ + (dx d’y — dy dx} ’ a result which agrees with the formule (1) and (4). If in the numerators of the expressions for the co-ordinates x, y', 2, of the centre of curvature, we restore the values of A, B, C, we shall have, in the case of the first, Bdz — Cdy = d’x (dy’ + dz’) - dx (dy d’y + dz d*z) = d’x (ds® — dx*) — dz (ds d’s —- dx d’zx) = d’z ds’ — dx ds d’s, analogous expressions resulting for the second and third: we shall have, therefore, ON THE CURVATURE OF CURVES IN SPACE. 293 As dx dsd*x-dzd’s_ , \ds Wa Boe & as wee Gk bs x -xz-=ds. ds d'y—dy ds, Z) |e Ge sete Dae es De oe y¥ Yy s By bat iN oe Se OL: P ds MODE . pn ds dz tade Wis ek \ ds Teele a NUT er ey rey 5, eal BE pctes (US Let a, (3, y, denote the angles which the radius of curvature p, estimated in the direction from z, y, z, to x,y’, 2’, forms with the co-ordinate axes: then z COS y = From these three formule, observing that cos’ a + cos’ (3 + cos’ y = 1, we obtain ds_ {/ ,dx\ dy \* dz\\3 —=/|d — d — = lipides 0 ( x) 3 ( ds ‘ (a a} (14), a result which is in agreement with the formule (1) and (2). (265) ‘The determination of the centre and radius of the osculating circle may be effected also by the following simple method.* Let PQ, QR, (fig. 30) be two consecutive elements of the curve, and let us suppose them to be equal, which is the same thing as considering ds constant or taking s for the independent variable. Complete the parallelograms PQRS, SQTR, and let M, N, M', N’, be the projections of P, Q, R, T, upon the * This method was given by Mr. Archibald Smith, of Trinity College, in the Cambridge Mathematical Journal for February 1838. 224 GN THE CURVATURE OF CURVES IN SPACE. axis of z. The centre O of the osculating circle, which passes” through the three points P, Q, R, will evidently le in the line QS' produced. Let Q' be the extremity of the diameter through Q: join RQ! and PR: QS will be bisected by PR in a point V. Then, QRV, QQ'R, being right-angled triangles, we have Oh QQ' x Ole or, putting QO0O=p, QS=\A, QR=ds, ds” = pi. Let a, 8, y, be the angles which QO makes with the co- ordinate axes; then QS cosa= RT cosa=M'N': but M'N' = NN'- NM'= MN - NM = dz: hence Acosa=d’2, d*x and therefore cosa=p a. Sunilarly cos 3 =p ay ; ds* ae cosy =p aa: Hence, observing that cos’ a + cos’ 3 + cos’ y = 1, Segue) all es ye (4) A (af If we change ‘af independent variable from s to any other quantity, we shall arrive at the formule obtained above for p. Thus, the formula (14) results immediately from this expression | for p, by introducing the alteration corresponding to the suppo- sition that ds is no longer constant. | Again, 2’, y', 2, being the co-ordinates of the centre of cur- vature, we get, putting for p its value just obtained in the | expressions for cos a, cos (3, cos y, Lat 0 COST = nce ae ne ag fe ds?’ d” y'-y =p cos B = ps PAZ z-z2=pcosy=p" ae s ON THE CURVATURE OF CURVES IN SPACE. 295 (266) To calculate an expression for dO, the angle of torsion. The equation to the osculating plane at the point z, y, z, being Az’ + By'+ Cz = D, the equation to the consecutive osculating plane will be Az'+ By +C2 =D, where Ara A dA. B' = B+dB; Ca Ca dG. Now, d@ being the angle between these two planes, which for brevity we will call 6, AA'+ BB'+ CC’ ; (A+ B+ Oh (A? +.B? 4 OV cos @, = and therefore (BC - CB’? +(CA'—- ACS +(AB'- BA'S £ CAR? SC Ca B eG ty _ (BdC — Cab)’ + (Cd A - AdCY + (AdB ~ Bd AY (4° + B+ CY / But B=dzd’x-dxd’z, dB=dzd'x- dz d’z, C=dzd’y-dyd’z, dC=dxrd'y — dy d*z, and therefore BdC'— CdB = dz {dx (d*y dz - d*z d°y) + dy (d*z d°x — dx d°z) + dz (d*x d*y ~ d’y d*x)\, analogous expressions existing for the two other binomial terms in the numerator of the formula for sin? 0. Hence, putting dO for sin 6,, and ds* for dz*+ dy?+ dz’, and taking the square root, we have dO dz (dy @z-@zd°y)+ dy (C2d'x-Pxd'z) + dz(@xd'y-d’y Fz) ds (dy d*z— dz d’y) + (dz d’x — dz d°zy + (dx d’y — dy d*xy Sette an te bone which is a measure of the rate of torsion at any point of the curve. (267) Points of Inflected Torsion. 'The total amount of the torsion of a curve, as we pass along any proposed length of the arc, is equivalent to an angle 9, that is, to the sum of all the successive values of d0: if the torsion, after having for a certain space taken place in one direction, then assume an opposite course, the torsion at the point of the curve where this change Q. sin’ 0, 226 ON THE CURVATURE OF CURVES IN SPACE. takes place may be said to be ¢flected, and the point itself ma be termed a point of inflected torsion. In passing along the are through such a point, the measure of torsion must evidently — change sign. Hence, by formula (15), we see that the condition for a point of inflected torsion coincides with a change of sign in — the expression dx (d’y d°z — d’z d’y) + dy (d*z d’z - &z d*z) + dz(d*x d*y — d’y d’x).... (16). If the change of sign take place so that the expression (15) pass through a zero value, the change in the character of the — torsion will be continuous: if this expression pass through infinity, the change of torsion will be abrupt, two consecutive — osculating planes including an angle of —180°. The point will therefore be cuspidal. (268) Points of Suspended and of Infinite Torsion. The expression for the measure of the rate of torsion given in (15), _ may be zero at a certain point, although it may not change sign as we pass through the point. The torsion does not in this case change its character, but is merely stationary for a small portion of the arc: the point may therefore be called ~ a point of suspended torsion. If the expression for the measure of the rate of torsion be infinite, and there be no change of sign, _ the increase of the torsion will be abrupt, and the angle between — two consecutive osculating planes will be + 180°: such a point — may be called, in relation to the rate of torsion, a point of — enfinte torsion. i If the expression (16) be satisfied identically by the equations _ to the curve, for all simultaneous co-ordinates, the curve will — le entirely in one plane; a conclusion which agrees with Art. (145). a (269) Points of Inflected Curvature. If, as we pass along a particular portion of a curve, the angle dé lie first on one side of the tangent at each point, and afterwards on the other, it is — evident that the nature of the curvature undergoes alteration, concavity and convexity being interchanged. ‘This will be clear on inspecting fig. (31), where three consecutive elements, PP’, PP’, P’P", are drawn, the angle ds, in the osculating’ ON THE CURVATURE OF CURVES IN SPACE. Q227 plane PP’P’, being below the tangent at P, and, in the oscu- lating plane P'P’P, above the tangent at P’. The point where the change takes place may be called a point of wnflected curvature. Suppose first, that the change of the character of the curva- ’ d ture 1s continuous: then, since * must change sign through S zero, we see, from formula (3), supposing s to be the inde- pendent variable, that goed ; d*y — ‘ dz de as) = ery a If we suppose 2 to be the independent variable, then, from formula (4), we see that, as necessary conditions, SANE ala d’y d’z dx* dz : Rah, et = Ov eee sees. (18). da’ dix? If = be finite, these two conditions evidently reduce z themselves to ay = 0 diz 2 dat i a a: When es is infinite, the conditions will occasionally be ke satisfied when d'y dz dei? da If the change of the nature of the curvature be abrupt, then dz ds curve will be inclined to each other at an angle of — 180°. Hence, in one or more of the equations (17), or of the equivalent equations (18), 0 must be replaced by . (270) Points of Suspended and of Infinite Curvature. If a will pass through w, and two consecutive elements of the change of sign do not take place in the value of = when It passes through zero, the curvature is merely suspended; and if, without changing sign, its value pass through infinity, Q 2 228 ON THE CURVATURE OF CURVES IN SPACE. the rate of curvature is infinite. Such points may be called respectively points of suspended and points of infinite curvature. (271) Points of Inflected and of Suspended Torsion are ordina- rily comprehended under the appellation of points of simple inflec- tion ; an essential property of such points being the coincidence of two consecutive osculating planes. Points of Inflected and of Suspended Curvature are ordinarily denoted by the common name of points of double inflection ; two consecutive elements at such points He in a single line. These points have been called points of double inflection, because their existence involves that of points of simple inflec- tion. I have adopted different appellations for such singular points, because I think the ordinary terms do not correspond with sufficient distinctness to the true geometrical peculiarities of the points. (272) There isanimportant distinction between plane curves and curves of double curvature, in regard to their radii of curvature. In plane curves the radii of curvature intersect each other con- secutively, the locus of these intersections being the evolute, to which all the radii are tangents; while in curves of double curvature the radii of the osculating circles do not meet each other consecutively. Through the middle points A, A’, & i. of the several elements PP’, P’P’, P’P’,....of the curve PP'P’,. ...(fig. 32), draw normal planes L, L’, L’,....intersect- ting each other consecutively in the straight lines AB, ‘ACB and thus forming a developable surface, the envelop of all the — planes. If we cut the planes L, L’,. .. .by the osculating plane PP'P’, which is at right angles to both of them, the lines of 2 intersection will be the normals AC and KA’'C, perpendicular to AB, and of which the former will be the radius of curvature of the curve at the point P. In the same way, cutting the normal planes L' and L’ by the osculating planes P’P’P", we shall have, for the sections, the two normals A’C’ and A’C’, perpen- dicular to .A’B’, the former of which will be the radius of curva- ture at the point P’. Now it is evident that the radius A’C" does not coincide with the other normal A’C, because these — two lines are the intersections of the same plane L’ with two ON THE CURVATURE OF CURVES IN SPACE. 229 different osculating planes: hence A’C’ will meet AB in a point J different from C, and consequently the two radii of curvature, AC" and A’C"’, situated in the planes Z and L’, have not a common point at the intersection of these two planes: it follows, therefore, that these two radii do not meet. From what we have said, then, it appears that the centres of curvature C, C", C’,....do not result from the successive inter- sections of the radii AC, K'C’, K’C’,. .. .and that consequently these radii are not tangents to the locus of these points: the radi of curvature cannot therefore be regarded as formed by the unwrapping of a thread wound about the curve CC'C’....: in other words, the CC’C’,.... is not an evolute of the curve PP'P’....whenever this latter curve has double curvature. (273) Although the locus of the centres of the osculating circles is not an evolute of PP’P’....; yet, as Monge has shewn, this curve may be shewn to possess an infinite number of evolutes. In fact, ifin LZ, the first of the normal planes, we draw arbitrarily a straight line AD (fig. 32), which will always be normal to the proposed curve ; and then, through the D and KA’, draw another straight line A’ DD’, which will lie in Z’, the second normal plane ; then a third line A’D'D’ situated in the plane P’, and so on successively, we shall obtain, by the successive inter- sections of these normals, a curve DD'D’....to which these normals will be tangents. The curve PP'P’....may evidently be described by unwrapping a string wound about the curve DD D',....which will thus be an evolute of the former. In proof of this, it is sufficient to observe that the portions DX and DK of the tangents to DD'D’. ...are equal to each other, or that the point D is at the same distance from the three points M, M', M’; for it is clear that each point of the line AB, which is the intersection of the two planes Z and ZL’ drawn at right angles to the elements PP’ and P'P’ through their middle points, must be at the same distance from P, from P’, and from P". Moreover, since the first normal AD was drawn arbitrarily in the plane LZ, we may, by varying the direction of this normal, obtain an infinite variety of evolutes all situated on the enyelop- ing surface of the normal planes. 230 ON THE CURVATURE OF SURFACES. CHAPTER XV. ON THE CURVATURE OF SURFACES. (274) A plane is said to be normal to a surface when it con- tains a normal line. If at any proposed point of a surface, a series of normal planes be drawn, the radii of curvature of the various normal sections of the surface at the point will generally vary: from a comparison of the curvatures of the different normal sections, we shall arrive at a conception of the nature of the curvature of the surface around the point in question. The radius of curvature at any point of a plane normal section of a surface is determined by the intersection of the normal to the surface at the point with the normal plane at a consecutive point of the curve of section. (275) Let the equation to the surface be Eas affizy 07: then, adopting the notation of Chapter x111., we shall have, for the equations to the normal at the point (z, y, 2), aw-x y—-y z2-z ii a | SACS, Now the direction-cosines of any tangent line are de dy ds ds’ ds’ ds’ and the tangent line may be taken as that which touches the curve of section at the point where the normal is drawn. Hence the equation to the normal plane, at the point (x, y, 2), will be (w'- x) da + (y'- y) dy + (2-2) dz=0......(2). Since, by differentiating the equation to the surface, we get Udz + Vdy + Wdz=0.........00, (3), ON THE CURVATURE OF SURFACES. 231 it is evident that the line (1) lies in the plane (2): the inter- section of (1) with the consecutive normal plane will there- fore lie in the line of intersection of (2) with the consecutive normal plane. ‘The equations to this line of intersection are (2), and an equation obtained by differentiating (2) with respect to Ws Y, 2 Viz. (a — x) d*x + (y'- y) d’y + (2 — 2) d®z = dz’ + dy’ + dz = ds’...(4). The equations (1) and (4), taken together, determine 2’, 7’, 2’, the co-ordinates of the centre of curvature of the section, and the radius of curvature is the distance between that point and (x, y, z). From (1) and (4), there is BiG yoy 2-8. ds® (5): U V W Ud’x+ Va’y + Wd’2 0 *” hence also, p denoting the radius of curvature, 2 2 2\1 2 pas {(e'- a +(y'-y) + - 2) a2 Ga phe (0) which is one expression for the radius of curvature. (276) We may eliminate the second differentials d*z, d’y, d*z, from the equations (5) and (6) in the following manner. Dif- ferentiating (3), we have Ud*x + Vd’y + Wd’z + udz’ + vdy’ + wdz" + 2u'dydz + 2v'dzdz + 2w'dxdy = 0: availing ourselves of this equation, and putting soe) Y= m, Tea we readily transform the equations (5) and (6) into Pu+ mo + n’w+ 2mnu' + Qnlv' + 2mw' ae ke Boa pay 2-20 +(U’+V"+ Ww’ PO” Pus mo + nw + Imnu + Qnle' + Zw! and (277) There is no condition which will enable us to select one _ of the two signs introduced by the radical into the expression Cae ON THE CURVATURE OF SURFACES. for p in preference to the other, although at each point of the — surface for an assigned normal plane p must have some determi- nate position. The ambiguity of sign indicates that it is quite arbitrary which side of the tangent plane we adopt for the — positive direction of p. ‘The actual position of p can be ob- — tained only from the equations (7), the equation (8) serving — merely to determine its magnitude. : (278) The equations (7) and (8) may easily be transformed — so as to involve the partial differential coefficients of z with respect to 2 and y, instead of the partial differential coefficients _ of F(z, y, 2) with respect to 2, y, z. Conceive the equation to the surface to be reduced to the form F(a, y, 2) =f(&, y)- z= 0, z being thus rendered explicit. Then it is easily seen that dz dz TEEN ee a ee rte la Celia d’z d*z Mimnleysa olla? Sea? Wi VU. a= o = 0 ys Be =§ es “se — dady — The equations (7) and (8) will therefore become L*r + 2lms + mt a F(l+p’+q°) and = 2 p l’y + 2lms + mt (279) For all normal sections passing through 2, y, z, the BES UV |W, use, oma, ws | are constant; but the expressions for z- 2’, y—y', z — 2’, p, will change as /, m, n, vary; the variation of 2, m, , taking place in accordance with the two conditions Ps+msnr= Cle penete es eevee vised ON LO PMV tn W =O, BA BAO ON THE CURVATURE OF SURFACES. 933 the latter of which expresses the perpendicularity of the tangent line to the normal. Since, at any assigned point of the surface, the quantities z-x,y-y', 2-2, m consequence of the variations of 7, m, n, have changes of magnitude, it is possible that they may likewise experience changes of sign. Should a change of sign take place in the value of any one of them, it will take place simultaneously in the values of all, the expression Pu +m'ov + nw + 2mnu' + 2nlv' + 2lmwi', which is common to all of them, being the only variable element in their values. Such changes of sign indicate that the centre of the circle of curvature must lie for different sections in oppo- site directions from the point (z, y, z), or that the surface in the vicinity of the point lies partly on one side and partly on the other side of the tangent plane. We proceed to ascertain under what conditions these changes of sign can take place. Put Me ieee nse A being a constant: hence we see that a change of sign in z2-x, y-y, 2-2, will take place simultaneously with a change of sign in &’: but, from (7), we see that R (Cu + mv + n?w + 2mnu' + Anlv' + 2lmw') = A; meepuune hi-2, m= y,, fin = 2, un” + vy, + we’ + Qu'y,2,+ 2v'2,2,+ 2way,= A... .(11): we have also, from (10), Uz, + Vy, + 2 Oeeicte ee turesie oe (12). Thus we see that a change of sign in the values of 2-2’, y—-y’, z-—2', is coincident with a change of sign in the square of the -radius-vector of a central conic section, of which the equations are (11) and (12). Ifthe conic section be an ellipse, all its radia are possible, and therefore their squares are always positive, and the values of x-z', y-y', 2-2’, have always the same sign. If the conic section be an hyperbola, some of the radu are pos- sible and some impossible; their squares may, therefore, be 234 ON THE CURVATURE OF SURFACES. positive or negative, and the changes of sign which we are considering may take place. The nature of the conic section will be best seen by sup- posing the surface to be referred to the tangent plane at the ‘ point (x, y, z) as the plane of zy, and then the equation to the | conic section is reduced to 4 ux,” + Qw'ey, + vy, = A, 2, = 0. The equation will be an ellipse if uv —-w"> 0, and an hyperbola, if uv —-w" <0. If it should happen that wo - w”? = 0, then the hyperbola will degenerate into two straight lines: also ah Au - Av (lat mY (ho! + mop’ | so that #’ never changes sign: if the relation between J and m — be such that Jy 4 gp’ = 0, or lw'+mv=0, these two relations being really one and the same, then z -— 2’, y-y', 2-2, p, all become infinite. The relation uv —-w"=0 | is satisfied in the instance of developable surfaces, the infinite — values of z — 2’, &c. having relation to sections along the gene- _ rating lines of the surface. (280) To find the greatest and least radii of curvature of the normal sections at any point of a surface. If we put +(U*+ V?+ Wy = P, we have to make P —=l*u + mv + nw + 2mnu' + Qnlv' + Zlmw'....(18) p a maximum or minimum; /, m,n, being supposed to vary in — agreement with the relations (9) and (10). From (10), we have mV? + ImnVW+rW? =U’, PU? —~ mV? —- nw 2mn = vw ON THE CURVATURE OF SURFACES. 235 mV? —nW? —-2v? similarly 2nl = wo : Sie a ””* Ww? 3 "E CV? nS mM ie T UV Substituting these expressions for 2mn, 2nl, 2/m, in (13), we get Lae LA es On ee Tarn ree iis, (14), p U / / U where ET rp (Ue —- Vo' - Ww') eee, “i V (Vi / Wrap! Ui ! 15 = wo (Ve - We'- eae (15). Ww 1b —— (Ww' — Uu' - Vo' Ww + Tyr | w — Uw — Vr') That p may be a maximum or minimum, we must equate the differential of ui to zero: hence, from (9), (10), (14), we have ldl + mdm + ndn = 0, Udi + Vdm+ Wdn = 0, Eildi + Kmdm + Indn = 0. Eliminating dl, dm, dn, by indeterminate multipliers, we have (7+ A)l-nwU0=0 (H+A)m-uV= ; (L a d) LEI I? W=0 ‘Multiplying these three equations in order by J, m, n, and adding, we have, by (9), (10), (14), ii +A=0. Pp Hence, from the equations (16), Fie EE wz P WATS it Pay 2 eee Cla} Pp uW 236 ON THE CURVATURE OF SURFACES. Multiplying these equations in order by U, V, W, and adding, we have, by (10), U? y? w oe =o Our tee (18) oes Ree ee p p p This quadratic equation determines, in terms of the quan- tities U, V, W, H, HK, L, P, which are known for any point of the surface, the greatest and least values of p at that point. By substituting either of these values in (17), we determine the ratios 2: m:n, which give the position of the corresponding normal section. (281) These equations also enable us to prove that the normal sections of greatest and least curvature are at right angles to one another. Thus, if J, m,, ,, and J,, m,,,, be the values of. /, m, n, corresponding to p,, p,, the greatest and least values of p, the equations (17) and (18) must be satisfied when each of these systems of values is substituted for J, m, m, and p. Hence, writing down (18) for each value of p, and subtracting one equation from the other, we get | (eo) ee | eal Cmrnl etre Cee) fe Py P2 P Po 72 cea Py Po i Hence, if p, and p, be different, the second factor must be equal to zero, or, which is the same thing, on account of equations( 17), Li,+ Mm, + nn, = 0; 4 which shews that the sections are at right angles to one another. The normal sections of greatest and least curvature at any point of the surface are called the principal sections, and the radii of | curvature the principal radi of curvature.* : | * The investigations which I have given for the determination of the principal radii of curvature and of the positions of the principal sections, were communicate 4 to me by Mr. Thomson, of St. Peter’s College. +. ON THE CURVATURE OF SURFACES. 257 (282) In the case of an ellipsoid, 2 2 2 Aa ae at yee 22% 2 22 we have U = a Vax, W=—, 2 2 2 OF iva? vag) pa? Gna? u=0, o =0, w =0, hall Vig cant 2 Pas2(Seh eS) =<, GauerU.l 6 p p being the perpendicular from the origin on the tangent plane. Hence the equation (18) gives us x ye 2 a’ (pp—a’) 8 (pp-5°) & (pp- ce) From the last term of this quadratic when cleared of its frac- Dds apo) ape tional form, which is equal to eae it appears that the product of the greatest and least radii of curvature is constant for all points for which p is constant. ‘The equation for p may be put into another remarkable form: for if we write it thus, a y 2 ar LN Sane ESSE TOSS oy Gini ON Ce) ie) aes) PP PP pe and subtract it from the equation to the ellipsoid, we see that x 9 2 2 a woe we “ a Ppa eC pp this is the equation to a concentric surface of the second order, which will be also confocal, since, if a’, 0’, c’, be its semi-axes, Ll: / yA 12 2 12 2 a’=a—pe, b°=b'-pp, ec =c -—pp, and therefore (283) The perpendicularity of the principal sections may be concisely established also by the following reasoning. 238 ON THE CURVATURE OF SURFACES. Putting ” = 0 in the equation (13), which amounts to taking the tangent plane at the point in question for the plane of zy, we have P — = l’u + Qlmw'+mo...... (19), Z and m being maigcer es the condition Dn 1 ee (20). Differentiating these two equations with regard to / and m, and putting dp=0, we get, by means of an indeterminate multiplier A, bi Die Ale Oe (21> mov +lw' =AM. 2... 000. .(22); whence ( — 0) lm + w' (mt — 2") = 0, mua m i ice hence, /,, m,, and /,, m,, being the values of /, m, for the principal sections, we see that m. m , cae ee eee d, é, which is the condition for the perpendicularity of the sections. (284) To prove that the curvature of any normal section is equal to the sum of the curvatures of the two principal sections, multiplied respectively by the squares of the cosines of the angles which the principal planes make with the normal plane. Multiplying (21) and (22) by 7 and m respectively, and adding, we get by (19), P > p and therefore, eliminating / and m between (21) and (22), (« ~ =| [> — =) = Udi wes Ae (28). Let p,, p,, denote the principal radii of curvature, and suppose that the planes of xz, yz, coincide with the principal planes. Then, from (19), putting 7= 1, m=0, p=p,, simultaneously, we have P —=U: P; similarly, putting 7=0, m=1, p=p,, Ih — = v. P2 ON THE CURVATURE OF SURFACES. 239 Thus « and v are the roots of the equation (23), which is . ° le ° , . a quadratic in —: hence it follows that w'=0. We obtain, p therefore, from (19), 120) Cop Lene — = bu +mo= P| +4+—}; p Pi —s Pa or, if /=cosa, m=sina, 1 cos’a_ sin’a = + p Py Pe which establishes our proposition. ere. COA), (285) We have shewn that if the planes of xz, yz, coincide with the principal sections, we must have, as a necessary con- dition, w'= 0. It may easily be seen that this condition is also sufficient; for, putting w' = 0 in (21) and (22), we get lu=Xl, mv=2Xm, and therefore mlu = mlb ; an equation which may be satisfied by 7=0, m=1, or by Z=1, m=0. (286) The formula (24) may be established also in the follow- ing manner.* Let (1, m,,”,), (4, m,, ”,), be the direction-cosines of any two lines through a point P, at right angles to one another, in a plane of which the direction-cosines are proportional to wey, We. ‘Then OE + Ven eee 0s... (a), Ele alae palin, =. ON toe aceee) (0): Again, let (@, m, n) be the direction-cosines of a line in the same plane passing through P, and making an angle 6 with the line (/,, m,, »,). We have then OL Vine a 0. og team heb ii, + mm, + nn, = cos a l,+mm,+nn,=sina;...... (ad). L/,+ mm, + 2,n, = 0 * This method of investigating the formula was communicated to me by Mr. Blackburn, of Trinity College. 240 ON THE CURVATURE OF SURFACES. Now, X\,, A,» being arbitrary multipliers, (c) — A, (a) — A, (4) } gives NTE | m =\,m,+ AM, : n= Xn, + AN} whence 2 2 2 i, + mm, + nn, = dr, (0 + m+ n°) + A, EE, + mm, + n,n), or, by (d), COSia = i\ re similarly, sin a = A,. Hence l=l,cosa+d,sina m =m, cosSa+ mM, sin + me tiie Pe (e), n=n,cosa+n, sina a geometrical theorem. Let the plane considered be the tangent plane to a surface at a point (z, y, z), and let (2, m,, ,), (4, m,, ”,), be they direction-cosines of the normals to the principal normal planes. Then, by the formule (17), Hence, by (c) and (d), Alli, + Kmm, + Inn, = Zs COS a, — 1 All, + Kmm, + Lnn, = af sin a: Po pA A CES Al + Km’? + Ln’ = ni : We have therefore, by (e), cos* a sin’ a S——- oe icceasinnnaresanateie pals Py P2 p (287) The formula (24), the discovery of which is due to Euler, is extremely valuable, as it enables us at once to calculate the radius of curvature of any proposed normal section when the ON THE CURVATURE OF SURFACES. 241 radu of curvature of the two principal sections are given. If instead of making use of the principal sections of the surface, we were to take any normal planes whatever at any point, it would be necessary to introduce the radii of curvature, Rh, R', R’, of three such sections, and the angles a’, a’, con- tained between them, into the expression for the value of p. Thus, taking one of the normal sections as the plane of zz, if we put in the formula (19), successively, fo m= 0, p=; l= cos a’, m=sina’, rene l= cos(a' +a’), m=sin (a +a’), p=R'; we shall have three equations from which we can find U, W, v, in terms of R, R', R’, a’, a’. Hence, by (19), we might deter- mine the value of p for any proposed normal section whatever, /in terms of the radii of these three normal sections and their contained angles. (287) If p, p', be the radii of curvature of any two normal sections at right angles to each other, to prove that 1d ee ar) ana Sh Se ae Pp PP, Pe Putting in the formula (24), a +} instead of a, and replacing p by p’, we have Ie sin’® a Rr COS’ a e Py Pe 1 1 1 1 hence —+ Ss + — i 6 0-8 WEF © a. @)0 @ Sie. 6 (25) ; pee Be Pity Po which shews that the sum of the curvatures of any two normal sections at right angles to each other is invariable. Cor. Combining this conclusion with the equation (18), which gives the values of the two quantities p,, p,, we see that, the system of co-ordinates being any whatever, P24 )-(k+ L) U?+ (L+H) V?+(H+ K) Ww’. p p (288) We will proceed to make a few remarks on the equa- tion (24). In the first place, it is important to observe that p, R 242 ON THE CURVATURE OF SURFACES. and p, are values of p in the equation(19) corresponding to the same sign of the radical P: this will be easily seen if, for distinctness of conception, we first take (19) with the positive, and secondly with the negative value of P. We shall find in both cases that in arriving at the relation (24), the same sign, whichever it may be, has been of necessity retained throughout. We must bear in mind, also, that the quantities p, and p, are not necessarily symbols of the mere magnitude of the principal radii of curva- “hal ture, for they may be either both positive, both negative, or the one positive and the other negative. The sign of p will depend upon that of p,, p,, and the value of a. As a consequence of (24), it will therefore follow that (25) signifies that the analytical sum of the curvatures at any point is constant, the geometrical sum not being subject to such limitation, except when p, and p, have the same sign. When p, and p, have the same sign, then it is evident that p will have always the same sign as either of them: this shews_ that all the normal sections at the point in question lie, in the neighbourhood of the point, on the same side of the tangent plane ; or that the surface is convex at the point. Suppose that p, is less than p,; then, writing (24) under the forms 1 1 (2 | a —=-—-(—--—]sin‘a, : Pants P, P, i € ) : —_=-—+{-— --— ] cosa; Pes Opa ies ne ; nN ae Ee | it 1s plain that =>-, —-<-, Periba PsP ae Pi Ps or that, in absolute value, p, is a minimum, and p, a maximum, among all the values of p. If p, = p,; then it is evident, from (24), that Tele eR ae whatever be the value of a: thus all the normal sections have at the point the same curvature, and may all equally be regarded - ON THE CURVATURE OF SURFACES. 243 as principal sections. A point of the surface possessed of this peculiarity is called an wmbilicus. We shall enter, below, more fully into the examination of these points. Next, let us suppose that the principal radii have opposite signs: for instance, let p, be positive and p, negative. In this case the curvatures of the principal sections will be opposite, so that the surface must lie partly above and partly below the tangent plane in the neighbourhood of the point. If we agree to denote by p, simply its geometrical magnitude, we shall have 1 costa sin’a e P; Pe Suppose that a’ is the least positive value of a which will satisfy the equation cos’ a’ sin’ a’ Py P2 then, as a increases continuously from - a’ up to 27 - a’, it is 3 é Foauk : : evident that — will be zero and therefore p infinite for the | p following values of a, viz. ‘ ‘ f , U It is clear also that p will be positive as a varies from — a’ to +4’, and from 7 — a’ to w+’; and that it will be negative as a varies from a to 7 — a’, and from 7 + a’ to 2a — a’, If therefore we draw, in the tangent plane at the point, two straight lines inclined at angles — a’, + a’, or, which comes to the same thing, at angles 7 — a’, w+ a’, to the axis of z, these straight lines will be the traces of two normal planes which separate the normal sections, of which the curvature has one direction from those of opposite curvature. It is easily seen from the formula that p, is the absolute mini- mum of all the positive radii, and p, the absolute minimum of all the negative radii. -Analytically speaking, p,, affected by its sign minus, is a maximum, being the least of the negative radii. In the case considered above, where p, and p, were supposed to have the same sign, one of these quantities is a maximum and the other a minimum, both geometrically and analytically. R 2 944 ON THE CURVATURE OF SURFACES. In the case of a developable surface, it is easy to see that one of the radii is infinite: thus, from (23), (1-2) (-- Eo but, in a developable surface, as we see by putting U and V both equal to zero in the equation of Art.(255), 12 uUv= Ww": hence = (us 0-2) a0; p p thus we see that p,=o. Euler’s formula is therefore reduced to 1 cosa. presen a result which shews that p has the same sign for all values of a, or that the surface in the vicinity of the point under considera- tion lies entirely on one side of the tangent plane. (289) There is a striking analogy between the formula connect- ing the radius of any normal section with the principal radii, and the relation subsisting between the diameters and axes of an ellipse or an hyperbola. In fact, putting a’ b? FF 15 Te 2 eee ee aa a the positive or negative sign being taken accordingly as p,, p,; have the same or contrary signs, we have 1 COS onli Fee ten the equation to a central conic section of which a, b, are the semi-axes, and # a diameter inclined at an angle a to the direction of a. ‘Thus we see that, if the principal axes of the conic section denote the square roots of the principal radii, the diameters will represent the square root of any radius whatever, the corresponding normal section being inclined at the same angle to one of the principal sections as the diameter of the conic section to one of its principal axes. (290) ‘The curvature of any surface at any of its points may always be assimilated to the curvature of an ellipsoid, or of an hyper- bade ame ON THE CURVATURE OF SURFACES. 945 boloid of one sheet at one of its real vertices. Suppose that in the planes of xz, yz, two ellipses are described, their centres being in the axis of z at a common distance ¢ from the origin, ¢ being a semiaxis of both: let the other semiaxes of the two ellipses be 2 2 , ; a b a, b, respectively, a and b being so chosen that a rns aCe An ellipsoid may be constructed, of which a, b, c, are the semi- axes in magnitude and position. Let R’ be the distance of a point in the ellipsoid from its centre, the distance of the point from the plane of zy being c. Then the radius of curvature of the elliptic section of the ellipsoid, made by a normal plane 12 through this point at the origin, will be p’ = a . Now the equation to the section of the ellipsoid by a plane through its centre at right angles to the axis of z, gives us ie cos ae. sit a BP adap cia alent hence, putting for a’, 4’, their values : CPi > Pos respectively, we get Cc cos’a__ sin’ a ee: hk” Py Po dine 1 cosa sin’a butc=—,; hence -= - p p mey pi P, But, in relation to the surface, 1 cos’a : sin’ a p Py P2 and therefore 0. = p> Thus we see that the radii of curvature of all normal sections of the surface and of the ellipsoid coincide both in magnitude and in direction; and, accordingly, the ellipsoid has a complete osculation with the surface. As we pass from point to point on the surface, the osculating ellipsoid will, of course, generally change, not only in form, but in all the circumstances of posi- tion. If we next suppose p, to be positive and p, negative, we must 246 ON THE CURVATURE OF SURFACES. have, preserving the same notation as in the preceding case, and making the sign of p, explicit, a b anes es ae Pa) P2 > c being taken positively as before: thus a will be real and 6 imaginary ; and consequently the ellipse in the plane yz will be — changed into an hyperbola, to which the axis of y 1s a tangent and the origin the vertex, and which lies in the negative — direction of the axis of z: the osculating surface will therefore — become an hyperboloid of one sheet, of which the ellipse in the plane of zz is the ellipse de gorge. ‘The identity of p and p’ for the surface and the hyperboloid may be shewn just as m the case already discussed, of the osculating ellipsoid. The osculating hyperboloid will therefore indicate the exact nature of the cur- | vature of the surface, both in magnitude and direction, for every normal section. In the case of a developable surface we know that one of the principal radii of curvature, p, for instance, is infinite. The axis b of the ellipse in the plane of yz will, therefore, become infinite; so that this ellipse will degenerate into two straight — lines parallel to the axis of y. The osculating surface will © accordingly, in this particular case, degenerate into a right cylinder upon the ellipse in the plane of zz as its base. It may be remarked that, in the case when p, and p, have the — same sign, we might have taken for the osculating surface, instead of an ellipsoid, either an hyperboloid of two sheets or an elliptic paraboloid, both of which surfaces are convex as well as the ellipsoid. We might also, when p, is negative, © p, being positive, have taken an hyperbolic paraboloid instead of an hyperboloid of one sheet, either of these surfaces being — equally capable of representing both the magnitude and the direction of the curvature. The two surfaces, however, which have been selected out of the five, will be sufficient for the purpose of illustration. | (291) To speak generally, any two surfaces are said to osculate at a point where they have a common normal, when all normal sections made by the same plane have mutual osculation at the - ON THE CURVATURE OF SURFACES. Q47 point. That this may be the case, it is sufficient and necessary that the principal planes of the two surfaces coincide, and. that their principal radii of curvature be equal and of the same nature. Suppose, in order to leave no obscurity on this point, that p, R, are the radii of the normal sections of the surfaces made by any normal plane. Let « be the angle between the principal sections of the two surfaces. Then, p,, p,, being the principal radii of one surface, and R,, R,, of the other, 1 cos’a sin’ a + fata? aa 3 p Py Po 1 cos’ (a + e) sin’ (a + €) R R, iieh ee But R& = p for all values of a; hence * 9 « 2 cos’ a , sinta cos’ (a + €) _ sin (a +e). p, Po ft Rh, 1 putting a = 0, a=37, successively, we have e 2 1 cos’ 8M ¢ and therefore COS’ a COS’ « ‘ cos’ a sin’ « y sin’ a sin’ « a sin’ a COS’ « Rk, h, hk, hi, cos’(a+¢«) sin’(a + €) = OO + R R 1 2 Qsinacosesinecosa 2cosa cose sina sine. hence 0 = ————____—————— — ———e: R R 2 1 since this is true for all values of a, we have Lowa Of intone ’ \R&, Rk, and therefore sin 2¢ = 0, a relation which shews that the principal sections of the two surfaces coincide. Putting <= 0, we see that 1 1 ti 1 =— = —_—_ = b] or p,= F,, p, = f,. 248 ON THE CURVATURE OF SURFACES. (292) Umbilict. The radii of curvature of the principal sec-_ tions, and therefore of all normal sections, at an umbilicus are equal, and have the same signs. The conditions, therefore, for a point of this nature are that the values of z-2', y-y', 2-2’, ; in (7), shall be the same both in magnitude and in sign for all Hl simultaneous values of 7, m,n, given by the equations (9) and 4 (10). That this may be the case it is sufficient and necessary — that Pu + mv + n’w + 2mnu' + Qnlv' + 2mw' = C, | where C’ is invariable under the conditions of the problem. : Now, from (9) and (10), we have (mVinwWy=?1?U*, 2mnVW =U’? -mV*?-n? WwW; similarly 2nkWU =m’? V* —- nv W? - U0’, 2mU0VaenW?- PU? -—mV’: combining these relations with the expression for C, we get CUVW = UVW (lus mv + nw) + Uw (PW - mV? — 2? W") + Vo (mV? - nv? W? —- PU?) + Ww' (nr? W* - PW - mV’), If we put 1 - m’- n’ in this equation for 2?, C will depend upon | two quantities m* and n*. Now, if between (9) and (10) we eliminate 7, we shall have a quadratic for determining m in terms of m, such that for any arbitrary value of n’, m? will have two different values; the converse being also, of course, true if we express n° in terms of m*. Hence, in the expression for Cin | terms of m* and n’, the coefficients of m? and n? must be each zero: this shews that, in the expression for C above given, the coefficients of 7°, m’, n, must be equal. Hence | Rh ppe YSO ON itn Tap ¢ U ; Vs Ob rack Ue = Aa a a0) 00 ts a WW ! ' / j = 0+ a (We'- Uu' - Vo')... (26). We have tacitly supposed above, in assuming that for each value of m* there are two values of n’, and vice versa, that no ON THE CURVATURE OF SURFACES. 249 one of the quantities U, V, W, is zero: suppose, however, that V is zero, W either finite or zero, and that U is finite; then, from (9) and (10), l=-—n ae mU" = U0? -(U*+ Wn’; and therefore Ww 2 C=n* ae te-(U+W) Wr’ Ww’ ~~ + 2mnu! — In? —— - 2mn —_—. + nw + 2mnu ne mn 7 Now for one value of ? there are two values of mn; hence the coefficient of mn must be zero: this being established, it is obvious also that the coefficient of 2” must also be zero, 2° being variable. Hence, when V = 0, we must have recourse to the conditions Ww' = Uv, W*u+ Uw =(W?+ U0") 04+ 2WUW...(27). If W=0, the conditions for an umbilicus will, in like manner, be Ou = Vo', U'v+ Vu=a(U' + V") w+ 2UVw'....(28); and, if U=0, Vo' = Ww', V?w+ W=(V? + W*) ut 2VWw...(29). In order to ascertain the existence of umbilici on a surface, we must combine the equation to the surface with the equations (26), and determine whether these equations can be satisfied by three real simultaneous values of x, y, z, so as not to make U, V, W, any of them zero; the values of the co-ordinates will in this case determine an umbilicus. If the two equations (26) are equivalent to only one really independent equation, then this equation, together with that to the surface, will deter- mine a certain curve on the surface of which every point is an umbilicus: such a line is called a line of spherical curvature, because at each of its points the surface possesses a uniform cur- vature like the sphere. We must try also whether any one of the equations (27), (28), (29), can be satisfied, when V, W, or U, respectively, is zero; the equation (26) being in such cases inapplicable. (293) We may readily transform our conditions for umbilici nto equivalent ones involving the partial differential coefficients 250 ON THE CURVATURE OF SURFACES. of z with respect to z and y. For this purpose we must replace U, V, W, u, v, w, u', v', w', respectively, by 5°97, 1,77, 0, 0,0 0,7 ume: Thus, (26) becomes ye oy eee A eer, 2 2 fi AS PP ee 7 8 t which will be the proper relation, unless p or g be zero. If p or g be zero, we must have respectively, as the requisite con- ditions, s=0, t=(1+q¢’)7, or S= 0, Fall +o) dt (294) In the case of the ellipsoid, its equation being 2 y” 2” = + Be + re Ly 22x 2 2: we have U=—, v=, Wie 2 2 2 “= a’ CS B ’ w= ’ ‘= 0, v=0, w' = It is evident that the equations (26) are not satisfied. Suppose — V = 0, so that y = 0; then, from (27), z ¥ ae re y =] 1 OD CEO UNG LAI Be 2 2 ty Z —=(0-¢)=-—(¢ -8 a ( ) C ? whence, from the equation to the ellipsoid, 2 2 ae, 0 rs a Uae —¢ = ey tks z= a-—c ie From these relations it appears that there are four umbilici on the surface of an ellipsoid, situated in the principal section through its greatest and least axes; and that they are the positions of its vanishing circular sections. ON THE CURVATURE OF SURFACES. O54 (225) * The conditions for the existence of an umbilicus may likewise be deduced from the consideration that the two roots of the quadratic equation UE 2 re H Ww’ ‘ (a) aaa Re I em BN cena, Se MUR Seer So (ae frp aT eee pe p p p must be equal. Now if H, KH, LZ, be not all equal when any values of z, y, 2, are substituted, let A be the mean. It 1s readily seen that one root of the equation must lie between H and K, and the other between A and ZL. Hence, if the roots be equal, they must either be each equal to A, or we must have re EF Uk ; ' or U@ + Vw (Uu — Vo - Ww’) aay plats (Veo' ~_ Ww' — U2) iW shelters (d), WU Ww ie WER Alin oe eS ! . w+ ae (Ww' — Uw - Vr') which are the general conditions for an umbilicus. The former condition, that each value of Is shall be equal to , can only be P satisfied if V=0. Now when any of the quantities U, V, or W, is equal to zero, the transformations of Art. (280), by which the products 2mn, 2nl, 2lm, are eliminated, is impossible, or rather nugatory; and therefore, to find whether in any given surface there are umbilical points for which any of these quantities vanish, we should have recourse to the original expression for 'y ~_. However, it is clear that if, making use of this expression, we had found by any other process an equation giving the required values of zi for any point of the surface, it would necessarily have p | been the same as (a) for general values of U,V, W, u, &c., though in a different form probably. Hence, if we can put (@) under a * This method of investigating the conditions for umbilici was communicated to me by Mr. Thomson, of St. Peter’s College, to whom I am also indebted for a knowledge of the formule themselves. 252 ON THE CURVATURE OF SURFACES. form which is not impossible or nugatory when any one of the . . ° . s e iP quantities U, V, W, vanishes, it will give the values of — at any P point where such cases occur. Thus, lett V=0. We may put (a) under the form Eaten ey p p + (2-2) [ya (z : = | + w(K =| ai p/ \ p p/J Now we have from the values of TTS IT VL+ WK =V'w+ We -2V Wu, generally. Also, when U = 0, OH 08 Uti 0 a O’°KL = -(Vo' - Ww'y. Hence the quadratic equation becomes, when U = 0 ; ~(Vo'- Ww'f+ (3 - =) {V0 W%0-2V Wu'-(V*4 W") |. 0, p p ae eae Po V*w+ We - 2V Wu _(Ve' - Wu'f p p Vs Ww e Vie This must be considered as supplementary to equation (a), giving the value of a for any point where U= 0. Symmetrical equa- p tions apply to the cases where V=0, or W=0. Now the condition that any equation of the form (2g-a)(e-b)=¢ may have equal roots, is (a + bY = 4 (ab — &), which requires that c=0 and a=6. Hence the conditions for an umbilicus, where U=0, are Vo —- Ww' = 0, ae whe Vw + Wy OV Wy Po (c). : Ves Ww? ; ON THE CURVATURE OF SURFACES. 953 Similarly the conditions for an umbilicus, where V=0, are Ww' — Uu' = 0 a): Wu it Tw B 2WU?' PROM x. ‘ js and v= ————__—__| W* + UV" j and, where W = 0, are Uu' —- Vo'=0 > + Vu 20 Vw Moameepoeninnl)- | Hence, to find all the umbilici of a surface, we must first satisfy i the general conditions (0) for as many points as possible, and we must then try whether there are any points for which any one of the special systems (c), (d), (e), is satisfied. (296) In the preceding Article it has been remarked that, although in certain cases the investigation of Art. (280) involves indeterminate operations, yet the equation (18) there obtained will be universally true, and that it must be essentially the same as if it had been found by a process involving no nugatory expres- sions. * We shall, however, now give an investigation which is at ' | ‘ : . all times free from nugatory operations, in order that every light may be thrown upon this important equation. We have to render fe Pu + mv + n’w + 2mnu' + 2nlv' + 2lmw'.... (a) p a maximum or minimum, subject to the conditions ieee le 4 WV = OS eee (d), he OR Ay aes ama regres £6) Differentiating (a), (6), (c), and putting dp = 0, we get 0 =(lu + mw’ + nv’) dl + (nv + nu! + lw’) dm + (nw + lv’ + mu’) dn, 0 = /dl + mdm + ndn, 0 = Udi + Vdm + Wdn. * This investigation was given by Mr. Greatheed, of Trinity College, in the Cambridge Mathematical Journal, for May, 1838. 254 ON THE CURVATURE OF SURFACES. From which equations we obtain, by indeterminate multipliers, lu + muo'+nv' =r + wU, mo+nu +lw' =rd\m + wV, nw + le + mu =dn + WwW. Multiplying these equations by 7, m, n, respectively, and adding, we have, by the aid of (a), (b), (c), nee P Hence (u-Z) 14 wm + on = WU, p (o- 2) mans wl= uP, p P (wT) navts um = pW. p If from these three equations we eliminate m and n by cross multiplication, we see that the coefficient of 7 is symmetrical. with respect to the coefficients ; so that if we call it S, and write TELA Gee) Ohi tea BAA aay abe ; Wi we (ia (° Q a iF “SL” o> 5) (8 a) ht ee ee so (e-o(o-2 aL le) eB) ee lore Bh | + V {ei - w' (« _ =} If we multiply these three equations by U, V, W, respec-_ tively, and add, we have, by the relation (0d), (Fle Boro 2) Boma £)(2 -2VWu' |u- =)= 2WU?' (> - 3 ~ 2UVw' (« — =) p p p — Wy? — Vy? — Ww 4 OV W's 4 2 WU el! + 2U0Vu'o' = 0. ON THE CURVATURE OF SURFACES. 255 e . >. . . P e . . This is a quadratic equation in —, with which the equation p (18) will coincide when cleared of its fractional form. (297) Having considered above the properties of the normal curvature of surfaces, we will now establish an important theorem, due to Meunier, by which the curvature of an oblique section at any point of a surface may be immediately obtained from that of a normal section at the same point. Differentiating the equation to the surface, we have Udxe+Vdy+Wedz=0, OUd’2+Vd’y¥+Wd2z+udz’+vdy’ + wdz + 2u'dy dz + 2v'dzdz + 2w'dzxdy = 0. Let B be the radius of curvature of the oblique section; then, by the theory of the curvature of curves in space (Art. 265), _ we know that, a, 3, y, being the angles which R makes with the co-ordinate axes, 2 d*z = an COS a, d. 2 a*y = — cos (3, 2 Died = Re 08 7: Hence we get 0= 5 (U cos « 4 V cos B+ W cos y) + &c:: but, 9 denoting the inclination of the section to the normal plane, mere Ucosa+ V cos B+ Weosy, 7 CG tor? 1 : henc 1-288 a It will be observed, that the remaining terms of the equation are functions only of the partial differential coefficients of dx dy dz F(z ANG. Obie ets a eo ds’ ds’ ds remain constant so long as the oblique section passes through these terms will therefore 256 ON THE CURVATURE OF SURFACES. the same tangent line to the surface at the point under con- sideration. Hence, under this condition, cos @ = constant. When 6=0, R=p, p being the radius of curvature of the normal section : hence R = p cos 0, or the radius of curvature of an oblique section is the pro- jection, upon the plane of this curve, of the radius of curvature of the normal section which passes through the same tangent line. (298) A line of curvature in any surface is the locus of a series of its consecutive points, such that the normals at each point shall meet the normal at the consecutive one. The equations to the normal are Zz-x y-y 2-2 U V W Let each member of this equation be represented by Q. Then 2=¢4QU; y=y+ QV, Zz=2+ QW. If the consecutive normals meet, z’, y', z', will remain con- stant when 2, y, z, and therefore U, V, W, Q, vary infinitesi- mally. Hence we have dz+ QdU + UdQ =0, dy+ QdV + VdQ =0, dz + QdW+ WdQ=0: or, putting dx, dy Te dz, | Use as 05 6 Ones hu + mic + nel + Be mos nil + Wo's 4 7 oo, Ripper: (TLL nwo + Io! + mul + = + 22 a0, mip te: ON THE CURVATURE OF SURFACES. Q57 Eliminating Q and dQ from our equations, we find U(dV dze—dWdy)+V (dWdzx-dU dz)+W (dU dy-dVdz)=0, as the differential equation for the lines of curvature. This equation, together with the equation to the surface, involves all the properties of lines of curvature. ‘The process of inte- gration, the differential equation being of two dimensions in dz, dy, dz, will introduce into our results an arbitrary constant so involved that, when z, y, z, are given, it will have two different values. If we substitute these two values of the constant in the integral, we shall thus get the equations to two lines of curvature passing through the given point. (299) Differentiating the equations (9),(10),(13), considering l, m, n, variable, and puttmg dp=0, which corresponds to the determination of the principal sections, we get ldl +mdm+ ndn =0, Udl + Vdm + Wdn = 0, (lu + mw' + nv’) dl+ (mv + nu' + lw’) dm+ (nw + l' + mu') dn=0. Multiplying these equations in order by AX, uw, 1, adding and equating to zero the coefficients of d/l, dm, dn, we obtain lu +mw'+nv' = 414+ 20, mo+nu +lw’ =rA\m+ uvV, nw+l' + mu =dAn+ uW. Now, if between these three equations A and y be eliminated, it is clear that we shall get an equation in /, m, m coinciding ‘with the equation resulting from the elimination of Q and ' dQ between the equations (30). From this it is evident that, 1, m, n, being in both cases subject to the same equations (9) and (10), the directions in which the lines of curvature start from any point on the surface coincide with the directions of greatest and } least curvature, and are therefore at right angles to each other. (300) That the two lines of curvature through any point of a surface are at right angles to one another, may be demonstrated also in the following manner.* * For this demonstration of the perpendicularity of the lines of curvature, I am indebted to Mr. Fischer, of Pembroke College. Ss 258 ON THE CURVATURE OF SURFACES. We have for the differential equation of the lines of curvature, (VdW-WadV)dz+(WdU-UdW )dy+(UdV -VdU ) dz=0. Now VdW-WdV=(Vo - Ww') dx+ (Vu - Wo) dy + (Vw —-Wu') dz: also Udz + Vdy + Wdz = 0. Therefore, eliminating dz, Vaw - Wav = {Pu Lara (Vo f Wie')| dy + {Vie - Wu - 7 (Ve Ww) dz = - WKdy + VLdz. Modifying the second and third terms of the above equation I similarly, and arranging the result, we find OU (K-L) dydz+ V (L-H) dzdx+ W(H-K) dxdy=0. — Hence, if 7, m,, be the direction-cosines of the tangent to — a line of curvature through any point of the surface, their ratios are determined by the equations Oh te IW ne eee (a), U(k-L)mn+V(L-A)nl+ W(A-K)lm=0 ...(6). ; It is readily shewn that two, and only two, systems of values — of the ratios may be deduced from these equations, and that — they are all real. Hence, if J, m,,”,, and J,, m,,,, be the values, we may write down equation (a) for each system; and ‘ we thence deduce, by a common process, 3 U V W —_________ = —_____ = _______...... (c). mn,-mn, nl,-nl, lm, - lm, Similarly, from (6) we deduce | U (Kk - L) V (L- #) W (HH - &) P RE ae Ll, (mn, -m,n,) mm, (nl,—nl,) nn, (Gm, -lm,) Therefore, dividing the members of equations (c) by the cor- ' responding members of (d), we have ti, mm, ~nn, —bi+mm+non,. K-L L-H H-K- 0 , therefore Li, + mm, + n,n, = 0. ON THE CURVATURE OF SURFACES. 259 Hence the two directions of the lines of curvature through any point are at right angles. (301) The identity of the equations for finding the positions of the principal sections and the directions of the lines of curvature at any point of a surface, points out to us a method of finding at once the principal sections and principal radii at any point of a surface of revolution. Let P be any point of the surface ; M the point where the axis of revolution is intersected by a perpendicular let fall upon it from P, N being the inter- section of the axis with a normal to the generating curve at P. The plane of the generating curve will evidently be one of the principal sections, since any two of its consecutive normals must meet; and the other principal section must pass through the normal PN at right angles to the area of the generating curve. ‘The principal radius of curvature of the former section will be the radius of curvature of the generating curve. The radius of curvature of the latter section will, by Meunier’s theorem, be equal to the product of PM, which is the radius of the circular section of the surface through P, and the secant of the angle MPN;; that is, it will be the line PW. (302) To find the lines of curvature on the surface of an ellipsoid. The equation.to the ellipsoid is Vr 2 2 ; gt oe tape Lee eng (Ys whence also xadz dy zdz ’ : T+ = 0.0 (2). Also, from the differential equation for the lines of curvature we have, in the case of the ellipsoid, cea a Aldi Pe ON BI at Aan ein i PE dene (b oD metas ON ea ike (3) 2 2 a Assume Meee ee = 0 BoA. oii Aare (4) to be an equation between 2, y, z, which, together with (1), shall determine the lines of curvature. The admissibility of this assumption depends upon the possibility of its satisfying (3) in such a way that the three constants A, B, C, shall $2 260 ON THE CURVATURE OF SURFACES. be equivalent to a constant with two values. That these constants should be subject to this condition will be clear when we consider, that if between (1), (2), and (8), we eliminate z and dz, we shall have a differential equation of the first order and second degree in x and y. From (4) there is whence, by the aid of (2), we see that Eee 4 ydy ; He 3: B-C:C-A:A-B....(85). a b c From (3) and (5) we get Ronse x 2 2) yf 2 72 z é (5°— c’) a(BO) +(¢-a’) (CoA) +(@ BOE =0. This equation will coincide with (4)', provided that b? — ¢’ c- ae a’ — B° CA dee A-B’ where A is any quantity whatever: these three equations establish only one relation between A, B, C, viz. | ee ne pale baie Teac A Wuiiapah ane ei nara tees (6) Sees t ay 8 okay &, If we put Anns, B=* = ———- then ifigt 9 tilt (0. ee neasiete eee Te b° a Ce x C Kas a y a 2? 2 Equation (8), together with (1)’, are the equations to the lines of curvature. If we substitute in (8)' the quantity ~(f +g) for h, by virtue of (7), the resulting equation will g involve only one arbitrary constant, viz.“. In order to determine the value of this constant, let us suppose that (z,, Y,» #,) 18 a point on the surface of the ellipsoid from which a conjugate pair of lines of curvature start. ON THE CURVATURE OF SURFACES. 261 Then, from (8)’ we have -—¢ xe iE We ne ye ta gq 765 (Maat and from (7) there is Mond ih multiplying Or two sauce eres we get BO oc Le a + (6* — a) rea) 4 = (By % eeee (9)’, which is a quadratic in 7 of which the roots, as may be readily ascertained, have real values. The double value of the constant & satisfies one of the two conditions on which the legitimacy of the assumption (4)’ is dependent. The equation (8) belongs to a cone of the second order, with its vertex at the origin, which shews that the lines of curvature through any point of an ellipsoid are the inter- sections of the surface with two cones, of which the vertices are both at the centre: the fact of there being two cones is shewn by the double value of the constant " : The equations (5)', putting for 4, B, C, the values obtained above, become x ee y dy zdz a 1G, ale ors): Cc De ONC ras 2 age a h St i g which are the differential equations of the projections of the lines of curvature on the co-ordinate planes: these projections, as the forms of their differential equations shew, are conic sections. Suppose that x? a a aml 6° 6B’, 2 ae ™eT (om ha a - Ce’ y= a—eé ( being a very small quantity. Then, from (9) we have B08) le = Ope Tos +(@-a) Fao Be 262 ON THE CURVATURE OF SURFACES. from which equation it is evident that, as (3 diminishes indefi- nitely without absolutely vanishing, the values of 4 approach indefinitely nearly to zero. Wemay see from (7) that the y. corresponding values of 7, are indefinitely near to zero. From ’ this it is evident that the equation (8), as the point (z,, y,, 2,) approaches indefinitely near to an umbilicus, without absolutely coinciding with it, degenerates indefinitely nearly into Jee or that, to proceed to the limit, the lmes of curvature through ~ an umbilicus coincide with the section of the ellipsoid made by a plane through its greatest and least axes. If in the equations (2)' and (3)’ we put y=0, they become xdx zd. a pyr oe Zz (6° o) SiS. ie —(a’ eS b’ nae ; Zz be — a— o> whence SS ee a c° a relation which is satisfied by the values of x, z, at an umbi- licus without subjecting to any restriction the ratios between the differentials dz, dy, dz. This shews that the property of the intersections of consecutive normals, as far as the first order — of differentials is concerned, is satisfied equally in whatever — direction an indefinitely small arc is taken on the surface of the — ellipsoid starting from an umbilicus. This résult may at first sight appear to be incompatible with our former conclusion, viz. that the lines of curvature through — an umbilicus, coincide with the plane of zz. Conceive, how- ever, an indefinitely small ring to be described on the surface around the umbilicus: from each point of this ring a pair of lines of curvature will start indefinitely nearly coincident — with the plane of zz. Hence we see, that in whatever direction ~ we may start from the umbilicus to this infinitesimal ring, the ON THE CURVATURE OF SURFACES. 263 subsequent course of the lines of curvature will be ultimately the same. The discovery that the lines of curvature on an ellipsoid are the intersections of the surface with cones of the second order, is due to Mr. Leslie Ellis: the more direct investigation by which he arrived at this conclusion may be seen in the Cam- bridge Mathematical Journal for May 1840. ‘The student is recommended also to consult Leroy’s Géométrie Descriptive, for graphic illustrations of the forms of the lines of curvature. (303) If there be three series of surfaces, such that all the sur- faces of each series cut the surfaces of the other two series at right angles, the lines of intersection of any one of the surfaces of the three series, with the surfaces of the two conjugate series, are its lines of curvature. This remarkable theorem was given by Dupin, in his Développements de Géométrie, Cinquuéme Mémorre. * Let O be any point in which three conjugate surfaces intersect, and let the rectangular axes OX, OY, OZ, be per- pendicular to the tangent planes of the three surfaces at O. Let TENE a en erent a eerste of Cotaeet (a), TREES ARON a ee rete rts sr e)s JEN (let Sue te et ne od EN be the equations of the three series; and, when proper values are attached to A, A,, A,, let (a) be the surface touched by YOZ, (a,) by ZOX, and (a,) by XOY. Hence, when z = 0, y=0, z=0, we have V=0, W=0, AS TETAS ee 1 ost. 1 ay the suffixes of the letters in (6) connecting them with the cor- responding surfaces. Now, since the system is orthogonal, we must have identically SREP: ARES Ms 0 U,U + ViVit WW Ope ee eres mes ied UU,+VV,+ WW, =0 * This demonstration of Dupin’s theorem was given by Mr, Thomson, of St. Peter’s College, in the Cambridge Mathematical Journal for February 1844. 264 ON THE CURVATURE OF SURFACES. Differentiating the first of these equations with respect to z, the second with respect to y, and the third with respect to z, putting <,Y, 2, each equal to zero, in the result, and making use of equations (4), we have (V,) (w,') + (W,) (v,') =10; (W,) (w’) + (U) @w,/) = 0, | (U)(@,) + (V,) (w/) = 0, the brackets denoting that, in the quantities enclosed, z, Ya are equated to zero. From these equations we conclude that (w’) = 0, (2,') ls (w,') = 0. The relation (w')=0, see Arts. (284, 285), shews that the planes of zy and xz contain the principal sections of (@) through O, and therefore the lines of intersection of (@,) and (a,) with (a) touch the principal sections of (a) at O. Now O may be any point in one of the surfaces (a), and therefore each of these surfaces has its lines of curvature traced upon it by the surfaces of the series (a,), (@,). Similarly, from the equations (v,')= 0, (w,') = 0, it follows that each surface (q@,) has its lines of curvature traced by (a,) and (a), and each surface (@,) by (a) and (a,), which is the theorem to be proved. It will be observed that in this demonstration only one sur- face of each series has been considered. Hence the theorem proved is, that if any three surfaces cut one another at right angles along each line of intersection, at any point where all three meet, the lines of intersection on each surface will be tangents to its principal sections. Dupin’s theorem follows - immediately from this result. ( 265 ) CHAPTER XVI. PROBLEMS. Pros. 1. To find the relations between the co-ordinates of the extremities of three conjugate diameters in an ellipsoid. Let the equation to the surface be 2 2 2 Sard Me Soy Ae Gan Ore This is satisfied by z=al, y=bm, z=cn, provided that Lerten Pip ann Ble eve Gls and therefore al, bm, cn, may be taken as co-ordinates of the extremity of a semi-diameter 7. In like manner we may take ‘'sal', y=bm, 2 =cn, under the condition 27+ m? +m" =1.....000+: err aie) and z=al', y'=bm’, z=en, Miider the condition 27+ m7 47 = 1 ....50. 2. eu ae (3), as the co-ordinates of the extremity of two other semi-diameters r' and 1’. Now if we change the co-ordinate axes so as to coincide with r, 7, and 7’, we shall have to put al al’ al’ cde Sort ELAS pf PU a and similarly for the others. If we substitute these values in the equation to the surface, and make the conditions that 7,7’, 7”, shall be conjugate semi-diameters, which involves the vanishing of 266 PROBLEMS. the terms containing the rectangles in the transformed equation, we find [7 + m'm' + n'n" = 0 [l+m'm+n'n = 0 The equations (4) shew that Il’ + mm + nn' = | 2 6° 2 Ca ey 7 eee. RA MEMS Te ear oh “ ’ A “Lu 4 22 aay te worg el a b C which are the required relations between the co-ordinates (z, y, z), (z, y', z), (2, y’, 2), of the extremities of three cone jugate diameters. Cor. 1. From the equations (4), we see that (d, m, n), (7, m', n'), (', m', n"), are the direction-cosines of three lines which are at right angles to each other. Coy. 2. Met (X96 ZX We Ay, (XK aye Z), be three such points and £# such a line that hi=X, Rm=Y, En = Z, R= X', Rm'=Y', Rr' =Z, Re Xx, ina Y's ain = 2 then, from the above conclusions, we easily see that if the points (KX) Y, 2), (XV, 25 x, YF, be the extremities of three radii, at right angles to each other, of a sphere erypt ea RP, the points aX bY cZ (> OP cZ) aX 0 Ya Mees eT JON IR OR) ON RA ae ome will be the extremities of conjugate diameters of an ellipsoid Yon 2 tat ao For further information on this subject the reader is referred to a memoir by M. Brassine, in Liouville’s Journal de Mathé- matiques, Av. 1842. PROBLEMS. 267 Pros. 2. The sum of the squares of the projections of any three conjugate diameters on a fixed line is constant. Instead of projecting the diameters on the line directly, it is better to project the co-ordinates of the extremities of each diameter, and add them. Now if A, p, », be the direction-cosines of the given line, the sum of the projections of the co-ordinates of the extremity of one diameter is alr + bmp + env: similarly, for the other two, we have al + bm'w + cn'y, al’) + bm'u + en'v. Squaring and adding, and observing that both the axes of co-ordinates and the lines of which the direction cosines are (J, m, »), (U', m', n’'), (U', m', n’), are rectangular systems, we shall have for the required sum, . ar’ + by? + er’, which is a constant quantity. Pros. 3. The sum of the squares of the perpendiculars drawn from the extremities of three conjugate diameters on a fixed diametral plane is constant. If the equation to the plane be At + py + vzZ=9, in which A, p, ”, are the direction-cosines, and he gay ir sey be three conjugate perpendiculars, p= alr + bmp + cnr, p =alr + bm'p + en, al'y + bm'p + env, Ss l and therefore pip’ +p =aNr + 24 cy", a constant. Pros. 4. The sum a the squares of the reciprocals of three diameters of an ellipsoid at right angles to each other 1s constant. The equation to the ellipsoid beg 2 2 2 x ee EYE 268 PROBLEMS. and r being any diameter of which the direction-cosines are l, m,n, z=rl, y=rm, z2=91n; therefore a ee Bie es SS 2 a 6? Cc Ten c Nee ioe and again se 5t+tsic. PI: c 2 Adding, then, and observing that, in consequence of the diame- ters being at right angles to each other, th +h = 1, mami +mi=1, Pint +n ed 1 1 ls Me ee Sal = i = 3 we have — + Pros. 5. To find the locus of the centres of the sections in a central surface of the second order made by planes which all pass through one point. Let the equation to the surface be Ax + By Chie AL at Ge BEGET a, 6, c, the co-ordinates of the fixed point through which all the planes pass. The equation to any one of them must be of the form W(e-a)+m(y - 6) +n(e-c)=0........(2). Now the equation to the line which is the locus of the centres of planes parallel to this one are Therefore the co-ordinates of the centre of the section must satisfy equations (2) and (8). Eliminating 7, m, n, between them, we have Az (x-a)+ By(y - 6) + Cze(e-0c)=0 as the equation to the required locus, which is evidently a surface similar to (1), and passing through the origin and the given point. PROBLEMS. 269 Pros. 6. To find the locus of the intersection of tangent planes to an ellipsoid drawn at the extremities of a system of conjugate diameters. Let the co-ordinates of the extremities be x, y, 20, 2, Y, %> x, Y, z,; then the equations to the tangent planes are Azz+ Ayy + A’zz2=1, Arz+ Ayy + A’zz2=1, Azaz+ Ayy+ Azzg=1. Adding, then, we have Az(4,+2,+2%,)+ Ayy,t+y,+ y+ Az, +2, +2) = 3. Now, as the diameters are conjugate, the tangent plane at the extremity of any one is parallel to the diametral plane contain- ing the other two, and therefore the point of intersection (xyz) is the extremity of the diagonal of a parallelopiped of which the three diameters are conterminous edges. Now the projection of the diagonal on any line is equal to the sum of the projections on the same line of the three edges terminated at one extremity of the diagonal. Hence, making the three axes in turn the lines of projection, we have B= Lot Lt lyy YHYtY tY, F=%y+%,4+24,- Substituting in the preceding equation, it becomes Ax’ + A'y’?+ A"# = 8, which is the equation to the required locus. It is obvious that this is an ellipsoid concentric with and similar to the original _ one, and that its axes are greater in the ratio of V3 to 1. Pros. 7. If at a point P in a curved surface a tangent plane be drawn, on which a perpendicular OY be drawn from a fixed point O; and ifin OY a point P’ be taken such that OP’.O Y=’ (a constant), the locus of P’ will be a surface such that the perpendicular from O on its tangent plane at P’ passes through P, and if the length of this perpendicular be OY’, there exists the relation OP.OY' = k’. Let the co-ordinates, measured from O, of P be (xyz), those of P’ (2'y'z'); then, if ¢ be the angle between OP and OY, 4 ee _oP.op, +t yy +z m=OP.OY = OP .0P.cos¢= OP'.OP. OP’.OP = 2x + yy + #2 ....(1). 270 PROBLEMS. Now if Fey, oF OA, a eae (2) be the equation to the locus of P’, and if p be the perpendicular from O on its tangent plane, x“ ee + y' sk z Be dx” dy * Ge a Cay di! dy/' dz } Differentiating (1) and (2), considering zyz, 2'y'z', as all variable, we have a'dxz + y'dy + 2'dz + «dz + ydy' + 2dz =0.... (4), aE). ak dF ay oH oneal a & = Cas alee}: to which, if f(z, y, z) = 0 be the equation to the given surface, we have to join of of of ae _- 1 hE =O ss iewtsl etnias : a Batis, dy + = z (6); A (5) + u(6) — (4) gives, on equating to zero the coefficients of each differential, dF dF dF = r = ad = ar pS 66S Sere A & d. te. r dy 3 @ dz' (7) i pk. Oe ye ie of de t= - EP 9 Y = Eh dy 9 ec. = Ae me Soro & (8). The equations (7) indicate that the line of which the direction- | cosines are proportional to z, y, z, coincides with that of which dF dF dF. dz” dy” dz’ that OP and OY’ coincide, or that OY' passes through P. Equations (8) merely indicate the original construction. Also from (3) and (7), we have Be Uy eh NiCr ee or De tg-ky inavds; OleO Tames Surfaces related to each other in the manner described above are called reciprocal surfaces by Professor Maccullagh, who, in the direction-cosines are proportional to — ;) thatoes PROBLEMS. pg ia the Zransactions of the Royal Irish Academy, vol. xvu., has investigated many of their properties, and applied them most ingeniously to researches in the Theory of Light. If the original surface be an ellipsoid, the reciprocal surface will also be one—such that the products of the corresponding axes are equal. Pros. 8. A straight lme moves so as to have three of its points constantly in three fixed planes; to find the surface traced out by any other point. Let the equations to the fixed planes be le+my+nz=0, Ue+my+nz=0, Ux+m'y+n'z=0. Let a, 6, c, be the distances of any assumed point (zyz) in the line from the three points which are to rest in the three planes. If the co-ordinates of these points be a, B, y; a’, B',y'3 a’, B's y's and the direction-cosines be A, p, v, the equations to the line may be put in three forms r — cee = @, Piso Y ANN ays ep r ph v , as ry tps / ee “ = > Fhe r iu v But the co-ordinates a, B, y, &c. must satisfy the equations to the planes ; hence, we have IX+ mu + nv = — (le + my + nz), VN +m t+ n'y (U'e+m'y + nz), tl Ba Khe Sle M4 m'wt+n'y = — (lz + m'y + n"2). C From these we can determine A, p, v, in the form h vt wt /} eis ty 412), w= %(U'x+my +2), y= ~ (2 L+m'y+n'z), a iis PROBLEMS. Fi 9, h, being functions of J, m,n, 1’, m', n', 1", m", n", of which the form is obvious. Hence, observing that \’+ p’+ >= 1, we haye 2 2 2 ee (la + my + nz) + - (Ue+my+n2zy+ <2 +m'y+n'zy =1, as the equation to the required locus, which is evidently a central surface of the second degree. Pros. 9. If through a fixed point O any three chords AA’, BB', CC’, be drawn in a surface of the second order, the locus of the intersection of the plane passing through A, B, C, with that passing through 4’, B’, C’, is a plane. Take O as the origin, OA, OB, OC, as the axes of 2, y, z; the equation to the surface being Az’+ A'y’+A’2+2By2+2B'2x4 2B xy +2Cxe+2C'y+2C"2z+ H=0 Now, if OA=a, OB=6, OC=c, OA=a, OB=0', OC=c’;; the equation to the plane ABC is eo, z Seete eeets Gi 0mac and that to A’B'C' is san Ea Ua oY t-—b oc When the planes intersect we may combine the equations in any way we choose: adding them, we have x a + tan; +e( eho aia) OO NBO B ot yaa This is a relation between the co-ordinates of the line of intersection, and it may also be considered as the equation to a plane in which that line les. We have now to shew that it remains fixed in position when the position of the chords is changed, the point O remaining the same. For this purpose we observe that a and a’, being the intercepts on the axis of z between the origin and the surface, are the roots of the equation Ag +2Czrz+ E=0, derived from the equation to the surface by making y=0, z=0. PROBLEMS. 273 Hence, by the known relations between the roots of an equation and its coeflicients, 1 f Be 2C Geta a Ae In like manner we find Lag] 2 Cen Loe 20" e+e, Hf oe. 6b b E Cac E Substituting these values in the equation to the plane, it becomes Ce + Cly + C'2+ H=0...... (2). Now, if we were to change the position of the chords passing through O, we should in fact be simply changing the direction of the co-ordinates without altering the origin. The substi- tutions for effecting this transformation are linear, or of the form z= an + by’ + cz’; and therefore the groups of the terms of the first and second degrees in equation (1) will change independently of each other, the constant term not being altered. Consequently the equation (2), which is the same as the last four terms of (1), will experience the same change from the transformation of co-ordinates as it would do if it were deduced from (1) after the transformation had been then effected. Consequently the position of the plane (2) remains the same when the co-ordinate axes are changed, or it is the locus of the lines of intersection of the planes passing through the extremities of the chords. Pros. 10. If there be two homofocal ellipsoids, a a 2 2 a y” 2 +8 4+5-1, ita hn aed and we take in one two points, P(z, y, 2) and Q(é,n, ¢), and in the other two points, P’ (2, y', 2) and Q'(é, 1, 2’), so connected, that = 1, Be Tale then shall PQ' = P'Q, (PQ =(e - a + (n' - y+ SO - 2y, (PQP=(E-eyst(n-y'y+ 6 -2/. 274 PROBLEMS. Eliminating 2’, y’, z', @,7', 2’, by means of the preceding relations, ae a’ 4 b 2 Cae : Pay (eZ -#) «(a5 -) +(25-2), ; oe ane 7 a bi pix (Par-(E-22)4(y v5) +(2 Za (P'QY -(PQ'y ve (=- wa) (=a) 4 (7 e y) (b° a b”) 4 2 er (4 — 5) (ce = oc) But, as the surfaces are homofocal, kG Ps ioe pa Be I ots Bb? = C ah c”; hence Fa me a x yf 2 (P'Q)’ - (PQY =(¢ - a”) (4+ pts) = (= 4 B +5) Since z, y, 2, & n, ¢, are co-ordinates of the ellipse, the second side vanishes, and we have (P'Q)'- (PQY=0, or PQ=PaQ. Pros. 11. To find the locus of the middle points of all the chords in a central surface of the second order, which pass through a given point. Let Ar By’ + C2 — D oe. 4.01) be the equation to the surface, and z-a_ y-b z-C¢ ee l m n a area (2), the equation to any chord passing through the fixed point a,6,c. ‘Then, if z,, y,, z,, 25 Yo9 2,5 be the co-ordinates of the points where the chords meet the surface, we have, from (2), Pies ese Ys Mitre ei 28 We aie l m n and, from (1), the two conditions Az,’ + By? +Cz'=D, Ax}+ By? + Cz2=D; whence A (,’- 2”) + B(y?-y,) + C@?-22) =0. Dividing each term of this last equation by the corresponding members of (8), we have Al (x, + %,) + Bm (y, + y,) + Cn (z, + 2,) = 0. PROBLEMS. 275 Now if 2’, y’, 2’, be the co-ordinates of the middle point of the chord, g/=3(@,+2,), y=3(Y,+4%), 2 =3(@,+%) and the last equation becomes Alz' + Bmy' + Cnz' = 0. But 2’, y’, z', satisfy equations (2), so that Br GREG) ONES l m n Hence, eliminating /, m,n, between the last two equations, we find Az' (2' - a) + By'(y' - 6) + Cz (z - ¢) = 0, as the equation to the required locus, which is evidently a surface similar to the original one, and passing through the origin and the fixed point. Pros. 12. To find the equation to the surface which is described in the following manner :— At the middle point of every central plane section of an ellipsoid a normal to the plane is drawn, and along this, in the same direction, are measured lines equal to the principal axes of the section. The extremities of these lines will trace out a surface of two sheets, which is the one in question. By Art. (122) the principal axes of a section of an ellipsoid 2 Eu Veen eas att Baba! ie: made by a plane Iz + my + nz = 0, are given by the equation al’ bm’ on ce es a ee eneee s Gp The equations to a normal to the plane through the origin are rea LN ee nd and if along this normal we measure distances equal to the principal axes of the section, p, being the length of the distance, must be a root of the preceding quadratic: or we may take it 276 PROBLEMS. as equal to 7: eliminating, then, 7, m, , between (1) and (2), we have ae ty ee | ie 7 C2 eee Foy Fi Bike ges bs as the equation to the surface. If we put for r* its value x+y + 2, and get rid of the denominators, the equation to the surface becomes (+y' +2) (Wa + by’ + &2*)-0(0 +0) 2 -B(C+a)y-(a +B)2 + @b’c? = 0 This is Fresnel’s construction for the Wave Surface in the Theory of Light. THE END. ——— PRINTED BY METCALFE AND PALMER, TRINITY-STREET. ea ee Gregorys 3 old Geometry. : ne — as Jd CWalker Sculp® GregoTys 50 lid Geometry. J Te CWalber Caetano Plate 2. : | Gregory's Solid Geometry. Metcalfe & Palmer, Lithog at Gregory's Solid Geometry. Metcalfe & Palmer, Lithog re Gregory's Solid Geometry. Metcalfe & Palmer, Lithog re Gregory's Solid Geometry. Metcalfe & Palmer, Lithog re RECENT PUBLICATIONS OF JI.&J. J. DEIGHTON, Agents to the Wnibersity, TRINITY STREET, CAMBRIDGE. PPPALIIIII I CLASSICAL. I, HSCHYLUS. Grece, recensuit J. ScHoLtertenD, A.M., Coll. SS. Trin. nuper Socius, et Grecarum Literarum Professor Regius. Editio secunda, 8vo. 12s. II. AESCHYLUS. Appendix ad editionem Cantabrigiensem novissi- mam. Confecit J. SCHOLEFIELD, A.M., &c. 8vo. Is. 6d. III. AESCHYLUS. Eumenides. Recensuit et illustravit J. ScHouE- FIELD, A.M., &c. 8vo. 5s. 6d. IV. JESCHYLUS. Prometheus Vinctus. The Text of Drvnporr, with Notes compiled and abridged by J. Grirritus, A.M., &c. 8vo. 5s. V AESCHYLUS. Septem Contra Thebas. The Text of Drnporr, with Notes compiled and abridged by J. Grirritus, A.M., Fellow of Wadham College, Oxford. 8vo. 5s. VI. ARISTOPHANES. Grece et Latine, cum Scholiis et Varietate Lectionis et Index locupletissimus. Recensuit J. BeKKERUuS. 56 vols. 8vo. 21, Large paper, 3. 10s. To be had separately : TExtTUus, 2 vols. NUEEs. ANNOTATIONES, 38 vols. PLUTUs. AVES. RANz. Vil. ARISTOTLE. A Life of, including a Critical Discussion of some Questions of Literary History connected with his Works. By J. W. BriakesLtey, M.A., Fellow and Tutor of Trinity College, Cambridge. 8vo. 8s. 6d. VIII. ARUNDINES CAMI. Sive Musarum Cantabrigiensium Lusus Canori ; collegit atque edidit H. Drury, A.M. Editio altera, 8vo. 12s. a a MESSRS. DEIGHTON’S PUBLICATIONS. Classical. IX. CAMBRIDGE CLASSICAL EXAMINATIONS. A Collection of Questions, &c. proposed to Candidates for Classical Honours from 1810 to 1823. By alate Recius Professor OF GREEK. Second Edition, 8vo. 4s. 6d. re CASTLE (Thomas). A Table for finding the Commencements, Characteristics, and Regular Inflections of GREEK VERBS. 4to. 2s. 6d. XI. DAWES. Miscellanea Critica. Ex recensione, et cum notis ali- quanto auctioribus T. Kipp, A.M., &c. 8vo. 3s. 6d. XII. DEMOSTHENES. Select private Orations, after the Text of DinpvorF; with the various readings of ReiskKe and BrexKker. With English Notes, for the use of Schools. By C. T. Penrose, A.M., Head Master of the Grosvenor School, Bath. 12mo. 5s. XIII. DEMOSTHENES. Translation of Select Speeches, with Notes. By C. R. Kennepy, A.M., Fellow of Trinity College. 12mo. 9s. XIV. DOBREE (Prof.) Adversaria, &c. edente J. SCHOLEFIELD, A.M., Gree. Lit. Prof. Reg. 2 vols. 8vo. 15s. Vol. 1. Containing Notes on the Greek Historians, Orators, Philoso- phers, &c. Vol. 2. On the Greek Tragedians, Aristophanes, Athenzeus, the New Testament, and Latin writers; with Notes on Inscriptions, the Lexicon Rhetoricum, &c. The following may be had separately :— ADVERSARIA, Vol. I. part 1, 6s. 5 vol. I. part 2, 5s. Hf vol. II. part 2, 10s. Lexicon RHETORICUM CANTABRIGIENSE, 1s, 6d. MiscELLANEOUS Notes ON INSCRIPTIONS, 4s. XV. DONALDSON (Rev. J. W.) The New Cratylus, or Contribu- tions towards a more accurate knowledge of the Greek Language. 8vo. 17s. SE. DONALDSON (Rev. J. W.) Varronianus. A Critical and His- torical Introduction to the Philological Study of the Latin LanouaGe. 8vo. 10s. 6d. XVII. ELMSLEIANA CRITICA. Sive Annotationes in Heracleidas, Medeam, et Bacchas. Selegit suisque et aliorum notis illustravit F. E. GrettTon, A.M., Head Master of Stamford Grammar School, 8vo. 7s. 6d. MESSRS. DEIGHTON ’S PUBLICATIONS. a Classical. XVIII. EURIPIDES. Tragcediz quatuor, cum notis Porsont. Recensuit, suasque notulas subjecit, J. SCHOLEFIELD, A.M., &c. LEditio Secunda, 8vo. 14s. RIX, EURIPIDES. Interpretatio Latina ex edit. Mus@ravit, passim refecta. 8vo. 6s. be EURIPIDES. Not Philologice et Grammatice in Euripidis, Tragedias, e variis virorum doctorum comment. select. 2 vols. 8vo. 12s. 50.46. EURIPIDES. Ezxcrra. Ad optim. edit. fidem emendavit et Annotationibus in usum Juventutis instruxit, H. Ropinson, A.M., Coll. S. Johannis Cantab. Socius. S8vo. 5s. 6d. adhe EURIPIDES. Iphigenia in Aulide. “ Effutire leves indigna Tragoediz versus.” Cambridge Edition, 8vo. 8s. XXIII. HEPHAESTION. Concerning Metres and Poems, translated into English, and illustrated by Notes and a Rythmical Notation; with Pro- legomena on Rythm and Accent. By T. F. BAnHam, M.B. 8vo. 8s. 6d. XXIV. HERODOTUS. Codicem Sancrortr manuscriptum denuo con- tulit reliquam lectionis varietatem commodius digessit annotationes vari- orum adjecit Tuomas GaIsFrorp, S.T.P. Editio altera, 2 vols. 8vo. 11. 1s. XXV. HORATIUS. Ad fidem Textus R. Benrien, plerumque emen- data et brevibus notis instructus. Edidit T. Kipp, A.M. 8vo. 5s. Large paper, 9s. XXVI. KENNEDY, D.D. (Head Master of Shrewsbury School), A Selection from the Greek Verses of SHREWsBuRY School, prefaced by a short account of Iambic Metre and Style of Greek Tragedy, and followed by progressive Exercises in Greek Tragic Senarii. 8vo. 8s. XXVII. LIVY. The History of Rome. By C. W. Stocker, D.D., late Fellow of St. John’s College, Oxford. Vol. II. Parts 1 and 2. 8vo. lJ, 4s. XXVIII. MALTBY (Bp.) A New and complete Greek Gradus, or Poetical Lexicon of the Greek Language, with a Latin and English Translation, &c. Second Edition, 8vo. 11. 1s. XXIX., PHILOLOGICAL MUSEUM. 2 vols. 8vo. il. 10s. Each of the sia Numbers contained in the 2 vols. may be had separately, 5s. 4. MESSRS, DEIGHTON’S PUBLICATIONS, Classical. EXE, PINDAR. Epinician or Triumphal Odes, in four books, together with the Fragments of his lost compositions. Revised and explained by J. W. Donatpson, A.M., Head Master of Bury School. 8vo. 16s. XXXI. PLATO. ScHLIERMACHER’S Introductions to the Dialogues. Trans- lated from the German by W. Dosson, A.M., Fellow of Trinity College. 8vo. 12s. 6d. XXXII. ® PLAUTUS. Aulularia. Notis et Glossario locuplete instruxit. J. Httpyarp, A.M., Coll. Christi apud Cantab. Socius. ditio altera, 8vo. 7s. 6d. XXXIII. PLAUTUS. Menechmei. Notis et Glossario locuplete instruxit. J. Hinpyarp, A.M., Coll. Christi apud Cantab. Socius. Editio altera, 8vo. 7s. 6d. XXXIV. PORSON. Note in Aristophanem, cura P. P. Dopren. 8vo. 9s. Large Paper, 15s. XXXV. SEALE (Dr.) An Analysis of the Greek Metres, for the use of Students at the Universities. Tenth Edition, 8vo. 3s. 6d. XXXVI, SOPHOCLES. With Notes Critical and Explanatory, adapted to the use of Schools and Universities. By T. Mitcuexi, A.M., late Fellow of Sidney Sussex College, Cambridge. CEpipus TYRANNUS. 8vo. 7s. ELectrRA. 8vo. 5s. QGipipus CoLoneEus. 8vo. 5s. TRACHINIZ. 8vo. 5s. ANTIGONE. 8vo. Os. PHILOCTETES. 8vo. 5s. AJAX. 8vo. 5s. XXXVII. TACITUS. Historize ex editione Brotier. Locis Annalium ab eo citatis, selectis et additis, quibusdam etiam notis subjunctis, ab editore R. Revuan, A.M. 8vo. 12s. XXXVIII. TACITUS. De Moribus Germanorum, et de Vita Agricole, ex editione G. Brorier, cura R. RevuHan, A.M., &c. With Maps of Ancient Britain and Germany. LEditio quarta, 12mo. 5s. 6d. XXXIX. TERENTIUS. Ex recensione F. LinDENBROGII, cum notis vari- orum, Scholiis, et Index verborum et Phrasium. Edidit J. A. Gites, A.M. &ec. 8vo. 9s. XL, THEATRE OF THE GREEKS. A series of papers relating to the History and Criticism of the GREEK Drama. With a new Introduction and other alterations. By J. W. Donatpson, A.M., Head Master of Bury St. Edmund’s Grammar School. Fifth Edition, 8vo. 15s. Ce ee MESSRS. DEIGHTON’S PUBLICATIONS. 5 Classical. XLI. THEOCRITUS, BION, ET MOSCHUS, Greece et Latine. Accedunt animadversiones Hrernporr, HArRuesil, &c., Scholia, Indices, et Porti Lexicon Doricum, edidit Kiesstine. 2 vols. 8vo. 10s. Large paper, 18s. XLII. THEOCRITUS, BION, ET MOSCHUS. Cum Notis variorum et suis T. Briaas, A.M. 8vo. 6s. XLIII, THUCYDIDES. Grece, ex recensione BEKKERI. 8vo. 14s. XLIV. THUCYDIDES. Illustrated with Maps, taken entirely from actual Surveys. With Notes chiefly Historical and Geographical. By Tuomas ARNOLD, D.D. 3 vols. 8vo. Separately—vol. 1. 12s.; vol. 11. 8s.; vol. 111. 10s. XLV. VIRGILIUS. Notis ex editione Heyntana excerptis illustrata. Accedit index MAITTAIRIANUS. 8vo. 14s. THEOLOGICAL. XLVI. H KAINH AIAOHKH META YIIOMNHMATON APXATON, EKAI- AOMENH YIIO GPEOKAHTOT ®APMAKIAOY. The New Testament, with the Commentaries of Euthymius, Gicumenius, Arethas, &c. Edited by the Rev. ‘THEocLITUS PHARMAKIDES, Professor of Theology and Greek Literature in the Royal University of Athens, &c. Vols, I. to 1V., com- prising the Gospels, the Acts, and the Epistles to the Romans and Corinthians. 4 vols. 8vo. 1J. 18s. XLVII. ACTS OF THE APOSTLES. With Notes original and selected. For the use of Students in the University. By Hastines Rosinson, D.D., formerly Fellow of St. John’s College, Cambridge. 8vo. 8s. XLVIII. ARTICLES (The XXXIX.) An Historical Account of, from the first Promulgation of them in M.D.LIII. to their final establishment in M.D.Lxx1. &c. By Joun Lams, D.D., Master of Corpus Christi College, Cambridge. 4to. 12. ds. pv) in BLUNT (Prof.) Sketch of the Church of the first Two Centuries after Curist, drawn from the Writings of the FaTHErRs, down to CLEMENS ALEXANDRINUS inclusive. $vo. 6s, 6d. 6 MESSRS. DEIGHTON 'S PUBLICATIONS. se nhc ce Ns lal ee Theological. ie BLUNT (Professor). An Introduction to a Course of Lectures on the Early Fathers, now in delivery in the University of Cambridge. Parts I. and II. 8vo. 2s. each. LI. BUSHBY (Rev. E.) Introduction to the Study of the Holy Scriptures. Fourth Edition, 12mo. 3s. 6d. LIl. BUSHBY (Rev. E.) Essay on the Human Mind. Fourth Edit. 12mo, 4s. 6d. LIII. BUTLER (Bp.) A Summary of the Argument on his ANALOGY of RELIGION. 8vo. Ils. LIV. BUTLER (Bp.) An Analysis of his Three Sermons on Human Nature, and his Dissertation on VirtuE. With a concise Summary of his System of Morats. 12mo. 1s. LV. CHEKE (Sir John). Translation of the Gospel according to St. MatTuew, and part of the first Chapter of the Gospel according to St. Mark, with original Notes. Also vir Original Letters of Sir J. CHEKE, Prefixed is an Introductory Account of the nature and object of the Trans- lation. By J. Goopwin, B.D., Fellow and Tutor of Corpus Christi College, Cambridge. 8vo. 7s. 6d. LVI. CHEVALIER (Rev. Temple). Translation of the Epistles of CLEMENT of Rome, Porycarp, and IGNATIUS; and of the Apologies of Justin Martyr and TEeRTULLIAN. With an Introduction and Notes. 8vo. 14s. LVII. CHRYSOSTOM (S. Joannis). Homilie in Mattheum. Textum ad fidem codicum MSS. et versionum emendavit, przecipuam lectionis varietatem adscripsit, annotationibus ubi opus erat, et novis indicibus instruxit F. Frerp, A.M., Coll. SS. Trin. Socius. 3 vols. 8vo. 21. 2s. Large paper, 41. 4s. LVIII. CRUDEN (Alex.) A complete Concordance of the OLp and New TESTAMENT; with a Life of the AuTuor. By ALEx. CHALMERS. 4to. portrait, 11, 1s. LIX, ECCLESIZ ANGLICANAZ Vindex Catholicus, sive Articulorum Ecclesize Anglicane cum Scriptis SS, Patrum nova Collatio, cura G. W. Harvey, A.M. Coll. Regal. Socii. 8vo. sVol. 1. 16s. Vol. 11. 16s. Vol. 111. 19s. Lx. FISK (Rev. G.) Sermons preached in the Parish Church of St. BoroteH, CAMBRIDGE. 8vo. 10s. 6d. MESSRS. DEIGHTON ’S PUBLICATIONS. 7 Theological. DMI; GARRICK. Mode of Reading the Liturgy of the Church of Enatanp. 4 New Edition, with Notes anda Preliminary Discourse. By R. Cuut, Tutor in Elocution. 8vo. 5s. 6d. LXII. GIBSON (Rey. J.) Four Sermons preached before the University of Cambridge in 1837. 8vo. 1s. 6d. LXIIt. GOSPELS or St. Marruew, St. Marg, St. Luxe, and Sr. JoHy, and Tue Acts oF THE APOSTLES. Questions on, Critical and Historical, for the use of Students in Theology. By the Rev. R. Wirson, M.A., Fellow of St. John’s College. 12mo. 3s. 6d. each. LXIV. GOSPELS. Questions on the Four Gospens, and the Acts of the ApostxEs, Critical, Historical, and Geographical. 12mo. 3s. 6d. LXV. HARE (Archd.) Sermons preacht in Herstmonceaux Church. 8vo. 12s. LXVI. HEY (Prof.) Lectures in Drvinrry, delivered in the UNIVERSITY of CamBripGk. Third Edition, 2 vols. 8vo. 11. 10s. LXVII. HILDYARD (Rev. J.) Five Sermons on the Parable of the Rich Man and Lazarus, preached before the University of Cambridge. To which is added a proposed Plan for the Introduction of a Systematic Study of Theology in the University. 8vo. ds. Wulsean Wectures. LXVIII. ALFORD (Rev. H.) For the Year 1841. The Consistency of the Divine Conduct in Revealing the Doctrines of Redemption. To which are added Two Sermons preached before the University of Cambridge. 8vo. 7s. LXIX. ALFORD (Rev. H.) For the Year 1842. The Consistency of the Divine Conduct in Revealing the Doctrines of Redemption. 8vo. 6s. LXxX. CHEVALLIER (Rev. T.) For the Year 1826. On the His- torical Types contained in the Otp TesTaMENT. 8vo. 12s. LXxI. CHEVALLIER (Rey. T.) For the Year 1827. On the Proofs of Divine Power and Wisdom derived from the study of Astronomy ; and on the Evidence, Doctrines, and Precepts of Revealed Religion. 8vo. 125. 8 MESSRS. DEIGHTON’S PUBLICATIONS. SI aD BN a POD an loans ere ite oe Theological. LXXII. HOWARTH (Rev. H.) For the Year 1836. Jesus of Nazareth the Christ of God. 8vo. 5s. 6d. LXXIII. HOWARTH (Rev. H.) For the Year 1835. The Truth and Obligation of Revealed Religion considered with reference to Prevailing Opinions. 8vo. 5s. 6d. LXXIV. PARKINSON (Rev. R.) For the Year 1837. Rationalism and Revelation ; or the Testimony of Moral Philosophy, the System of Nature, and the Constitution of Man, to the Truth of the Doctrine of Scripture. 8vo. 9s. 6d. LXXV. PARKINSON (Rev. R.) For the Year 1838. The Constitution of the Visible Church of Christ considered. S8vo. 9s. 6d. LXXVI. ROSE (Rev. Henry John). For the Year 1833. The Law of Moses viewed in connexion with the History and Character of the Jews. 8vo. 8s. LXXVII. SMITH (Rev. Theyre T.) For the Year 1840. The Christian Religion in connexion with the Principles of Morality. 8vo. 7s. 6d. LXXVIII. JONES (Rev. W. of Nayland). An Essay on the Church. 12mo. 1s. 6d. LXXIXx. LEIGHTON (Arch.) Prelectiones Theologice; Parzneses, et Meidtationes in Psalmos IV. XXXII. CXXX. Ethico-Critice. Editio nova iterum recens J. SCHOLEFIELD, A.M., Greec. Lit. apud Cantab. Pro- fessore Regio. 8vo. 8s. 6d. LXxXx, LITURGIE BRITANNICA; or the several Editions of the Book of Common Prayer of the Church of England, from its compilation to the last revision ; together with the Liturgy set forth for the use of the Church of Scotland; arranged to shew their respective variations. By W. KEELING, B.D., Fellow of St. John’s College. 8vo. 1. 1s. The Rubrics in these Liturgies are printed in red. LXXXI. MARGARET, Countess of Richmond and Derby, and Foundress of Christ’s and St. John’s Colleges, Cambridge. The Funeral Sermon, preached by Bisnor Fisurr in 1509; with Baxer’s Preface to the same, &c. Edited by J. Hymers, D.D., Fellow of St. John’s College; with illustrative Notes, Additions, and an Appendix. 8vo, 7s. 6d. a MESSRS. DEIGHTON’S PUBLICATIONS. 9 Theological. LXXXIT. MARTYN (Rev. H.) Controversial Tracts on CHRISTIANITY and MouHAMMEDANISM. Also of some of the most Eminent Writers in Persia. Translated and explained by the Rev. S. Len, A.M., Professor of Arabic in the University of Cambridge, &c. &c. 8vo. portrait. LXXXII. : MERIVALE (Rev. C.) The Church of England a Faithful Witness of Curist ; not Destroying the Law, but Fulfilling it. Four Ser- mons preached before the University of Cambridge in 1838. 8vo. 4s. LXXXxIv. MERIVALE (Rey. C.) Sermons preached in the Chapel Royal at Whitehall. 8vo. 10s. 6d. LXXXV. MIDDLETON (Dr. T. F.) The Doctrine of the Greek Article applied to the Criticism and Jllustration of the New Testament. With Prefatory Observations and Notes. By Hucu JAMES RoskE, B.D. 8vo. 13s. LXXXVI. MILL (Dr. W. H.) The Evangelical Accounts of the Descent and Parentage of the Saviour, vindicated against some recent Mythical Interpreters. 8vo. 4s. LXXXVII. MILL (Dr. W. H.) The Historical Character of St. Luke's first Chapter, vindicated against some recent Mythical’Interpreters. 8vo. 4s. LXXXVIII. MILL (Dr. W. H.) Prelectio Theologica in Scholis Cantabri- giensibus habita Kal. Feb. a.p. M.DccC.xLIII. 4to. 2s. LXXXIX. MILL (Dr. W. H.) Observations on the attempted Application of Panruetstic Principles to the Theory and Historie Criticism of the GosPEL. 8vo. 6s. 6d. ee NEALE (Rev. J. M.) Ayton Priory, or the Restored Monastery. 12mo. 4s. XCl. PALEY. Analysis of the Principles of Moral and Political Philo- sophy. By S. FennevL, M.A., Fellow of Queens’ College. 12mo. 2s. 6d. xCII. PALEY. Analysis of the Evidences of Christianity. By S. FENNELL, M.A. &c. 12mo. 2s. 6d. XCIII. PALEY. Examination Questions on the Evidences of Christianity, 12mo. 2s. 6d. Z 10 MESSRS. DEIGHTON’S PUBLICATIONS. Theological. XCIV. PEARSON (Bp.) Exposition of the CrEED. An Analysis of, with some additional matter occasionally interspersed. By W. H. Mixx, D.D., Chaplain to the Archbishop of Canterbury. 8vo. 5s. XCV. PORSON (Prof.) A Vindication of his Literary Character from the Animadversions of BisHop Burgess, on 1 John, v. 7. By Crrto CANTABRIGIENSIS. 8vo. 11s. XCVI. ROBINSON (Prof.) The Character of St. Paun the Model of the Christian Ministry. Four Sermons preached before the University of Cambridge in 1840. 8vo. 3s. XCVII. ROSE (Rev, Hugh James). Christianity always Progressive. 8vo. 8s. 6d. XCVIII. ROSE (Rev. Hugh James). Eight Sermons preached before the University of CAMBRIDGE in 1830 and 1831, Second Edition, 8vo. 7s. 6d. XCIX, ROSE (Rev. Hugh James). The State of ProresTantism in GERMANY Described. Second Edition, 8vo. 14s. c, SCHOLEFIELD (Prof.) Hints for an Improved Translation of the New Testament. Second edition, 8vo. 4s. cl. SCHOLEFIELD (Prof.) Scriptural Grounds of Union, con- sidered in Five Sermons preached before the University of Cambridge in 1840. Second Edition, 8vo. 38s. 6d. Ci: | SEDGWICK (Professor). A Discourse on the Studies of the University. Fourth Edition, 8vo. 4s. CIIl. SMITH (Rev. C.) Seven Letters on Nationan Rewieton. Ad- dressed to the Rev. Henry MELvinu, M.A. 8vo. 7s. 6d. CIV. TAYLOR (Bp. Jeremy). His Whole Works, with a Life of the AvTHOoR, and a Critical Examination of his Writings. By Bisuor Huser. 15 vols. 8vo. £6. Cv. TAYLOR (Bp. Jeremy). Holy Living and Dying, together with Prayers. Containing the whole Duty of a Christian. 8vo. 12s. MESSRS. DEIGHTON’S PUBLICATIONS. ll Theological. CVI. TAYLOR (Bp. Jeremy). Rule and Exercises of Hoty Livine. Svo. 4s, CVII. TAYLOR (Bp. Jeremy). Rule and Exercises of Hoty Dyine. 8vo. 4s. CVIII. TAYLOR (Bp. Jeremy). Life of, with a Critical Examination of his Writings. By Bisnop Heser. Third Edition, 8vo. portrait, 6s. CIx. TERTULLIAN. The Apology. With English Notes and a Preface, intended as an Introduction to the study of Patristical and Eccle- siastical Latinity. By H. A. Woopuam, A.M., Fellow of Jesus College, Cambridge. 8vo. 8s. 6d. CX. THORP (Archd.) Four Sermons, preached before the University of Cambridge in May 1838. 8vo. 3s. 6d. CxXI. TURTON (Dean). Naturau THEOLOGY considered with refer- ence to Lorp BrouGHam’s Discourse on that subject. Second Edition, 8vo. 8s. CxII. TURTON (Dean). The Roman Catholic Doctrine of the Eu- charist considered, in Reply to Dr. Wiseman’s Argument from Scripture. 8vo. 8s. 6d. CxXIII. TURTON (Dean). Observations on Dr. Wiseman’s Reply to Dr. Turton’s Roman Catholic Doctrine of the Eucharist considered. 8vo. As. 6d. CXIV. USHER (Archbp.) Answer to a Jesuit. With other Tracts on Porery. 8vo. lds. 6d. CXV. WELCHMAN. Articuli Ecclesie Anglicane. Textibus sacre Scripturee et Patrum Primevorum Testimoniis confirmati, brevibusque notis illustrati. Appendicis loco nune primum adjiciuntur Catechismus Epvarpi VI. et Articuli a. D. 1552 approbati. 8vo. 6s. 6d. CXVI. WILSON (Rev. Wm.) An Illustration of the Method of Explain- ing the New Testament by the early Opinions of the Jews and Christians concerning Christ. New edition, 8vo. 8s. 12 MESSRS. DEIGHTON’S PUBLICATIONS. MATHEMATICAL. CXVIL. AIRY (Astronomer Royal). Mathematical Tracts; or the Lunar and PLANETARY THEORIES; the FicureE of the Eartu; PREcEssION and Nutation; the Catcutus of Variations; the UNpULATORY THEORY of Optics. Third Edition. 8vo. Plates, 15s. CXVIIL. ASTRONOMICAL OBSERVATIONS, made at the Observatory of CAMBRIDGE. By Professor Airy, By Professor CHALLIS. Vol. I. for 1828, 4to. 12s. Vol. IX. for 1886, 4to. 12. 5s. Voli; “y 1829, 12s. Vol. X. oe 1837, 12. 11s. 6d. VOLT Gass) bi So0, 14s. 6d. Vole XoLgai. gieobs 21. 2s. VOL GV cicukrsc mn ghos 1, 14s. 6d. Vol XT © yo. 18302 1l. 11s. 6d. Vol. V. ey ae, 15s. Vol. XIII, .. 1840, 21. 12s. 6d Vol? V Wii, Hivesss; 15s. 1841, pratt ys Vol. VII... 1884, 15s. Vol. VIII. .. 1836, 15s. Oxi: BROOKE (C.) A Synopsis of the Principal Formune and Resuuts of Pure Matuemarics. 8vo. lds. CXxX, BROWNE (Rev. A.) A Short View of the First Principles of the DIFFERENTIAL CALCULUS. 8vo. 93. CXXI, CAMBRIDGE PutuosopnicaL Socrery, (TRANSACTIONS of). 4to. Plates. Vol. I. Part: i, sli. Vol. V. Part 2, 14s. Vol. I. Part 2, 11. 10s. Vol. V. Part 3, 12s. Vol-uiiy 2) Bartle 18s. Vol. VI. Partl, 13s. VolslLy nParti2; 18s. VoL Vii 4 PAartaia slas: Vol. III. Part 1, 12. 11s. 6d. MoLAViD a. Barto stan Volall (aeerartes, 2s. 6d. V Ol. Vill abPart leis cGd: Vola LI. «Parts, 3s. Vol.-ViAi, (Rarts2. 7s: Volt Veam Parte. 15s. Vol. VIL Part's.2e135, Vol 1Ve PPart2; Git. its. Vol. VITI. Part 1, 12s, Vousl Vee Part. 3; 12s. Vol. VIII. Part 2, Vol. V. Part.1,; 10s. CXXII. CAMBRIDGE ProsBiems; being a Collection of the Questions proposed to the Candidates for the Degree of BacHELOoR of Arts, from 1811 to 1820 inclusive. 8vo. 5s, MESSRS. DEIGHTON’S PUBLICATIONS. 13 Mathematical. OXXIII. CARNOT (M.) Reflexions on the Metaphysical Principles of the INFINITESIMAL ANALYSIS. Translated from the French. 8vo. 3s. OXXIV. CODDINGTON (Rev. H.) Treatise on the Reflexion and Re- fraction of Lignt. Being Part I. of a System of Optics. 8vo. plates, 15s. CXXV. CODDINGTON (Rev. H.) Treatise on the Eyx, and on OPTicaL Instruments. Being Part II. of a System of Oprics. 8vo. plates, 5s. CXXVI. CODDINGTON (Rev. H.) Introduction to the DirrerentiaL CaLcuLus on Algebraic Principles. 8vo. 2s. 6d. OXXVI. COLENSO (Rev. J. W.) Elements of ALGEBRA, designed for the use of Schools. Fourth Edition. 12mo. 4s. 6d. OXXVII. COLENSO (Rey. J. W.) AniTHMeETIC, designed for the use of Schools. Second Edition. 12mo. 4s. 6d. CXXIX: CONIC SECTIONS, the Elements of the, with the Sections of Conoids. Third Edition. 8vo. 4s. 6d. CXXX. CRESSWELL (Dr.) Elementary Treatise on the GEOMETRICAL and ALGEBRAICAL Investigation of Maxima and Minima. Second Edition. Svo. 6s. OXXXI. CRESSWELL (Dr.) Problems in GEomETRY deduced from the First Six Books of Evcurp, with their Solutions; forming a Supplement to the Elements of Eucuip. Second Edition. 8vo. 6s. CXXXII. CRESSWELL (Dr.) Treatise on GoEMETRY. 8vo. 6s. CXXXIII. CRESSWELL (Dr.) Elements of Linear PERSPECTIVE. 8vo. plates, 3s. 6d. : CXXXIV. CUMMING (Prof.) Manual of Erectro-Dynamics, chiefly trans- lated from the French of J. F. DEMONFERRAND. 8vo. plates, 12s. CXXXV. EARNSHAW (Rev. 8.) Dynamics, or a Treatise on MorIon; to which is added, a short Treatise on Attractions. Third Edition. 8vo. plates, 14s. OCXXXVI. EARNSHAW (Rey. S.) Treatise on Stratics. Second Edition. 8vo. plates, 10s. 14 MESSRS, DEIGHTON'S PUBLICATIONS. Mathematical. CXXXVI. EUCLID; the Elements of, by R. Simson, M.D. 25th Edition, revised and corrected. 8vo. 8s.; 12mo. 5s. | CXXXVII. EUCLID; the Elements of. From the Text of Stmson. Edited, in the Symbolical Form, by R. BLakELocx. 12mo. 6s. CXXXVIII. EUCLID; a Companion to. With a set of improved Figures. 12mo. 4s. OXXXIX. FENNELL (Rey. S.) Elementary Treatise on ALGEBRA. 8vo. 9s. CXL GREGORY (D. F.) Examples on the Processes of the DirFE- RENTIAL ana INTEGRAL CaLcuLus. 8vo. plates, 18s. CXLI. GRIFFIN (W.N.) Treatise on Optics. Second Edition. 8vo. plates, 8s. CXLII. HAMILTON (H. P.) Principles of ANAtyTIcAL GEOMETRY. 8vo. plates, 14s. CXLIII. HEWITT (Rey. D.) Problems and Theorems of PLANE TRrI- GONOMETRY. 8vo, 6s. CXLIV. HIND (Rev. J.) The Elements of AterBra. Fifth Edition. 8vo. 12s. 6d. CXLY. HIND (Rev. J.) Introduction to the Elements of ALGEBRA. Second Edition. 12mo. 5s. CXLYI. HIND (Rey. J.) Principles and Practice of ArtTHMETIC. Fourth Edition. 12mo. 4s. 6d. CXLYII. HIND (Rey. J.) Elements of PLane and SPHERICAL TRIGONO- METRY. Fourth Edition. 12mo. 7s. 6d. CXLVIII. HUTTON (Dr.) Mathematical Tables. Edited by Oxynruus GREGORY. Royal 8vo. 18s. CXLIX. HYMERS (Dr.) Elements of the Theory of Astronomy. Second Edition. 8vo. plates, 14s. CL. HYMERS (Dr.) Treatise of ANALYTICAL GEOMETRY of Three Dimensions. Second Edition. S8vo. 10s. 6d. MESSRS. DEIGHTON 'S PUBLICATIONS. 15 Mathematical. CUE. HYMERS (Dr.) Treatise on the Integra CatcuLus. Third Edition. 8vo. plates, 10s. 6d. CLII. HYMERS (Dr.) Treatise on the Theory of ALGEBRAICAL Equa- TIONS. Second Edition. 8vo. plates, 9s. 6d. CLII. HYMERS (Dr.) Treatise on Dirrerentian Equations, and on the CatcuLus of Finite DIFFERENCES. 8vo. plates, 10s. CLIV. HYMERS (Dr.) Treatise on TRIcoNoMETRY, and on the TRI- GONOMETRICAL TABLEs of LoGcaritHMs. Second Edition. 8vo. plates, 8s, 6d. CLY. HYMERS (Dr.) Treatise on SpHERIcAL TRIGONOMETRY. 8yo. plates, 2s. 6d. CLVI. INTEGRAL CALCULUS; a Collection of Examples on the. 8vo. 5s. 6d. CLVII. \ JARRETT (Rey. T.) An Essay on ALGEBRAIC DEVELOPMENT ; containing the principal Expansions in Common ALGEBRA, in the DIFFERENTIAL and InrEGRAL CatcuLus, and in the CaLcuLus of Finite DirrERENCES. 8vo. 8s. 6d. CLVIII. KELLAND (Rev. P.) Theory of Hear. 8vo. 9s. CLIX. LA CROIX. Elementary Treatise on the Mathematical Principles of ArnituMetic. ‘Translated from the Frencu. 8vo. 3s. GLX. MECHANICAL PROBLEMS, adapted to the course of Reading pursued in the University of CAMBRIDGE. 8vo. 7S. CLXI. MILLER (Prof.) The Elements of Hyprosratics and Hypro- pynamics. Third Edition. Svo. plates, 6s. CLXII. MILLER (Prof.) Elementary Treatise on the DIFFERENTIAL CaucuLus. Third Edition. 8vo. plaies, 6s. CLXIII. MILLER (Prof.) Treatise on CRYSTALLOGRAPHY. 8vo. Plates. 7s. 6d. CLXIV. MILLER (Prof.) Table of Mineralogical Series ; being a Syllabus of the Lectures on Mineralogy. 8vo. 1s. 6d. 16 MESSRS. DEIGHTON’S PUBLICATIONS. Mathematical. CLXV. MURPHY (Rey. R.) Elementary Principles of the Theory of ELEcTRICITY. 8vo. 7s. 6d. CLXVI. MYERS (C. J.) Elementary Treatise on the DIFFERENTIAL CatcuLus. 8vo. 2s. 6d. CLXVII. O’BRIEN (Rev. M.). Mathematical Tracts. On La Puacr’s Co- efficients; the Figure of the Earth; the Motion of a Rigid Body about its Centre of Gravity ; Precession and Nutation. 8vo. 4s. 6d. CLXVIII. O’BRIEN (Rev. M.) Elementary Treatise on the DirFERENTIAL CatcuLus. 8vo. Plates, 10s. 6d. CLXIX. O’BRIEN (Rev. M.) Treatise on PLane Co-orpDINATE GEOMETRY; or the Application of the Method of Co-ordinates to the Solution of Problems in PLANE GEOMETRY. 8vo. plates, 9s. CLXX. PEACOCK (Dean). Treatise on AtemBRA. Vol. I. ARITHMETICAL ALGEBRA. 8vo. 1ds. CLXXI. PRATT (Rev. J. H.) The Mathematical Principles of MecuanicaL PuHiLosopHy. Sccond Edition. 8vo. Plates, 11. 1s. CLXXII. STATICS (Elementary); or a Treatise on the Equitiprium of Forces in One PLANE. 8vo. Plates, 4s. 6d. CLXXIII. STEVENSON (R.) Treatise on the Nature and Properties of ALGEBRAIC EquaTions. Second Edition. S8vo. 6s. 6d. CLXXIV. TRIGONOMETRY. A Syllabus of a Course of Lectures upon, and the Application of ALGEBRA to GEOMETRY. Second Edition. 7s. Gd. CLXXV. VENTUROLI (G.) Elements of the Theory of Mxrcuanics; translated from the Italian of D. CresswExL, D,D., Fellow of Trinity Col- lege, Cambridge. 8vo. Plates, 5s. CLXXVI. WEBSTER (T.) Principles of Hyprostarics. Second Edition. 8vo. 10s. 6d. CLXXVII. WEBSTER (T.) The Theory of the Equilibrium and Motion of FLuIpDs. 8vo. Plate, 9s. MESSRS. DEIGHTON'’S PUBLICATIONS. LY y Mathematical. CLXXVIII. WHEWELL (Dr.) Elementary Treatise on Mecuanics. Siath Edition. 8vo. Plates. 7s. 6d. CLXXIX. WHEWELL (Dr.) On the Free Morton or Ports, and on UNIVERSAL GRAVITATION. Including the principal Propositions of Books I. and III. of the Principia. The first part of a Treatige on DYNAmics. Third Edition. 8vo. Plates. 10s. 6d. CLXXXx. WHEWELL (Dr.) On the Morton or Points constrained and resisted, and on the Motion or a Rictp Bopy. The second part of a Treatise on Dynamics. Second Edition. 8vo. Plates. 12s. 6d. CLXXXT. WHEWELL (Dr.) Doctrine of Limits, with its Applications ; namely, Conic Sections; the Three First Sections of Newron; the DIFFERENTIAL CALCULUS. 8vo. Qs. CLXXXII. WHEWELL (Dr.) Awnanytican Statics. 8vo. Plates. 7s. 6d. CLXXXIII. WHEWELL (Dr.) Mechanical Euclid, containing the Elements of Mecuanics and Hyprostatics, demonstrated after the manner of GEOMETRY. Fourth Edition. 12mo. 4s. 6d. CLXXXIV. WHEWELL (Dr.) The Mecuanics of ENGINEERING, intended for use in the Universities, and in Colleges of ENGINEERS. 8vo. 9s. CLXXXYV. WILLIS (Prof.) Principles of Mecnanism. 8vo. 15s. CLXXXVI. WILSON (Rev. R.) A System of Puane and SPHERICAL TRIGONOMETRY. 8vo. 6s. CLXXXVII. WOOD (Dean.) Elements of ALGEBRA. Revised and enlarged, with Notes, additional Propositions, and Examples, by T. Lunp, B.D., Fellow of St. John’s College. 8vo. 12s. 6d. CLXXXVIII. WOOD (Dean.) Elements of Mecnanics. Revised, re-arranged, and enlarged by J. C. SnowspatL, M.A., Fellow of St. John’s College. 8vo. 8:. 6d. CLXXXIX. WOODHOUSE (Prof.) Treatise on Astronomy, Theoretical and Practical. Second Edition. 2 vols. 8vo. 10. 10s. CXc. WOODHOUSE (Prof.) Elementary Treatise on Puysicau Astronomy. 8vo. 18s. CXCI. WOODHOUSE (Prof.) Treatise on PLANE and SPHERICAL TRIGONOMETRY. Fifth Edition. 8vo. 12s. c 18 MESSRS. DEIGHTON'S PUBLICATIONS. MISCELLANEOUS. CXCII. ALFORD (Rev. H.) The School of the Heart, and other Peems, 2 vols. 12mo. 8s. CXCIII. ARCHAEOLOGICAL Journal. Published under the direction of the Central Committee of the British AkCHAEOLOGICAL ASSOCIATION for the Encouragement and Prosecution of Researches into the Arts and Monuments of the Early and Middle Ages, Published Quarterly. Nos. 1, 2, 8, Plates, 2s. 6d. each. CXCIV. ARNOLD (Dr. Thomas) History of Rome. 8vo. Vol. I. 16s. Vol. II. 18s. Vol. III. 14s. CXCV. BIOGRAPHICAL Dictionary. A New General One, projected and partly arranged by the late Rev. Hucu James Ross, B.D., Principal of King’s College, London. Edited by the Rev. Henry J. Rose, B.D., late Fellow of St. John’s College, Cambridge. 8vo. vols. I to VI. 18s. each. A volume is published Quarterly. CxXCVI. BUTLER (Bp.) Memoirs of his Life, Character, and Writings. By Tuos. BartTLeTtT, A.M., Rector of Kingstone, Kent. Portrait. 8vo. 12s. CXCVII. CAMBRIDGE. A complete Collection of the Prize Poems, from the institution of the Premium by the Rev. T. Seaton, in 1750 to 1806. 2 vols. Svo. 6s. CXCVIII. CAMBRIDGE UNIVERSITY ALMANACK for the Year 1845. Embellished with a Line EneGravinec by Mr. E. CuHatuis, from a Drawine by W. G. Dopeson, of the Gate of Honour Caius CoLLEeGE, SenaTE Housg, and New University Lisrary, 4s. 6d. Continued Annually. CXCIX. CAMBRIDGE University CaLenpar, for the Year 1844. 6s. Continued Annually. CC. CAMBRIDGE GUIDE, including Historical and Architectural Notices of the Pusiic BuiLpinGs, &c., new Edition, illustrated by numerous beautiful Engravings, from Drawings by Mackenzie and Rudge, and a New Plan of the Town. ccl. CAMBRIDGE, [Illustrations of. Being a series of Views of the Pustic Buitpines of the University and Town. Engraved by Messrs. STORER. 4to. and 8vo. Each plate is sold separately. CCII. CAMBRIDGE PORTFOLIO. Consisting of Papers illustrative of the principal features in the Scnoxtastic and Sociat System of the University. Edited by the Rev. J. J. Smiru, M.A., Fellow of Gonville and Caius College. 2 vols. 4to. with numerous engravings, &c. 41. 4s. MESSRS. DEIGHTON’S PUBLICATIONS. 19 Sa a EC a Meee Ccill. CAMBRIDGE ANTIQUARIAN SOCIETY’S PUBLICATIONS. No. I.—Containing a CATALOGUE of the ORIGINAL LL BRARY of St. CATHARINE’S HALL, mccco.uxxy. By G. E. Corriz, B.D., Fellow and Tutor of St. Catharine’s Hall, and Norrisian Professor of Divinity. 4to. sewed, 1s. 6d. No. I.—ABBREVIATA CRONICA ab anno 1377, usque ad annum 1469. By the Rev. J. J. Smirn, M.A., Fellow and Tutor of Gon- ville and Caius College. 4to. sewed, 2s. 6d. No. II—AN ACCOUNT of the RITES and CEREMONIES Which took place at the CONSECRATION of ARCHBISHOP PAR- KER, with an Introductory Preface and Notes, By the Rev. JAMES Goopwin, B.D., Fellow of Corpus Christi College. 4to. sewed, 3s. 6d. Nos. IV. and. V.—AN APPLICATION of HERALDRY to the Illustration of various UNIVERSITY and COLLEGIATE ANTIQUI- TIES. By Henry ANNESLEY WoopuaM, Esq., A.B., F.S.A., Classical and Divinity Lecturer of Jesus College. Part the First, 4to. sewed, 6s. Part the Second, 4to. sewed, 4s. 6d. Nos. VI. and VIII.—A DESCRIPTIVE CATALOGUE of the MANUSCRIPTS and SCARCE BOOKS in the LIBRARY of Sr. JOHN’S COLLEGE, CAMBRIDGE. By the Rev. Morcgan Cowie, M.A., Fellow of St. John’s College. Parts I. & II. 4to, sewed, 4s. 6d. each. No. VII—A DESCRIPTION of the SEXTRY BARN at ELY, lately demolished. With Illustrations. By Rogert Wituis, M.A., Jack- sonian Professor. 4to. sewed, 3s. No. Ix. — ARCHITECTURAL NOMENCLATURE of the MIDDLE AGES, By Roserr Wiuuis, M.A., F.R.S. &c., Jacksonian Professor, 4to. Plates, sewed, 7s. FIRST REPORT, presented to the CamBRiIpGe ANTIQUARIAN Society, at its General Meeting, May 6, 1841. S8vo. sewed, 1s. SECOND REPORT, presented May 13, 1842. 8vo. sewed, 1s. THIRD REPORT, presented May 24, 1843. 8vo. sewed, 1s. CClv. CORRIE (Prof.) Brief Historical Notices of the Interference of the Crown with the Affairs of the ENGLIsu UNIVERSITIES. 8vo. 8s. 6d. CCV. DYER (G.) History of the Untversrry and CotuEcEs of Cam- ) BRIDGE ; including Notices relating to the Founders and Eminent Men. 2 vols. 8vo. Plates, 18s. Large Paper, 1. 11s. 6d. 20 MESSRS. DEIGHTON 'S PUBLICATIONS. Miscellaneous. CCVI. ENCYCLOPEDIA METROPOLITANA. — Parts I. to LVIII. Or in Volumes: Pure Sciences, 2 vols. Mixep and AppLiep Sciences, 6 vols. History and BrocraPuHy, 5 vols. MiscELLANEOUs and LExIcoGRAPHICAL, 12 vols. The Index, which will complete the work, will appear shortly. ccVii. FULLER (Dr. Thos.) History of the University of CAMBRIDGE, from the Conquest to the Year 1634. Edited by Rev. M. Prickett, M.A., F.S.A., of Trinity College, and. Thos. Wricut, Esq., M.A., F.S.A., of Trinity College, with illustrative Notes. With two Plans of Cambridge. 8vo. 12s. CCVIiII. JENYNS (Rey. L.) Manual of British VERTEBRATE ANIMALS : or Descriptions of all the Animals belonging to the Classes MAMMALIA, Aves, Reprinia, AMpuisia, and PIscEs, observed in the BriTisH IsLANDs. 8vo. 13s. CCIX. RELHAN (R.) Flora Cantabrigiensis, exhibens Plantas Agri Cantabrigiensis indigenas, &c. ditio tertia. 8vo. 6s. ccx. SMITH (Rev. Charles Lesingham). Poetical Works. 8vo. 5s. ccXl. SMYTH (Prof.) Lectures on Mopsrn History, from the Irrup- tion of the Northern Nations to the Close of the AMERICAN War. Fourth Edition. 2 vols. 8vo. ll. Is. CCXII. SMYTH (Prof.) Lectures on History. Second and concluding Series. On the Frencuw RevonuTion. Second Edition. 3 vols. 8vo. il, J 1s. 6d. CCXIII. WHEWELL (Dr.) Architectural Notes on GERMAN CHURCHES ; with Notes written during an Architectural Tour in Picarpy and Nor- MANDY. Third Edition. To which are added, Translation of Notes on Cuurcues of the Rute, by M. F, De Lassauvx, Architectural Inspector to the King of Prussia. 8vo. Plates, 12s. CCXIV. WHEWELL (Dr.) History of the Iypuctive Sctenczs, from the Earliest to the Present Time. 38 vols. 8vo, 2. 2s. CCXV. WHEWELL (Dr.) The Puttosopuy of the INDUCTIVE SCIENCES, founded upon their History. 2 vols. 8vo. 1l. 10s. CCXVI. WHEWELL (Dr.) On the Principles of Enexis UNIvErsITy Epucation. Including Additional Thoughts on the Study of MatTue- matics. Second Edition. 8vo. ds. CCXVII. WILLIS (Prof.) Remarks on the Architecture of the MIDDLE AGEs, especially in Ivany. 8vo. Plates, 10s. 6d. Large Paper, ll. 1s. ge MESSRS. DEIGHTON’S PUBLICATIONS. A | Re ater gegen Se et het PR AEE tT, VIEWS OF THE COLLEGES anp otuER PUBLIC BUILDINGS Yn the Wnibersity of Cambridge, Taken expressly for the Universtry ALMANACK, (Measuring about 17 inches by 11 inches). Onn men nn ee No. Year. Subject. 1...1801 TRINITY COLLEGE —West Front of Library. 2...1802 KING’S COLLEGE and CHAPEL—West Front, and Clare Hall. 3...1803 Sr. JOHN’S COLLEGE—Bridge and West Front. 4...1804 QUEEN’S COLLEGE taken from the Mill. 5...1805 JESUS COLLEGE —taken from the Road. 6 ..1806 EMMANUEL COLLEGEW—West Front. 7...1807 PEMBROKE COLLEGE—West Front. 8...1808 TRINITY HALLtaken from Clare Hall Garden. 9...1809 SIDNEY SUSSEX COLLEGE — taken from Bowling Green. 10...1810 CHRIST’S COLLEGEW—taken from the Garden. 11...1811 CAIUS COLLEGE —Second Court. 12...1812 DOWNING COLLEGE— Master’s Lodge. 13...1813 ST. PETER’S COLLEGE—taken from the Street. 14...1814 CATHARINE HALL—Interior of Court. 15...1815 CORPUS CHRISTI COLLEGE—Interior of Old Court. 16...1816 MAGDALENE COLLEGE—Front of Pepysian Library. 17...1817 SENATE HOUSE and UNIVERSITY BANS 18...1818 TRINITY COLLEGE—Great Court. 19...1819 ST. JOHN’S COLLEGE —Second Court. 20...1820 MAGDALENE COLLEGE—First Court. 21...1821 EMMANUEL COLLEGE —First Court. 22...1822 KING’S COLLEGE—Old Building, 23...1823 JESUS COLLEGE —taken from the Close. MESSRS. DEIGHTON’S PUBLICATIONS. ha rr es Fe es 2 24... 25... 26... Wises 28... P20.0, 30... Ol... 32... 34... 35.00 36... OS] ce Dose 39 i. 40... 41... 42... 43... 44 45 QUEENS’ COLLEGE —taken from the Grove. OBSERVATORY. CORPUS CHRISTI COLLEGE—West Front, New Building. TRINITY COLLEGE—Interior of King’s Court. ST. PETER’S COLLEGE—Gisborne’s Court. KING’S COLLEGE NEW BUILDINGS and CHAPEL —taken from the Street. ST. JOHN’S COLLEGE—New Building. TRINITY COLLEGE — West Front of King’s Court and Library. CHRIST’S COLLEGE—New Buildings. KING’S COLLEGE CHAPEL—Between the Roofs. PITT PRESS. . SIDNEY SUSSEX COLLEGE —taken from an Elevation. KING’S COLLEGE—CHAPEL, &c. West Front. ST. JOHN’S COLLEGE—New Bridge, &c. FITZWILLIAM MUSEUM. THe NEW UNIVERSITY LIBRARY. CAMBRIDGE—from the top of St. John’s College New Buildings. CLARE HALL—from the Bridge. FITZWILLIAM MUSEUM. Entrance Hall and Statue Gallery. TRINITY COLLEGE. Interior of the Hall. ST. SEPULCHRE’S CHURCH, as restored by the CamBripce Campen Society. CAIUS COLLEGE Gate of Honour. Senate House and New University Liprary. . 1 to 25, inclusive .... Price, Plain Impressions.... 2s. 6d. Proofs Via Peo} ds. Od. - 26 to 45, ——— .... —— Plain Impressions.... 5s. Od. Proofstge pies. Sia es 8s. Od. on India Paper 12s. Od. ( 23 ) . WORKS PREPARING FOR PUBLICATION BY J. & J. J. DEIGHTON. eee eee i nbrh—The BOOK of SOLOMON, called ECCLESIASTES, translated from the original Hebrew, chiefly in accordance with the arrangement and elucidations of R. Moses MENpDLEssoHN. With copious Notes philological and exegetical ; accompanied by a literal translation from the Rabbinic Hebrew of his Commentary and Preface; and Prolegomena, containing various Observations illustrative of Hebrew and Rabbinic Literature. By THEoporeE Preston, M.A,, Fellow of Trinity College, Cambridge. 2. AIZXYAOY IKETIAEX. ASSCHYLI SUPPLICES. Recensuit, emendavit, explanavit, FrepERIcus A. Patey, M.A., Coll. Div. Joh. Cant. 3. AHMOZXOENOYS O TEPI TH ONAPAIIPEZSBEIAS AOTOS, with CRITICAL NOTES in LATIN, ENGLISH EXPLANATORY NOTES, PHILOLOGICAL and HISTORICAL, and APPENDICES. By R. SuitueTo, M.A., of Trinity College, Cambridge. Nearly ready. 4. TREATISE of ALGEBRA, Vol. II. By George PEacocr, D.D. Dean of Ely, late Fellow and Tutor of Trinity College. 5. _ A TREATISE on the APPLICATION of ANALYSIS to SOLID GEOMETRY. By D. F.Grecory, M.A., late Fellow and Assistant Tutor of Trinity College, Cambridge. Nearly ready. 6. An INTRODUCTION to the DIFFERENTIAL CALCULUS being an ELEMENTARY TREATISE on the METHOD of LIMITS and the THEORY of SERIES, with several simple Applications By the Rev. M. O’Brien, M.A., Professor of Natural Philosophy and Astronomy in King’s College, London, and late Fellow of Caius College, Cambridge. the A TREATISE on the THEORY of DEFINITE INTEGRALS. By J. Epteston, M.A., Fellow of Trinity College, Cambridge. 8. AN INTRODUCTION to the PRINCIPLES of LINEAR and ISOMETRIC PERSPECTIVE. By the Rev. C. Pritcuarn, M.A., F.R.S., late Fellow of St. John’s College, Cambridge. 9. The CAMBRIDGE GUIDE, including Historical and Archi- tectural Notices of the Pusiic Buitpines, &c. A New Edition, illustrated by numerous beautiful Engravings, from Drawings by Mackenzie and Rudge. 12mo. 10. A COLLECTION of ANTHEMS used in the Chapels of Ki1ne’s, Trinity, and St. Joun’s Cotteces, CamBripGE. Compiled, and arranged in chronological order, by Tomas ATTwooD WALMISLEY, M.A., Professor of Music in the University. 11. An APOLOGY for the Greek Cuurcn, or Hints towards the Religious Improvement of the Greek Nation. By J. Epwarv Masson, late Professor in the University of Arnmns. Edited by J. S. Howson, M.A.,, Trinity College, one of the Masters of the Liverpool Collegiate Institution. INDEX. *.* The Works in each Class are arranged alphabetically. PAGE Claksical Works (io Gcscscsss-ccssserscerebessnncvesccrccoscccncursasatns 1 Theological Works........sssccescssesecneceseesecnsceeeseeescescenscues 5 Hulsean Lectures .......esesees voices ts sh elteCosetan ert aseaaerene fe Mathematical Works .......sscccccceccccccccscccessssescsecscccscosess 12 Cambridge Observatory Astronomical Observations ......... 12 Cambridge Philosophical Society’s Transactions .......+..+. 12 Miscellaneous Works 20... .nc0s-0c000 0s 50 ncidceee tiation ce oles saat eade aids 18 Cambridge Antiquarian Society's Publications ............+ 19 Views of the Colleges and other Public Buildings ............... 21 Works preparing for Publication .cc.sssessssecesseececesensenecneens 23 i) - a) 4 NS * APY, ies 4 UNIVERSITY OF ILLINOIS-URBANA 516.33G86T C001 A TREATISE ON THE APPLICATION OF ANALYS| MUN UA 12 017262384