pee Bey iL hk ean "4" Veg i] Wich ub es 4 tate f ii KG SUA by ws Tiel, ay ee ie Star ah pin hoa i ii sae rt ie i eet eu iy a i ae ted Hah 4 aa iit bi aie! Ms at 4 a aed Gite ia Hy 1 ‘ Paste WuiPe Gy v foisth ied 8 y iy he: ae al ee thegs wh F i dain vi sy on 7m Pe feats + a4 east tH a i 8 U et 7 al a 4 + peutery Ca «dj ior Aa SR ye be beh ull dah oh it ' yah 4 godt a Asi rie ett aye ae i Pt Pate pera ne rl its ‘aio i j hades Ast dele ee ings patie i je \ i ae Dit ane iu He ig h hate elite fk) Ase! is Haare on % ie i of abt Tie, icicle i ae a . eae at ae APR 14 hi Hh) pO Mat ‘ be fash: i iH, ied faded fan ad HAY it ish Hhipa Mi it iy ae ae P iit ihe ati be Pate ae aA Bie Lay ie ce (J ae i iii Ae sila ¥ } 7 cay " fl tah ind ay" WHT HSEA| ) jigedoteltbe te IAF se SH LES ran WL Hie Apple a i a i hs Hs bbe nd i} Aly ii ti: i ol a, t ih AE, ft 1 thats th i i ‘al ‘ Brak j 4 he ne ip wie ts i ee Hath aia! bho ty 4 Hit AV ivi 4 eit 4 esl Qo tete ohn a i) ytiea} ae Svea Hi HH Wet hdr jab Guede! Nani ne Deore ve hh 4 REARING ee ia oh a ie es iE ee aan io ORLA it Gaede neadeis in ay wih i Ws belt il abil ihegets 4 {li taht re 4 feta 4 vic whadaileh Hi MORO Ma WUE TR aL A ‘ iia ii de aah He Fy ARRAY 4 AGA nu iy Ms) ft ws, Wit ten BB iy a ty 4 HH fh f i" (ing 4 4 I: 8 Ths (hale i Pclba hy ru } : ‘46 4) ena i} bib i it A oe ae i ie ce il H ial 4 ihe | i a i pe ri ue mi iets } Gctrecrarg bebo ibeay ie ih at } iia! re pai i i hy at fh. 19; ate 4 a I a fo oe a ae ] i) int (i HN Nie i ia ibaa AO ‘ anh ine) oma tu Wy hy : sate Ny ynak i a a ae Vit iigiael Dt ti en AE id ae iy i eae PAR Wi sO ae HOU rh A at ik vi ise i a ae SHEEN sate sity nit ‘ant Oe ane EN aM Mi) OU es) a that HOR Peer MESURE Hi Henan ER ie a 7 an ea ea est i ae sue Ati hohe Fitna aah eT HO a HL | oF s Ri eae ae it i % i vio ue ite i ie oe ney a ee ue ratty ii Mat tin ith i: iil He ie tae f js ti aN iit Bry \ wath BS yaad MR ory Aaron ie ee a Pans ea q i } th i) noi My vk Wy uy hy i a a at na Ht V 14 eat ie it Ht wn i a Ls iii if ig ) i ini i q i Pac SL EAL bi FAO sitpo abl ncuconm danse viz ry Reyer tae i HA sha iene Pak ri ade ce | ua eet ; ifiie maint “4 iit ant 2 uae } tat btonsate abate ake 8 sted ih in ‘3 int batt rate iret a ' aby it i ide coe i is ei “ahi i} Mien an Ht iy i BUH wi wi TRH ‘ i PNW ONT Le ay i My f ain i tise ie a is) Watt a “Get te ee ae Bs da iy “i HF iy! ti aiin ag) ih iad veh eat Hie) ie Hf Tea eR iy bots} eth hb, ini Tea it ait an iia alsheia i a en See ala ae LL arate e ys be ait i Pe wi ae ri agit a’ a Aue He i fies i a iF fie PH a Alt : i eas oy ey J Dy nd Hn va i i i : a ae 2 . i zn ct Mh i 4 ai i 3 ris i i ih ds Hy a sf si ite a. iit a ‘t ie Hh vi Nia} | ; uh 4 Ait Wvid SON os) Ran : ns va ? Mitel ' H ie Ms ij srk ta AH i h Hehe aie ate ‘i rt apna } j diet) ay ENG ' i ity i Int ebay A ts shia a x taht ae Has i j y\ ite eee "i ‘ cnn rear ts babe tt, So aitate AAW dit ie vii isn ing iN Haat i A Aeon i in ai shlay i a : ia 4 ei a ' HL Ya it ia My ata trai ny ; Hi a Tbe oe oh me uch 3 ata Nei wi aHAN Tatty i ita ad) AW MRR TELE EYRHE Wii pias iso ii alata hai oH ¥ 4 i ii at Ai sath ith i ie Sea Ms My) he i iM cane 4 pai on wt ants oh RIEL i inns i ce ei M4 Al Haan oc ae hae ti : ish ee i638 Hy Psi - ih ee pa ae ie ‘ AMR i int a i ‘ . nuke a . i aes i t+ sf q i weit ti ck anit Be rt i ie Hens PE ae i lit ‘it a i ee hes i taht nba Git : a Ronnies Rut mht LA Rap Fea a bake it ura ih ih #) iN H ae fedsnel i Hejchites Pet at ae ln ie eh nh i us — the ne — sn a ipa wile . eta! ine My ss ae ahi eae ‘i ‘ H MateRead ely ie . aN Why 4, Mitek 0 wey THAT) 159 0 DG eae bed is rie ra et " ney ata heh nia MT tay Died aans at te , Ls i « “ iit cia iF BA Mets net i vi fe 3 Ha 1 ut . eb Mili? i Paigisihst * Math my ! seit ra oy ut i i . aa ae it inne heb hr a sats ‘? Br fettede y Nery Meet fee oA he ted Maa} Ns dae A itu Neieh he Vere de tek tee TAHA I Hibs Hs in itp lat i Ht nt VO d Dota i ol Be a anes wat Bi My my ie ae Has at a Hf iy a a. iH lye ANY i Fhe ot ‘ it, Hele bs oe Mi th Wns Rk aH tite siete it" bala ar Wii 4 nacht Hit i gigas rah igee ? rt te af ; i ih tit Hila a Hsiek ie a mates anal) | ee a te eae ot A At doko de eet UA UR eS ety Naty ded JOA ALS. Cite on We ne Pc} APN ey eA Ah Roh ite Reg Yee tae Dale ar A benthic 8 59: 9) ym 9 2 ee eo ait Aidt fi nary ; 4 wh 1 ae sini ke K rth) 4 Ben ra 3 ae ; i (ehete tad tbe tet lati: te ah 4 he ies it n na Vekey vitidetay ails A ae si ith oF i ie He ait | : ah oe ae i ive on at ae tity" LT Ba tau iit at , a) oy c an ty oy ne q i Deleas : soba Celt abe ASI, . 4 oh os tte r.4 matey . \" ihe nh ‘ it i) y nahn! ists v Na! Pues ft si uti ies it) H gta Vite ny Is) FWeiks 1s *{ Pe Heth f iy re. sa th ee at y ) beyetee wie biihek, ( ah 2A ae haha diWaone tet stots aw ohearas te 1 { aan ately me ‘ oN i carne Oca mye shee bei alle china. staat fh Set AYA ii Pua Nisgqesen hited t 4 b9. e a es " ran pe | Ait ait if aera “af ( aa! rt Wieite, is ; CrheDpePutiny SMH. si Lt 1: a Laas 1 ett NA a sant A nigite " anes \ Ni sant Meda acaati hs stay 4 tena 4 hy ahad es tt ORR tit oe r Wire tA Bae 77 Ai asad at 49, yeh fhe fa ae ateh a MATHEMATICS LIBRARY as %e Return this book on or before the Latest Date stamped below. University of Illinois Library JUN 4 1964 JUL 1 1% L161—H41 Digitized by the Internet Archive in 2022 with funding from University of Illinois Urbana-Champaign https://archive.org/details/treatiseonanalytOOhyme_0O ANALYTICAL GEOMETRY , THREE DIMENSIONS. B A TREATISE ON ANALYTICAL GEOMETRY THREE DIMENSIONS, CONTAINING THE THEORY OF CURVE SURFACKS, rpAND,OF. ,, CURVES OF DOUBLE CURVATURE. BY J. HYMERS, D.D. FELLOW AND TUTOR OF ST JOHN’S COLLEGE, CAMBRIDGE. THIRD EDITION, ALTERED AND REVISED. CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS, FOR J.& J.J. DEIGHTON, AND MACMILLAN & CO., CAMBRIDGE. WHITTAKER & CO., LONDON. M.DCCC.XLVIII. eats (Hild Ad (yaaa TA AI Ds ee a ~~ 2 i SORA Ma) hd % Vis yy a ee ie : “ . Mia red me Gi "1, ga 7 eign? Seo Ble Li PRACt fT te Oa ; 4 tia, o* , J om i } sacl ie oe j 3 | / 7 g } \ V } wr 7 'y ; 12 An i$ fs e ¢ t _Y CONTENTS. a i t ‘of é d SECTION LI. ON THE PLANE AND STRAIGHT LINE. ART, PAGE 1—16. Equations to a plane and straight line............... 1 -17—41.. Problems on the plane and straight line............... 18 SECTION ILI. ON SURFACES OF THE SECOND ORDER. 42—-52. Equations to a sphere, and to a cylinder, cone, and BH TIRCe COTE YUM ence ae rr iste ate dng soa e eaten pin tee cats 32 53—65. Equations and forms of surfaces that have a center.... 39 66—72. Equations and forms of surfaces that have not a EEC ee eee OCT AG. coe oiet Sth een EN ha Ot ay CRE ol 55 cA 50 SECTION III. ON THE PROJECTIONS OF LINES AND PLANE SURFACES, AND ON THE TRANSFORMATION OF CO-ORDINATES. 79—82. Length of the projection of a line, and area of the Rrorccholm Ole Ay DIANE sSUtt acts eee cats eo eit ce vb ps as SRR Beene) PL OU CUCL CO-OLUINALCS niwerer: svar copes Scr cle ss a's se 64 88—98. Transformation of co-ordinates. Polar co-ordinates... 68 99—106. Plane sections of surfaces............. S25 snk Some: a2..0070 vi CONTENTS. SECTION IV. ON TANGENT PLANES AND NORMALS TO CURVE SURFACES, AND THEIR VOLUMES AND AREAS. ART. PAGE 107—111. Equations to the tangent plane and normal......... 86 112—120. Differential coefficients of the volume and area of a curve surface in rectangular and polar co-ordinates ....... 90 SECTION V. ON TANGENT LINES AND NORMAL AND OSCULATING PLANES TO CURVES, AND THEIR LENGTHS. 121—130. Method of assigning the form of a curve in space by equations. Tangent line, and normal and osculating planes..... eR eer ea roe ee re a 96 131—132. Differential coefficient of the arc of a curve in rect- angular or polar co-ordimates...c24 ae. 2 eens oe 104 SECTION VI. ON THE DISCUSSION OF THE GENERAL EQUATION OF THE SECOND ORDER. 133—138. Center, and diametral planes of a surface......... 106 139—144. Reduction of the general equation of the second Oe Ee hac Fe ers Pre cat A RE Nya ony eee eee 112 145—148. Mode of determining the species and form of the surface represented by a proposed equation...... Ait AcioN 118 149—150. Case of surfaces of revolution.............c.cc00- 2 ie 151—155. Properties of conjugate diametral planes............ 1 156—157. Position and magnitude of paraboloids, when repre- sented by the general equation of the second order....... 133 CONTENTS. Vil SECTION VII. ON CYLINDRICAL, CONICAL, AND CONOIDAL SURFACES, AND ON SURFACES OF REVOLUTION. ART, PAGE 158—159. Number of constants in the equations to surfaces. Mode of representing the intersections of surfaces...... 137 160—172. Finite and differential equations of cylindrical, coni- cal, and conoidal surfaces ; and surfaces of revolution...... 141 SECTION VIII. ON SURFACES HAVING MORE THAN ONE ARBITRARY CURVILINEAR DIRECTRIX, AND ON DEVELOPABLE SURFACES, AND ENVELOPES. 173—177. Twisted surfaces generated by a straight line that constantly passes through two curves and remains parallel to a fixed plane, or which constantly passes through three BIVVCUACUL VES ty giaoate fect ne clei stain tare) Ge ae sib aries oe it'6 6 148 Ps =10l-sDevelopable Surtaces eras ta ess ake te ecko sine ass « 153 TOS eaol OG MORVElOPes at te hears -reres pety a icler ahr ssa «amie fee 167 SECTION IX. ON THE CURVATURES OF CURVES IN SPACE. 197—201. Intersection of consecutive normal planes of a curve. 172 202—206. Radius of spherical curvature ; radius of absolute CLE CULES oon ee oa Scone te ee Pe ot en iets eee Ti elchs a bhe alate te eNetete oles 174 907—210. Evolutes. of a curve in*space.....:.....2..se00s.0: 178 SECTION X. ON THE CURVATURE OF SURFACES. 211—214. Conditions for a contact of any order between two THEIACESUR. ties she < BPO be DOC OOOO Pare Pees ecatet a a's . 181 Vill CONTENTS. ART. PAGE 215—218. Radii of curvature of oblique and normal sections... 184 219—226. Normal Sections of greatest and least curvature... 189 227—229. Intersection of consecutive normals of a surface... 197 230—237. Lines of curvature, Principal radii of curvature, and RMpilict of surlaces 4s: ine acre ksl. daahs Mein 5 2 AG 200 PPPODIEINS cts ciate hk 8a he lin pa ce ete hie ales 3 ae ee 209 Students reading this subject for the first time may confine their attention to the following Articles. 1—41, 42—52, 57—71, 79—84, 107, 109—115, 121—126, 131. ERRATA. PAGE LINE ERROR. CORRECTION. 107% 3 YR LL 21 22 before this assertion supply the converse of 109 15 L,Y, z h, k,l 118 ] vy HS 121 9 +75+38 +255 -9 so that the surface is an ellipsoid. 133 12 2A’ x 2 A’, x 136 5 (> 1) ey kh) 7 a dr dr 19] last line Peps h tan? 6 i7;:? 205 24 +4uv +4p?g?uv which causes the two succeeding equations to be incorrect; the equation at the top of p. 206 should be 2p? g° (1+ Raa +92)—p292 lip)e-(149%=" a, (L+p?)(1 +9°)-p°¢" p”) ( +9° 1+ p? u f+ (l+p?)? 4u*p?9?=0, 231 li (rrr?) (rr'r’)2 251 1 : a gn ANALYTICAL GEOMETRY OF THREE DIMENSIONS. SE GDTON I. ON THE PLANE, AND STRAIGHT LINE. 1. Iw order to determine the position of a point in space, some fixed point is taken for the origin of co-ordinates, and through it are drawn three fixed planes, called the co-ordinate planes, at right angles to one another, and intersecting one another two and two in straight lines, which are also at right angles to one another, and are called the awes of the co- ordinates. Then, if the perpendicular distances of a point from each of the co-ordinate planes be given, its position will be completely determined. Let O (fig. 1) be the origin of the co-ordinates, and y Oz, Ov, wOy the co-ordinate planes; M any point, and ME, MF, MN the perpendiculars let fall from it upon the co- ordinate planes; these perpendiculars are called the rectan- gular co-ordinates of M, and, as their values change for the different points of space, are denoted by the variables a, y, z. The point WM will be determined in position, if we know the values of its three co-ordinates; that is, if we know that for that point Oe 0.8 SC, For, if along Ov we measure OA =a, and through A draw an indefinite plane parallel to yOz, this plane will contain all points whose distance from yOx is a, or for which w=a, 1 2 and, therefore, the point in question. Similarly, if we mea- sure along Oy, Ox, the distances OB=b, OC =c, and through B, C, draw indefinite planes respectively parallel to Ox, wOy, each of these must contain the point in question; therefore the three planes will, by their intersection in M, determine one single position for the point whose co-ordinates are w =a, y=b, x =c; which position coincides, as we see, with the angular point, opposite to the origin, of the rectangular parallelopiped constructed upon the edges OA, OB, OC, equal to the three given co-ordinates. 2. The feet of the perpendiculars L, F, N, are called the projections of the point M upon the co-ordinate planes. Of these if any two be given, the third can be found; thus if E, & be given, draw EB, FA, parallel to Oz; then BN, AN, drawn respectively parallel to Ox, Oy, will by their concourse determine JN. Instead of the perpendiculars ME, MF, MN, the three lines OA, AN, NM, which are respectively equal to them, are commonly called the co-ordinates of M, and denoted by @, Y, 3; also the axes of the co-ordinates Ox, Oy, Oz, are often called the axis of w, the axis of y, and the axis of x; and the co-ordinate planes vOy, yOx, xOw, the planes of wy, yx, xa, respectively. 3. The determination of the point M will not however be complete, unless we take into account the signs of the quantities a, 6, c, in the equations U =10e Yi Ds AAC in order to measure these distances, when they are positive, along the positive parts Ow, Oy, Ox, of the co-ordinate axes ; or along the negative parts Ow’, Oy’, Oz’, of the axes pro- duced in the contrary direction, when they are negative; as is explained in Plane Geometry. For since the co-ordinate planes, which must be supposed to be prolonged indefinitely, form among themselves eight solid angles, there are eight positions in which M might be situated at absolute distances 3 a, b, c from the co-ordinate planes; and it is only by attending to the signs that we shall be enabled to select the true one. Without going through the combination of signs which belongs to each of the solid angles, it may be sufficient to observe that, if a point be situated in the compartment wys, all its co-ordinates will be positive; if in the compartment a’y’x’, all negative; and that for points in the compartments vy'z, a’y2’, we must have, respectively, t= dy, Y=—b, S=C¢; #=—a, Y=b, sie, 4, Sometimes it is requisite to take the co-ordinate planes not at right angles, but inclined at given angles, to one another, in which case the system of co-ordinates is called oblique. Thus (fig. 2) if wOy, yOx, sOw be three planes drawn through the point O, and intersecting one another two and two at given angles, in the lines Ov, Oy, Ox; and if from any point M, MN be drawn parallel to Ox meeting the plane vw Oy in N, and NA be drawn in that plane parallel to Oy ; OA, AN, NM, are the co-ordinates of the point M referred to the oblique axes Ox, Oy, Ox; and WM is called the pro- jection of M made parallel to the axis Ox. In the present chapter we shall confine our attention to the case of rectangular co-ordinates; noticing however, as they arise, several relations of lines and planes to one another, the analytical expressions of which, and mode of investigation, are the same, whether the axes of the co-ordinates be rect- angular or oblique. 5. To find the distance of a point from the origin in terms of its co-ordinates. Let IW (fig. 1) be the point, OA=2, AN=y’, NM =» its given co-ordinates ; join OM, ON, and let OM = d. Then from the triangle ONM, right-angled at N, OM? = ON’? + NM’; ra 4 but the triangle OAN, right-angled at A, gives ON* = OA’ + AN’; “. OM? = OA? + AN’? + NM’; Cor. Let a, 8, y be the angles which OM makes with the axes of a, y, x; then, joining AM, since angle OAM is a right angle, OA x cosa = —— = —; OM da’ .. # =dcosa; similarly, y’ =dcosB, zx’ =d cosy. n r2 12 Also (cos a)’ + (cos (3)? + (cos vy)’ = a = 1. This is the condition which the three angles, made by a line through the origin with the co-ordinate axes, must satisfy ; they cannot, therefore, be assumed at pleasure, but two being given, a, §, for instance, the third is determined by the equation cosy = + 4/1 — (cosa)* — (cos 3)”, and will therefore have two values, y and t—-y; the value a — yy corresponding to a line OM’ in the plane OM, which is inclined at an angle y to Oz’, and visibly makes angles equal to a, B with Ow, Oy, the same as OM does. 6. To find the distance between two points in terms of their co-ordinates. Let M’ (fig. 3) be a point whose co-ordinates are x’, y’, 2’, and NM any other point whose co-ordinates are wv, y, #3; join . MM’ =d, and upon MM’, as diagonal, describe a rectangular parallelopiped having its edges parallel to the co-ordinate axes. Then from the triangle M’MK, right-angled at K, MM” = M'K? + KM? = M’K? + (2 - 2’). 5 But the triangle NW’LN, right-angled at L, gives M'K? = N'N? = (#- af + (y-y')’; MM? = (a#-2')? + (y-y) + (@- 2’), or d=/ (wv — a’)? + (y—y)? + (#—8') Both in this formula, and in that of Art. 5, we take the radical with a positive sign; as the question only relates to the absolute distance of the points. Cor. Let a, 8, y be the angles which MM’ makes with the edges of the parallelopiped, and which are manifestly the same as those made by a line OP, drawn through the origin parallel to MM’, with the co-ordinate axes. Then from the triangle MM’H, right-angled at H, we have MH «v-« Mis adi i cosa = similarly, cos 3 -2? (Sttrigh Z—-z 7. To find the equation to a plane. We may consider a plane as a surface generated by a straight line which moves so as always to remain parallel to itself, and to intersect a straight line given in position. Let AC (fig. 5) in the plane of xa be the line to which the generating line PQ is always parallel, and BC in the plane of yz, intersecting the former in C, the line along which it moves. Let s = 4v+ec be the equation to the line AC referred to the axes Ov, Oz, in the plane in which it lies; similarly, let s = By +c be the equation to BC referred to the axes Oy, Ox, where in both cases c= OC. From P any point in the generating line, and from Q where it meets BC, draw PN, QM, parallel to the axis of z, and join MN. Then the plane QN is parallel to zw, and therefore MN is parallel to Ow (Euc. xt, 15, 16); and consequently PQ is inclined to MN at the same angle that AC is toOwv; and since MN, NP, are the co-ordinates of P in the plane QN, PN=A4A.MN + QM. 6 But, Q being a point in the line BC, QM = B.MO+ OC, . PN=A.MN+B.MO+0C; or, if v, y, s denote the co-ordinates of P, s=Av+ By +e, the equation to the plane. Cor. The lines in which a plane intersects the co- ordinate planes are called its traces on those planes; thus AC, BC, are two of the traces of the plane BCA, the third being the line in which, if prolonged, it would intersect the plane of wy. Hence, we perceive the meaning of the constants in the equation to a plane, s=Av+Byt+e; for A, B, are the tangents of the angles at which the traces on #&,Yy%, are respectively inclined to the axes of w and y pro- duced in the positive directions; and ¢ is the portion of the axis of x produced in the positive direction, intercepted between the plane and the origin. It is important to observe, that the foregoing method of finding the equation to a plane, applies equally to the case where the co-ordinates are oblique, and leads to a result of the same form; so that whether a plane be referred to rect- angular or oblique axes, its equation may be represented by x= Awxv+ By+c; but in the latter case, A will signify the ratio of the sines of the angles at which the trace on xa is inclined to the axes of w# and x; and B the ratio of the sines of the angles at which the trace on yz is inclined to the axes of yand x. Hence, all results which involve no other as- sumption than s = 47+ By +e, for the form of the equation to a plane, will be equally true for oblique and rectangular co-ordinates. 8. To investigate the equation to a plane under the form ve 41 x weet AE as) a C 7 Let AC (fig. 4) in the plane of za be the straight line to which the generating straight line is always parallel, BC in the plane of yz that along which it moves; Ga point in the generating line DE in any position, ON=a, NH=y, HG =z, its co-ordinates. Draw DF, FE parallel to OC, OA, and let OA, OB, OC, which are called the intercepts of the co-ordinate axes be re- spectively denoted by a, b,c. Then since G is a point in DE, & wv p+ at erin DF'F (1) but ere Se aie ye Cc b a b 5 ae DF therefore, multiplying the first term of (1) by ——, the second c FE , and the second member by the equal quantity term by a 1 = and transposing, we get —$- 4 5=] hig tal ok ; a relation holding between the co-ordinates of any point in the generating line in any position, and consequently the equation to the plane generated, and expressed by the intercepts of the co-ordinate axes produced in the positive directions to meet the plane. Cor. It is easily seen that this result might have been deduced from Art. 7. For if we take s = dv + By +c to be the equation to the plane ABC (fig. 4), and in it make z = 0, y=0, «=a for the point 4, we get 0= Aa+c; and making s=0, w=0, y=b for the point B, we get 0= Bb+c; there- fore, substituting for A and B their values, the equation to the plane becomes, as above, a 8 9. To investigate the equation to a plane under the form wcosa+y cos + x cosy =p. Let OQ =p (fig. 4) be the perpendicular from the origin upon the plane 4BC, making angles a, (3, y with the axes of &, Y, ¥, respectively ; join AQ, then, OQA being a right angle, OA = OQ sec AOQ, or a = pseca; similarly, 6b = psec3, c =p secryy; consequently, by substi- tuting in the equation oy cD we get xcosa+ycos 3 + x cosy =p, the equation to a plane in terms of the perpendicular let fall upon it from the origin, and the angles which that perpen- dicular makes with the co-ordinate axes produced in the positive directions. It will be observed that neither in this article, nor the preceding, are the co-ordinates required to be rectangular ; only, in the case of rectangular axes we must have (Art. 5) cos’ a + cos’ 3 + cos’ y = 1; but in the case of oblique axes the cosines, which fix the position of the normal to the plane, will be subject to a different condition. Cor. If with the above equation we compare the general form of the equation to a plane Ax+ By+ Cz=D, im) b) p cosa cosp cosy which shew that the constant term bears the same ratio to the perpendicular on the plane, that the coefficient of each variable 9 does to the cosine of the angle which its axis makes with the perpendicular. Hence the perpendicular on a plane from the origin constant term s/ sum of squares of coefficients of variables likewise the square of its reciprocal equals the sum of the squares of the reciprocals of the three intercepts of the co- ordinate axes. 10. The particular cases of the equation to a plane, when it involves only one, or only two of the variables, require a separate notice. Thus the equation #2 =a, since it belongs to all points whose distance from the plane of yz, measured parallel to the axis of #, is a, represents an indefinite plane parallel to yz. Similarly, the equation w = 0, characterizes all points situated in the plane of yz, or is the equation to that plane; and y=0, x=0, are respectively the equations to the planes of SL, LY. Again, the equation — + 5 =1, a represents not only the line 4B (fig. 6) in the plane of ay, but all points situated in the plane RA drawn through 4B parallel to the axis of x ; for any one of these points P, what- ever be the value of PM or x, will have the same wv and y as its projection M, and therefore its co-ordinates will satisfy the equation which does not involve x. Similarly, the equations # = mz+a, y =nx +, represent planes parallel to the axes of y and 2; i.e. each equation represents a plane parallel to that axis whose co-ordinate it does not involve. This also appears from the equation z = da+ By+c; for if the plane be parallel to the axis of y, then its trace on 10 yz will be parallel to the axis of y, and therefore B =0, or x = Awv+cis its equation, the same as the equation to its trace on yz; similarly, if the plane be parallel to the axis of x, 4=0. In order to deduce from it the equation to a plane parallel to the axis of x, the equation requires to be reduced, as in Cor. Art. 8, to the symmetrical form now let ¢ be infinite, then the plane becomes parallel to the axis of x, and its equation 1s — + : =1, the same as that a to its trace AB. If the plane pass through the origin, c= 0, and its equation is 7 = 4v+ By. 11. Every equation of the first order involving three variables is the equation to a plane. Let the equation be reduced to the form = Awv+ By + ¢;3 and to find the trace of the surface which it represents on yx letv =0; .. x = By+ ce, which is the equation to a straight line ; let this be CB (fig. 5), Next to determine the sections made by planes parallel to va, let y=b,0’, &c. «1. x= Av+Bb+t+e, z= Axv+ BY +c, &c. which represent straight lines QP, OP’, &c. parallel to one another. They likewise all intersect the trace CB; for making w«=0, we have Bb+c, Bb’ +c, &e. for the values of QM, Q’M’', &c. and these are the ordinates of CB corresponding to y=b, y= 6, &c. Therefore the locus of the proposed equation is a system of parallel straight lines all passing through a fixed line; i.e. it is a plane, 12. The equation x= Ar+ By +e, where A, B denote numbers, and ¢ a line, has the defect of not being symmetrical; to compensate for which, it contains only three constants, each having an obvious signification so as to enable us readily to interpret any results to which its use may lead; and it furnishes the analytical expressions of the more useful relations of lines and planes to one another, under — 11 forms the simplest possible ; and it is readily adapted to planes in all particular positions, with the single exception of a plane parallel to the axis of x. But it may often be advantageously replaced by the symmetrical form expressed by the intercepts of the co-ordinate axes (which however cannot represent a plane passing through the origin) or by the equation depending on the magnitude and position of the perpendicular from the origin upon the plane, which may be written lv+my+ns=D, if we take 1, m, m to denote the cosines of the angles which the perpendicular p forms with the axes of a, y, #3; and which may represent a plane in every position. ‘The above three forms of the equation to a plane, as has been observed, hold equally for rectangular and oblique co-ordinates. 13. To find the equations to a straight line in space. A straight line is the intersection of two planes, and con- sequently is given when the equations to any two planes which contain it are given. Among the various planes which may be drawn through a given straight line, those are employed, for the sake of simplicity, to determine it, which are parallel to the co- ordinate axes, since their equations will involve only two of the variables. From all the points of the line MM” in space (fig. 6, bis) let perpendiculars MC, M’'C’, &c., be drawn to the plane of xy, meeting it in the points C, C’, &c.; the assemblage of these points is called the projection of MM” on this plane. This projection will manifestly be a straight line, since all the perpendiculars are situatad in the same plane, which is parallel to Ox, and is called the projecting plane of MM”. Similarly, if we project IM” upon the other co-ordinate planes by per- pendiculars upon them from its various points, we shall have 12 the three projections 44’, BB’, CC’, of which any two are sufficient to determine MM’. For suppose Ad’ and BB’ are given; if through these we draw planes respectively perpen- dicular to ga, yx, each of them must contain the line MM, and will therefore fix its position in space by their intersection. But the positions of the projections NA, QB in the planes za, yx, will be determined by their equations 7 =mz +a, y=nx-+b6, in which m and m denote the tangents of the angles at which they are respectively inclined to the axis of zg, and a=ON, b=OQ. Hence the system of equations G=IMS+a, y=nsv4+, which belong to the projections of the line, or rather to its projecting planes, will completely determine it. Cor. 1. Any values of x, y, x which satisfy the above equations are co-ordinates of some point in the line; for sup- pose wv’, y’, x’, were such values, and let M’ be the point in MP for which x = 2’; then for each of its projections 4’, B’, g =’, and therefore a’ = ON’, y’ = OQ’; that is, if x’ = C'M, then a and y’, provided they be determined from the above equations, will equal ON’, N’C’. Hence the equations =mz+a, y = nx +b, may be called the equations to the line MP itself. Cor. 2. Corresponding to the value M’C’ of x, the values of a and y in the simultaneous equations 7 = mz +a, y=nz+b, are ON’, NC’, the co-ordinates of the point C’ of the projection PC; and therefore the equation to the third projection may always be deduced from the two others by eliminating x, which gives w-a y-b = » or nv —- my = na—mb. m 14. In the same manner it might be shewn, that if the axes of the co-ordinates were oblique, the equations to a straight line would still be of the form L=ME+a, Y=nsv4+db; the only difference would be that the projections would be made by planes parallel to the axes of the co-ordinates; and 13 m would denote the ratio of the sines of the angles which the projection on x# makes with the axes of x and w respectively ; and similarly for . 15. The point in which a line meets any one of the co-ordinate planes, is called its trace upon the plane. The positions of these points are determined by putting, in the equations to the line, a, y, x, successively = 0; thus the co- ordinates of the traces. of the line whose equations are v-a y-b L=ME+a, y=nz4+b, = —_ m n 3 upon the planes of yz, zx, wy, are respectively a b a — = m n rie =b yaa 48| re A 16. Asin the case of a plane, it is sometimes attended with great convenience to have the equations to a straight line expressed in symmetrical forms. If a, 2, y be the co-ordinates of a given point in a straight line, and w, y, x those of another point in the line at a distance r from the former, and /, m, n, be the cosines of the angles at which the line is inclined to the axes of vw, y, =, we have (Cor. Art. 6) and as these relations subsist amongst the co-ordinates of every point in the straight line, they are its equations. This will still be true when the axes are oblique, if J, m, m are the ratios of the projections of a line on the co-ordinate axes, to the line itself. Problems on the plane and straight line. These principles being laid down, we proceed to the reso- lution of several Problems relative to the plane and straight 14 line, the results of which are of great use. As the equation to a plane contains three disposable constants, a plane may be drawn so as to fulfil various conditions; as, for instance, to pass through three given points; to pass through a given point and be parallel to a plane, or to each of two given straight lines; or to pass through a given point and be perpendicular to a given straight line. Similarly, a straight line, since its two equations contain four disposable constants, may be drawn so as to satisfy various conditions. 17. To express that a plane passes through a given point. Let zs = da + By +c, be the equation to a plane passing through a point av’y’s'; then the equation to the plane must be satisfied by the co-ordinates of the point, .. s’= Aa’+ By'+c¢; therefore, subtracting this equation from the former to elimi- nate c, we have w-2'= A(w-a') + BYy-y), in which A and B remain undetermined, so that the plane may still be made to satisfy two other conditions. Similarly, if a, 8, yy be the angles which the perpendicular on the plane forms with co-ordinate axes, and the plane pass through a point w’y’s’, its equation will be (w — aw’) cosa + (y—y') cos B + (x — 3’) cosy = 0. 18. To find the conditions that two planes, whose equations are given, may be parallel to one another. Let x= dv + Byt+c, x= A'x+ By +c’, be the equa- tions to two planes; then if they be parallel to one another, their common sections with any plane which cuts them are parallel; therefore their traces on the co-ordinate planes are parallel. But, making y = 0, the traces on the plane of sa# have for their equations s = dv+c, s = Aa +c, which will be parallel if 4 = 4’; similarly, since the traces on yz are parallel, B = B’. Hence the conditions of parallelism of two planes are, that the coefficients of w and y in their equations are respectively equal. 15 Cor. The equation to a plane drawn through a point a yx’, parallel to a given plane x = 4’7+ By +e, is s—-x =A (w-wv)+B(y-y). 19. To find the equation to a plane which shall pass through three given points. Let 2, 91215 %oY2%2y V3Y3%3, be the co-ordinates of the given points, and s = dv + By +c the equation to the plane ; then the co-ordinates of the points must satisfy it ; 2 =Av,+ By, +c, %= Av, + By. +, %3 = Axv;+ By; + ¢, which are the three equations for finding A, B,c. Multiply the second and third respectively by indeterminate coefficients t, w, and add them to the first ; °, 8 + t%. + U%3 = A (a, + FH, + UH5)...... Clie provided y, +¢y,+ uy,;=0, 1+t+u=0O. These two latter equations give YWi-Ystt(Y2—Ys)=% Yr—Y2t U(Ys — Y) = 0, and so determine ¢ and zw; and substituting these values in equation (1), we find A; and then B may be obtained by interchanging a,,7, and Y,Y2y3; so that #—#,=A(w-4,)+ Biy-y), the equation to the plane, will be completely determined. 20. ‘To determine the line of intersection of two planes whose equations are given. Let the equations to the planes be s=Av+By+e, x=Av+By+e. _ Then the line of intersection is, as we know, sufficiently de- _ termined by these two equations taken simultaneously; that is, by supposing w, y, x to receive only such values as satisfy 16 both at the same time. If, for instance, we substitute any series of values for x, we shall have for each of these values two equations to find the corresponding values of w and y; and thus we may determine as many points in the line as we please. But the line of intersection is more conveniently de- termined by its projections on the co-ordinate planes. To find these projections, let a’y's’ denote the co-ordinates of any point in the intersection, then 2’, x’, are also the co-ordinates of the corresponding point of the projection on za; . &=Aa't+ Byte, # = Aa t+ By +c; therefore, eliminating y’, (B’ — B)s' = (AB — A'B)a' + Be - Be; which is the relation between the co-ordinates of any point in the projection on zw. Hence, suppressing the accents, (B' - B)z = (AB — AB) & + Be — Be’, and, similarly, (A’ — A)x = (BA'—- B’A)y + A’c - AC,~z are the equations to the line of intersection of the planes; which, as we see, result from separately eliminating y and x from the equations to the planes. The projection on wy has evidently for its equation (A’— A)v + (B’- B)y+c-—c=0. 21. To find the equations to a straight line which shall pass through two given points. VE OF OP Let a’y 2’, v’y”z”, be the co-ordinates of the given points ; L=MS+a, y=nsx +b, the equations to the line, which must be satisfied by the co-ordinates of the points ; * v0 =ms +a, yons + D....., (1), 4} ” ” 4? C=HamMs+a, y=ns +d, the four equations to determine the four constants. First, for the values of m and n, we have uv’ med a” =m (x' rl 2"), y wr y” = n (2 2") ; 17 also subtracting equations (1) from the assumed equations in order to eliminate a and J, v-v =m(x-2), y-y =n(e-%); therefore, substituting for m and n, the required equations are av —- a” y y” , ar ’ t “hal , Gig Bilan Ae = *); Sas Unc a ten Pig S ds Cor. 1. It is easily seen that if we take abe, a’b’c’, for the given points, and call their distance d, and the distance of abe from wyz, r, the equations to the straight line may be written @-a y-b B8-C¢ Qls eed ah Oe ah Oo Cor. 2. If the line is required to pass through only one point a’y’s’, its equations will be e-al=m(z-2), y-y=n(s-2), where m and m remain undetermined, as might have been foreseen ; for this condition may be satisfied by an unlimited number of lines. If, besides, it is to be parallel to a known line of which the equations are c=mzta, yans4+0, since the projecting planes (uc. x1. 15) and therefore the pro- jections of these lines must be parallel, we have m = m', n =n’; therefore the required equations are e-xv=m(e-2%) y-ya=an (x —-2). 22. To determine the point of intersection of two given lines. Let their equations be L=Ms+a c=omz+a eae eapeaety For any given value of z, the values of w and y furnished by these two systems of equations will in general be different ; 2 18 but if s be the ordinate of their point of intersection, the cor- responding values of w and y furnished by them, will be the same. Therefore the co-ordinates of the point of intersection are the values of wv, y, %, which simultaneously satisfy the above four equations ; and as the number of equations exceeds the number of quantities to be determined, there will be an equation of condition, without which the problem would be impossible; since two straight lines do not, in general, intersect. Hence, eliminating # and y separately, O=(m—-m)zx+a-a, 0=(n-n')x+b—-8; and equating these two values of z, a-a 6b-D' it, m—m rm—-nN 3 an equation which must be made identically true by the con- stants entering into the equations to the two lines, in order that they may intersect. The two remaining co-ordinates of the point of intersection, will be found by substituting the value of a-a 6-0 3 = = = eit ? Le 9 m—-m wu— 7 in either of the above systems of equations. Cor. When m=m’, and n=7’, the equation of con- dition is satisfied, and yet the lines do not intersect, being parallel; therefore the above condition will not ensure the intersection of the lines, without the restriction that the value of zs is not to be infinite. Taken without any restriction, it expresses that the lines are in the same plane and may ~ therefore intersect; without deciding at what distance that — intersection will happen. 23. To find the conditions in order that a straight line and plane whose equations are given, may coincide, or be parallel to one another. Let s = da + By +, be the equation to the plane, V=MS+a, y=nz+), a 19 the equations to a line coinciding with it. Since every point of the line is in the plane, if x be the ordinate of any point, and ... mz +a, nz +, the values of the other co-ordinates, these values must satisfy the equation to the plane; . O=(4m+ Bn-1)24+Aat+Bb+e; and since this equation must be true for all values of z, “~. Am+Bn-1=0, Aa+Bb+c=0, which are the required conditions. If the plane and line are parallel, then a plane and line drawn through the origin respectively parallel to them, coincide; therefore the above equations must be satisfied when we suppose ¢, b, a, to vanish; hence the remaining equation Am+ Bn-1=0, expresses the condition in order that a line and plane may be parallel to one another. 24, If we employ the symmetrical forms of the equations to the straight line and plane, then in order that the straight line may be parallel to the plane da + By + Cz=D, we must have lA+mB+nC=0; and if it coincide with it, the further condition is Aa+Bb+Cc=D. It will be observed that the results of the preceding Articles from (17) inclusive, are also true, when the axes are oblique ; since the only assumptions made, are the forms of the equations to the plane and straight line; but in that case, the constants will have different significations. 25. ‘To find the conditions that a straight line and plane may be at right angles to one another. 2—2 20 Let the equations to the straight line and plane be S Gnt hg and te 4 wor Bane y=ns + b, then the equations to the straight line and plane respectively parallel to them through the origin are (1). s=Auv+By (2). eC=MsZ y= ns Now let OA (fig. 8) be the straight line (1), then since it is perpendicular to the plane (2), it is perpendicular to every straight line meeting it in that plane, and consequently is perpendicular to the trace OB of (2) on the plane of #z whose equations are z= Ax, y=0 (38). In straight lines (1) and (3), take two points, 4, B whose co-ordinates are respectively a’, y’, 3’; #”, 0, 2°; and join those points; then by equating the two values of the hypo- thenuse of the right-angled triangle 4OB so formed, we get AB’ = AO’ + OB’, or (a — a") + yy? + (eX — 2" Poa ey? + ev? te? ee”; U0 +22 =0, or since a’ = mz’, 2" = Av”, m+A=0. Similarly, for the trace on yz, we get 2+ B=0; hence the required conditions are m+A=0 n+B=0. Cor. 1. Ifit were required to find the conditions for the line and plane represented by the equations DONT oy oT hesom CNIS Say l a ee 5) Auv+ By+Cz=D being perpendicular to one another, since the line is parallel to the normal to the plane, those lines must form equal angles — with the co-ordinate axes; but the cosines of the angles which the line forms with the axes of x, y, x, are proportional to 5 l,m, 2; and the cosines of the angles which the normal forms 21 with the same axes are proportional to 4, B, C (Arts. 9 and 16) ; Lia ed Cor. 2. Hence we can find the equations to a straight line which shall pass through a given point, and be perpen- dicular to a given plane. Let a’, y’, x’, be the co-ordinates of the given point, x = Aw + By +c the equation to the plane; then » — a =m(z —), y-y =n(z% —2’) are the forms of the required equations (Cor. 2. Art. 21); and since the line is perpendicular to the plane, m=-— A, n=-—B, - 2—-v' + A(z—2) =0, y-y + B(z -2%) =0, are the required equations. 26. Since the trace of the given plane on vw has for its equation xs = Aw + ¢, and the equation to the projection of the . e . av a given line on sw may be written x =— — —, the result mm 1 A+m=0, or Ax —+12=0, shews that the trace and the m projection on sa, and consequently on any plane whatever, are perpendicular to one another. The geometrical interpretation of the result of Art. 25 consequently is, that when a straight line is perpendicular to a plane, the projection of the line and the trace of the plane on each of two co-ordinate planes, are perpendicular to one another. ‘The only exception which this assertion is liable to, is when the plane is parallel to one of the co-ordinate axes, as the axis of x; for in that case the two projecting planes become coincident, and a line CQ (fig. 6) may have two of its projections CS, CR perpendicular to AS, BR the parallel traces on x@ and yz, without being perpen- dicular to the plane AR. b Suppose that y = —— w + b was the equation to the plane; : a then every line perpendicular to it would be in a plane parallel 22 to wy, and therefore have « =c for one of its equations, a ° : Ale and y = ,et b’ for the other, since its projection on vy must be perpendicular to the trace on that plane. 27. If through CO (fig. 7) the axis of z, a plane COQ be drawn perpendicular to AB, the trace of the plane ABC on wy, and OP be drawn perpendicular to CQ, it is evident that OP is perpendicular to plane ACB and z CQO is the inclination of the plane ABC to the plane of wy, and is equal to the 2 COP, which OP makes with the axis of x, each being © equal to the complement of 2ZPOQ; in other words, the angle between two planes is equal to the angle between two — lines respectively perpendicular to them. If the perpendiculars should not meet, then by the angle between them, and gene- rally by the angle between any two lines that do not intersect, we understand the angle formed by two lines drawn through any the same point respectively parallel to them. 28. Having given the equations to a straight line, to find the angles which a line parallel to it through the origin, makes with the axes. Let v= mz+a, y=nz +b, be the equations to the given line. Through the origin draw a line OA (fig. 8) parallel to — it; then the equations to OA are V=EMZ YHNS, Let a, 3, ry, denote the angles which OA makes with the © axes of x, y, #3 also let OA = 1, and let 2’, y’, ', be the co- ordinates of the point A, then (Art. 5) a =cosa, y’ = cos P, — % = cosy; also since the point is in OA, Ud , , ’ Cz MB 5 Y = 72S. But OA? = v7? +y? +2", or 1 = (m? +n? +1) 2”, ; 1 1 aa ES i ee COS SS Ji +m +n y /1 +m +n? n m cos 3 = ——————., COs | /1+m + n° Y/1l+m +n* 23 The radical in these expressions admits of a double sien + or —; but as it must be taken + in all, or — in all, we have only two systems of values for a, 6, yy; viz., the angles which OA makes with Ow, Oy, Ox; or their supplements, which AO produced or OA’ makes with the same lines. It is usual to take the positive sign; then cosy is +, and therefore -y acute; and the angles determined are those which the part of the line above the plane of wy makes with the co- ordinate axes produced in the positive directions. Cor. Conversely these equations give the values of the coefficients m and nN, cos a cos (3 m = | 7 ae e cos y cos y Hence the equations to a line passing through a given point (v’, y’, 2’) and making angles a, 3, y with the axes, are, as we have already found at Art. 16, Uy , U 0-@ Y-Y 8-8 cosa cos B cos 29. To find the angles of inclination of a plane whose equation is given, to the co-ordinate planes. Let s = Av + By +c be the equation to the plane ; then v=-— Av, y= — Ba, are the equations to a line through the origin perpendicular to it; and if a, 3, y denote the angles which this line makes with the axes of a, y, z, then (Art. 27) a, 8, y are also the angles at which the plane is inclined to the co-ordinate planes of yx, xv, wy respectively ; and by the preceding article we have -—A -B cos @ = cos: BP = —— /1+4+ A? + B? /1+ 47+ B 24 30. Having given the equations to two straight lines, to find the angle which two lines parallel to them drawn through the origin include. Let , y=ans+ a. v=ms+a’ ‘ Y ? lines to which OA, OB, (fig. 8) are respectively parallel ; then L=ms, y=nsz are the equations to OA, and r=m sz, y=n's, those to OB. Take OA, OB, each=1; and let the co-ordinates of A and B be denoted respectively by L,Y, %3 @,y,2%3 join AB, and lett AB=d, 2 AOB=8. Then from the triangle AOB, ~ be the equations to the ns +b 124+1?-2cos0=ad’ =(w — a’)? 4+ (y—-y')? + (x — 2’) or, expanding, and reducing by the relations | e+y+e2?a1, w+ y? 4+ 2% = 1, cosO0 = wa’ +yy' + 22". But since 4 and B are situated respectively in OA, OB, L=Ms, y=ne; 2 =m, yY =n's; 1 J 1 +m +n? . Ll=(m'? +n’? +1)2", or v= 1 Sia larl ve ver es ap pe ae /1+m? +n?’ , , . cos@ = (mm'4 nn’ +1) 23 = nT DE OES : Si tm +n /1 +m? +n’ This expression, on account of the two radicals it con- tains, will furnish for 6, four values equal two and two, corresponding to the four angles formed by the indefinite lines Ad’, BB’. It is usual to take the denominator positive, — i. e. to take the radicals with the same sign, so that the points 4, B, are both above or both below the plane of wy; and the angle determined is AOB or its opposite 4’OB, contained between the two portions of the lines which form 25 each an acute angle with Ox or Ox’; this angle, moreover, will be acute or obtuse according as 1 + mm’ + nm’ is positive or negative. Cor. By means of the formula sin = rv gee (cos)? we may shew that /(m — m')? + (n— n+ (mn' — m'n)? sin 9 = fa Si tm +n? /1 +m? +n? 31. The angle between two lines may also be ex- pressed in terms of the angles which each makes with the co-ordinate axes. Let OA make with the axes of w, y, x the angles a, 8, y; and OB make with them the angles a’, 8’, y’; then, proceeding as in the last Art., we may shew that cos 0 = wa’ + yy + 22". But vx=cosa, y=cos, % = cosy; , , ? ‘ , : , @ =cosa, y =cosf, % =cosy ; *. cos@ = cosacosu + cos 3 cos 3’ + cos y cosy’. 32. Hence the condition in order that two lines may be perpendicular to one another is cos @ = 0, or mm +nn +1=0; at which we may arrive immediately by observing that the equation to a plane through the origin perpendicular to the first line is 7 + maw +ny=0, which must necessarily contain a line through the same point parallel to the second whose equations are w= m'z, y=; “St mms+nn'z=0, or mm+nn'+1=0. This condition expressed in another form, is cos a cos a’ + cos $ cos B’ + cosy cosy’ = 0. 26 Also, in order that the lines may be parallel to one another, we must have sinO=0, or (m—m’)? + (n—17')?+ (mn' — m'n)’= 0, which can only be satisfied by m=m’, n=n’'; this agrees with Cor. 2, Art. 21. ee ———— 33. To find the angle of inclination of two planes whose ~ equations are given. Let s=Av+ Byt+e, s= Ax + By +c, be the equa- tions to the two planes. Then the equations to two lines respectively perpendicular to them through the origin are (Art. 25) e=-Ax, y= —-Bz; w= -— A's, y= —- B82; and (Art. 27) the angle of inclination, 0, of the two planes is equal to the angle between these two perpendiculars ; therefore (Art. 30) 9 1+ 44+ BB COs = ee ee ee af/1+ + B/1 +A 4 BE the double sign belonging to the acute and obtuse angles between the two indefinite planes. 34. The angle of inclination of two planes may also be expressed in terms of the angles at which each is inclined to the co-ordinate planes. For if a, B,y3; a, By’; denote the angles at which the planes are inclined to the co-ordinate planes of ys, za, xy, these are also the angles which the perpendiculars on them from the origin respectively make with the axes of #2, y, #3; therefore (Art. 31) cos 9 = cosa cos a’ + cos 2 cos (3 + cosy cos ¥’, 35. Hence the condition that two planes may be per- pendicular to one another (since in that case cos @ = 0) may be expressed in the two following ways: 27 1 a AALS BB’ = 0, cos a cosa’ + cos 3 cos 3’ + cosy cosy’ = 0. We may arrive at the former of these results immediately by observing that v= — dz, y = — Bx, are the equations to a perpendicular through the origin to the first plane which _ must coincide with a plane through the origin parallel to the second plane whose equation is s = 4’x + By; * AA t BBA Ie 0: If the second plane be parallel to the axis of x so that its equation is y=atan@+ 6, the condition becomes B= Atan ¢. 36. To find the angle between a straight line and a plane, of which the equations are given. The angle, 0, here meant, is the angle between the line and its projection upon the plane, and is therefore equal to the complement of the angle made by the line with a perpendicular to the plane. Let s = dv + By +c be the equation to the plane; then the equations to a line through the origin perpendicular to it are e=—-Azx, y= — Bz. Let v=ms+a, y=nx+b, be the equations to the : : uu : : given line; then since aoa is the angle between it and ~ the perpendicular to the plane, mute ae 1—-Am— Bn cos (= _ ) or sing = /1 tne + 8 Viepeee Be ° 37. To find the length and the co-ordinates of the extremity of the perpendicular from a point upon a plane. Let s=Axv+By+ec be the equation to the plane, x, y’, x the co-ordinates of the point from which a perpen- 28 dicular is to be dropped upon it; then the equations to the line passing through the point and perpendicular to the plane | are (Art. 25, Cor. 2) e-av =—A(xsx-2), y-y=-B(s-2); and combining these with the equation to the plane, which may be written s-s'=A(ea—-v)+B(y-y) -(' -Aa' - By -0), in order to deduce the co-ordinates XY, Y, Z of the point where the perpendicular meets the plane, and putting x — Aa’ — By’ -—c=P, we get Z—-2# =-A(Z-3') -B (2-2) -P, 075 — is a AP ; BP Xe pete Ee igs == ——________ : = ieee ee J 1 + A424 Be” values from which it is easy to deduce X, Y, Z, the co- — ordinates of the foot of the perpendicular. If we denote by D the length of the perpendicular inter- — cepted between the point and the given plane, we have P* 1+ A? + B’ 2’ — Aw’ — By -c £4/14 4° +B” the radical being taken with that sign which makes the whole expression positive. D?=(Z- x)? +(¥ -y)’+ (X-2)’ = . D= Cor. If we use the symmetrical form of the equation to a plane Av+ By+Czs=D, D / A? + B+ C?’ and a plane parallel to it through (abc) has for equation Axvx+ By+ Cz =Aa+ Bb+ Ce, the perpendicular upon it from the origin = 29 the perpendicular upon which from the origin equals Aa+ Bb+Cc Aa+Bb+Cc~D which is the perpendicular from (abe) upon the proposed plane. : the difference of these perpendiculars = ? 38. To find the length and the co-ordinates of the extremity of the perpendicular from a point upon a line. Let the equations to the given straight line be V=ME+a, y=nsth, and a’, y’, x’ the co-ordinates of the given point from which a perpendicular is to be dropped upon it; then the equation to a plane passing through (a’, y’, ’) and perpendicular to the given straight line is (Art. 25) #—s'=—m(x-a)-n(y-y); and combining this with the equations to the line in order to get the co-ordinates X, Y, Z, of their point of intersection, we have, putting x +m(a —a)+n(y -b) =P, Zax’ —m(mZ+a)+ma—n(nZ +b) + ny’; x +m (uv —a)+n(y —b)_ » 2 1 +m? +7? H Reelily sno. i'd.) 1+m’ +n? m P aC nr cae oO) m n FNC aii py Die fists Ss el Va Ga —Y =NnN4+ Pe hie rapes reece) Consequently, if D denote the length of the perpendicular intercepted between the point and the given straight line, D = (Z - 2%)? +(X -wv)’?+(V-y’)’, 30 ie CA ae lt m+n 1+m+n S24 m (a —a)+n(y'— b)}? Hy 1+m+n +? 4 (a -— a)? + (y' - 5)’, ae re) ae) 39. The perpendicular distance of a point from a straight line may also be expressed in terms of the angles which the line makes with the co-ordinate axes. Let AB (fig. 10) be the given line passing through a point 4 whose co-ordinates are a, b, c, and making angles a, 3, y with the axes of a, y, and x; also let AP make with the same axes, angles a’, 9’, y’, P being the point with co- ordinates a’, y’, x’, from which the perpendicular PB is to be drawn. ‘Then cos PAB = cosa cos a’ + cos 3 cos 3’ + cosy cosy’ _ (#—a)cosa+ (y'—b) cosB+-(z" W AP . BP? = AP* — (AP cos PAB)? = (wv — a)? + (y'— b)’ + (x — e)? — {(w -— a) cosa + (y' = 6) cos B + (2’— c) cosy}* Or, if we denote the cosines as usual by /, m, n, the distance of the point w’ y’ x’ from the line neyeoay SC Coe ©-a y-b w-e — —_ —_—_—____ is »/[(#—a)’+ (y’—b)? + (x'—c)?— {1(w'—a) + m(y'—b) +n(x’—c) }?]. If the line passes through the origin, this becomes J w+ y? +8? — (la’ + my’ + nz’). 40. As two straight lines in space, although not parallel, may never intersect, it is a Problem which sometimes arises to determine the shortest distance between them; it may be solved by means of the following proposition. The shortest line which can join any two points in two straight lines in space, is the line which meets both of them at right angles. 31 Take one of the lines OC (fig. 11) for the axis of #, and let the other 4B meet the planes of sw, vy, in A and B, in which planes draw AO, Oy perpendicular to OC, and let these be the axes of = and y. Draw By parallel to Ow, join Ay and draw OM perpendicular to it; draw MN parallel to By, make Ow equal to MN and join Nw. Then MN being parallel to Ow is perpendicular to MO; therefore OM is perpendicular to each of the lines WN, Ay, and therefore to the plane 4yB; also MN being equal and parallel to Oa, the figure VO is a rectangle, and Nw is equal and parallel to MO; therefore Nw is perpendicular to Ow, and also to the plane AyB and therefore to 4B; hence Naw meets each of the given lines at right angles. It is also their shortest distance; for take any point K in AB, draw KL parallel to By and join OL; then the distance of K from O# is equal to OL, because a perpendicular from K on Ow would be parallel and equal to OL, zLOw being a right angle, and OL is manifestly greater than OM. 41, Any plane passing through the shortest distance will be perpendicular to the plane to which both the given lines are parallel; for since Na is perpendicular to the plane Ay B, any plane through Nw is perpendicular to dyB; and AyB, since it contains one and is parallel to the other, is itself parallel to every plane to which both the lines are parallel. Also if through OC we conceive a plane to be drawn parallel to AyB, it will be parallel to 4B; and MO = Ne will be its perpendicular distance from the plane 4yB; i. e. the shortest distance between two lines not in the same plane is the per- pendicular distance of two planes, each of which is drawn through one of the lines parallel to the other. The complete determination of the shortest distance be- tween two lines, by finding the position and magnitude of the shortest distance when the equations to the lines are given, is effected in Probs. 4 and 5. (Appendix, Section I.) SECTION II. ON SURFACES OF THE SECOND ORDER. 42. Tue foregoing section comprehends all the more useful results relative to lines and planes referred to rect- angular co-ordinates. We shall now proceed to investigate the equations, and forms, of surfaces of the second order, which, after the plane, are next in simplicity; reserving for the subject of the following section (after the reader has become familiar with the mode of representing surfaces by equations, and thence deducing their forms) one of the prin- cipal means of modifying and simplifying those equations, viz. transformation of co-ordinates. We shall next have oc- casion to introduce the projections of lines and plane sur- faces; and the more useful results, relative to the line and plane referred to oblique co-ordinates, may be then more conveniently given than at present. It will be necessary however to make some preliminary observations on the geo- metrical meaning of equations containing only one or two of the variables wv, y, x, when we embrace in our enquiries the three dimensions of space. 43. It was remarked (Art. 10) that the equation =a, when three dimensions of space are regarded, repre- sents an indefinite plane parallel to yz; similarly, the equa- tion y? + py+q=0, which gives for y two constant values y=a+b, represents two planes parallel to sw; and in general, every equation containing only one of the variables, represents a system of planes parallel to the two axes whose co-ordinates it does not involve. Again, the equation f(#, y)=0, which, in Plane Geo- metry, belongs to a curve’ CC" (fig. 12) in the plane of vy, in Geometry of Three Dimensions, represents a cylindrical surface formed by drawing lines through all the points of CC’ ae ae ee 33 parallel to Ox; for any point P in this surface, whatever be the value of PC or x, will have the same 2 and y as its projection C; and therefore the co-ordinates of every point in the cylindrical surface will satisfy the relation f(a, y) = 0, which does not contain x; and any point whose projection is not in CC’ cannot by its co-ordinates satisfy the relation Sf (@, y) = 0. The same observations apply to equations of the form f(a, x) =0, f(y, x) =0. Hence, an equation con- taining only two of the variables represents a cylindrical surface parallel to that axis whose co-ordinate it does not involve; and of which the trace upon the plane containing the other two axes, is in Plane Geometry given by the same equation. 44. The locus of an equation f(a, y, z) = 0, containing all the variables, is a surface; that is, if all the points be taken whose co-ordinates satisfy it, they will not comprehend all the points of a solid figure, but only those situated in a surface. In the equation f(x, y, s) = 0 we may assume any values for two of the co-ordinates; and, deducing the value of the third from it, we have the co-ordinates of the cor- responding point in the locus; and similarly, we may de- termine as many points in the locus as we please. Instead of this, however, let us assume the value of only one co- ordinate w=a; then (Art. 43) f(y, x, a) =0, represents a cylindrical surface parallel to the axis of w; but since we must only take the points in this surface which satisfy the condition vw =a, it follows that by this assumption we obtain a curve, namely, the section of the cylinder made by the plane =a parallel to yx. Let this be PQ (fig. 13), where ON =a; then of all the points in the indefinite plane PNQ it is only those in the curve PQ which are deter- mined by the equation f(y, x, a) =0; similarly, if we take «= ON'=a, we shall have f(y, x, a’) = 0, which equations determine the curve P’Q’; and, proceeding in this manner, we may obtain an infinite number of curves all situated in planes parallel to yx, and succeeding one another at intervals as small as we please, the assemblage of which will form a surface which is the locus of the equation f (7, y, x) = 0. 3 34 45. We conclude, therefore, that every equation, whether containing one, two, or three of the variables, represents a. surface; if however the equation can be satisfied by no system of real values of the co-ordinates, the surface will be altogether imaginary; or, if it can be satisfied only by breaking it up into two or three others, the surface is reduced to a limited number of lines or points. In other cases the equation will enable us to determine the figure and properties of the surface which is its locus, as will be seen. In discovering the figure from the equation, the traces on the co-ordinate planes, determined by putting #, y, # separately = 0 in the equation to the surface, and the sections made by planes parallel to the co-ordinate planes, determined by putting the co-ordinates each separately equal to a constant, are the principal means. Thus (fig. 13) 4B, BC, CA are the three traces of the surface; and PQ is a section made parallel to yx, the equation to which is f(y, x, a) = 0, where ON = a, and QN, NP are to be regarded as the axes to which the curve PQ is referred. As in Geometry of ‘wo Dimensions a curve may have several branches, and an ordinate may have several values corresponding to the same value of the abscissa; so a surface may have several sheets, and points in different sheets may have the same projection on the co-ordinate planes. As many real values as x has for given values of w and y, so many sheets will the surface have; if the value of x is imaginary, there 1s no point in the surface corresponding to those values of wv and y. 46. Surfaces, in the same manner as lines, are divided into orders, according to the degree of their equations; the degree being determined by the sum of the indices of the three variables in that term of the equation (supposed to contain no fractional or irrational term) where it is greatest. The plane is the surface of the first order, being the locus of the equation of the first degree between three variables ; the sphere, the common cone and cylinder, the ellipsoid, &c., are surfaces of the second order, because their equations are 35 of the second degree. We shall at present obtain the equation to each variety of the surfaces of the second order from some known properties of them, and determine their figures; re- serving the discussion of the general equation of the second order, and the closer consideration of the properties of the surfaces which it represents, to a more advanced part of the work. ‘This arrangement, which corresponds to that usually followed with regard to curves of the second order, will gradually introduce the student to the more difficult parts of the subject, and put him at once in possession of the more important results. 47. 'To find the equation to the surface of a sphere. The characteristic property of this surface is, that every point in it is at the same distance from the center. Let #, y; x be the co-ordinates of any point in the sur- face, a, 6, c the co-ordinates of its center, and 7 its radius; then (vw -— a)? + (y— by +(e - cP? =7”", is the relation which the co-ordinates of every point must satisfy, and therefore the equation to the surface. Cor. This equation will assume different forms accord- ing to the position of the origin; thus if the origin be in the center, a=b=c=0, and fie equation becomes Pty tear; if the origin be in the surface of the sphere, a+bP+e=r", and the equation becomes v+y +s? =2 (av + by +c). 48. To find the equation to the surface of an oblique cylinder, the base being circular. This surface is generated by an indefinite line which is carried round the perimeter of a given circle, always remain- ing parallel to a given straight line, 3—2 36 Let r=mz, y=ne, be the equations to a line OR (fig. 14) through the origin, to which the generating line is always parallel; and r the radius of the circle OPA described in the plane of wy with its diameter coinciding with the axis of x, along which the generating line moves. Let PQ be the generating line in any position, and wv, y, the co-ordinates of any point M in it, and therefore in the cylindrical surface. Then the equations to PQ are X-wv=m(Z—2), Y-y=n(Z—-s), (Cor. 2, Art. 21); therefore, making Z = 0, we have the co-ordinates of Pjror X =ON=2-mz, Y=NP=y- nz. But, P being a point in the circular base, Y?=2r X'— X”"; therefore, substituting for Y’ and X” their values, (y — nx)? =2r(# — mz) -—(# -— mz)’, a relation among the co-ordinates of any point in the surface, and therefore its equation. Cor. We may suppose the curve OP, instead of a circle, to be any curve of the second order determined by the equa- tion Y°= 1X’ — (1 — e?) X”; we shall then obtain the equation to any cylindrical surface of the second order, viz. (y —nz)?=1(e — mz) —- (1 -— e&) (@—- mz)’. And it is easily seen that whatever the curve OP be, if its equation be Y’ = f(.X’), the equation to the cylindrical surface generated by a straight line carried along it, and of which it is called the directrix, will be (y — nz) = f(#@ — mz). 49. To find the equation to the surface of an oblique cone, the base being circular. This surface is generated by an indefinite line which is carried round the perimeter of a given circle, always passing through a fixed point. 37 Let a, b,c be the co-ordinates OR, RQ, QV, (fig. 15) of the vertex V, and r the radius of the circle OPA described in the plane of wy with its diameter coinciding with the axis of xv, along which the generating line moves. Let PV be the generating line in any position, and a, y, x the co-ordinates of any point M in it, and therefore in the conical surface ; then the equations to VP are (Art. 21) =f; 2.33) We ea rm ay 7A Sc) z—C€ xv X-a= x therefore making Z = 0, we have the co-ordinates of P, or z Sf PON ae ere VON oe ee wot C bs a But, P being a point in the circular base, Y= or XX"; therefore, substituting and reducing, (bz —cy)’ = 2r (x — c) (ax — cx) — (ax —c2)?, the equation to the conical surface. Cor. We may suppose the curve OP, instead of a circle, to be any curve of the second order, determined by the equation Y? = 1_X’— (1 — e) X”; we shall then obtain the equation to any conical surface of the second order, viz. (bz —cy)’=1(% —c) (az -— ca) — (1 - &) (az - € 2)’. And it is manifest that whatever the curve OP be, if its equation be Y’=f(X’), the equation to the conical surface of which it is the directrix will be bz—cy aS— Cx ra alereces 50. To find the equation to the surface of a spheroid. This surface is generated by the revolution of an ellipse about one of its axes. 38 Let CA (fig. 16) be a quadrant of an ellipse which, by revolving about its axis OC coinciding with the axis of x, generates a spheroid. Let A’C be any position of the gene- rating curve, and ON =7, NM =y, MP’==3z the co-ordinates of any point P’ in it; AO=a, CO=c; then from the right- angled triangle ONM, OM = a aa y’. But, since P’ is a point in an ellipse with 3 axes a and ec, OM? 3 weg e om opt. o the equation to the surface; which is called an oblate or prolate spheroid, according as the axis 2c, about which the ellipse revolves, is the less or greater of its axes. Cor. Similarly, if the hyperbola 4Q (fig. 18) revolve about its conjugate axis OC coinciding with the axis of gz, the equation to the surface of the hyperboloid generated, consisting of one continuous sheet, will be 2 2 very Pe: and if it revolve about its tranverse axis OA (fig. 19) co- inciding with the axis of aw, the equation to the surface generated, consisting of two disunited portions or sheets, will be | yt DB) ° a~ Ce Ya 51. To find the equation to the surface of a paraboloid. Let OQ (fig. 20) be a parabola which by revolving about its axis, coincident with the axis of w, generates a paraboloid ; OP any position of the generating curve, ON =2, NV HW, eM Lee, 39 the co-ordinates of any point P in it; then PN’ =1.0N = la, 1 being the latus rectum of OP; but from the right-angled triangle PMN, PN? me y” a 2", YY te = le, the equation to the surface. 52. When, as in the preceding instances, a plane curve revolves about an axis coincident with one of the co-ordinate axes, the equation to the surface generated may be readily obtained, whatever be the generating curve. For let OC (fig. 16) the axis of x be the axis of revolution, CPA the generating curve, and PR =f(OR) its equation; and let it revolve about OC into the position CP’4’; ON =a, NM = y, MP’ = x, the co-ordinates of any point P’. Then by the equation to the curve, MP’ =f(OM), or x= f(a" + 9"), the equation to the surface. 53. The surfaces which we have hitherto considered, are the simplest cases of surfaces of the second order; their equations are all contained in the general equation of the second order between three variables, which is av +by? +e 42a yr+20 sx4+2cuyt2a v4+2b y+20"s+4+d=0. Tt will be shewn hereafter that this equation, by giving a proper position and directions to the origin and axes of the co-ordinates, can always be reduced to one of the forms Av? + By + C2? =+ D, By + C2? =2A'e, which represent two distinct families of surfaces; the former 40 those which have a center, the latter those which have not a center; meaning by center, a point such that all chords of the surface drawn through it are bisected in it. 54. The origin is the center, and the co-ordinate planes are the principal planes, of the surface represented by the equation Aa’ + By? + Cz’ = + D. For let P (fig. 22) be a point in the surface whose co-ordinates are v=h,. y=k, x =1; then the equation is satisfied by these values, and therefore it is also satisfied when —h, —k, —/ are written for w, y, and x. But if we produce PO to P’, and make OP’= OP, the co-ordinates of P’ are evidently equal to —h, —k, —1; therefore P’ is a point in the surface, and PP is a chord, and it is bisected in O; i.e. every chord is bisected in O, therefore O is the center of the surface. Also the surface is situated symmetrically with respect to the co-ordinate planes. For if in the equation we make v«=ON=h, and y= NM=k, it will give for = two equal values with contrary signs PM, QM; so that for every point of the surface, situated above the plane of wy, there will be a corresponding point situated at an equal distance below it. Similarly, the other co-ordinate planes may be shewn to bisect all their ordinates at right angles. Planes which have this property are called the principal planes of the surface, and their intersections (which in this case coincide with the axes of the co-ordinates, and with respect to each of which also the surface is symmetrically situated) the aves of the surface. Also every chord passing through the center is called a diameter of the surface. 55. In the equation to surfaces that have a center Aw’ + By’ + Cx2? = +D, having taken care to make the second member positive, since the coefficients of the variables cannot be all negative together, 41 we can only have three varieties of form, (1) all the coefficients positive, (2) one negative, (3) two negative; so that the equa- tion may assume the three following forms : 2 2 2 Lv yo nee + F + re ! 9 oO 5 ian Lene x’ y 2 ates POR a pa: y? oe? The surfaces represented by them are called respectively the ellipsoid, the hyperboloid of one sheet, and the hyperboloid of two sheets; the two latter surfaces being-also sometimes called the continuous, and discontinuous hyperboloid. 56. In the family of surfaces, represented by the equation By’? + Cz? = 242, the origin is not the center, since the equation does not remain unchanged when w, y, x are changed into —w, —y, —#; and it will be seen hereafter that no other point can be its center. Also since only even powers of y and x enter the equation, the planes of sw and wy are principal planes, but the plane of yx not a principal plane; consequently the surface has only one axis, coinciding with the axis of vw. In the equation, if Band J’ are not both positive, let the coefficient of y? be made positive; and then, if necessary, change the sign of & in order to make the second member positive, for the change of vw into —@ will merely alter the position of the surface ; the equation will then offer two varieties according as C is positive or negative, under the forms 2 2? Tipe y” 2 1 ie v 3 and the surfaces represented by them are called respectively the elliptic and hyperbolic paraboloids. 42 57. To find the equation to the surface of an ellipsoid. This surface is generated by a variable ellipse which moves parallel to itself with its axes in two fixed planes, and vertices in two ellipses in those planes having a common axis coincident with the intersection of the planes. Let BC, CA (fig. 17) be quadrants of the given ellipses traced in the planes yx, sv”; OC =e their common 3 axis coinciding with the axis of zy, OA =a, OB =b, the other 4 axes; QPR a quadrant of the generating ellipse in any position, having its plane parallel to wy, its center in OC, and two of its vertices in the ellipses AC, BC, so that the ordinates QN, RN are its 4 axes; also let ON=x, NM =a, MP =y be the co-ordinates of any point P in it; then x” y 2 in ere ae 1 NQ? NR () NQ? zs NR Pte but AG es ——=1--—. a’ Ce : Cc TL)? Multiply the first term of (1) by » the second term by ae and the second member of the equation by the equal Pei quantity 1 aKa and transpose and we get 2 2 2 (ite ATs ie =+3+- =}, ‘Tie 8 SSO pe the equation to the surface. 58. To determine the form of the ellipsoid from its equation. Since in the equation, w can only receive values between a and —a, y between 6b and — 6, and z between ec and —e, the surface is limited in all directions. If we put x = 0, we obtain 43 for the equation to the trace on wy, which is therefore an ellipse AB; also, from the mode of generation, all sections parallel to vy are ellipses, and since their axes are in the ratio of a to b, they are all similar to AR. If we make x = +h, we have aphTe Bigs® h? ~=+—sl-- bh? Ce a’ for the equation to any section parallel to yz, which is an ellipse similar to the trace BC (since its axes are in the ratio of b to c whatever be the value of h) and which becomes imaginary when h>a. In the same manner it may be shewn, that all sections parallel to xsw-are ellipses, similar to the trace AC. The ellipses, of which 4B, BC, CA are quadrants, in which the surface is intersected by the co-ordinate planes, which are its principal planes, are called the principal sections of the surface; and the parts of the co-ordinate axes inter- cepted within the surface, are called its axes. Hence a, b, c represent the J axes of the ellipsoid, and also the 4 axes of its principal sections. The extremities of the axes such as A, B, C are called the vertices of the ellipsoid, of these it has six, one at the extremities of each axis. The whole surface consists of eight portions precisely similar and equal to that represented in the figure. Cor, If a=b, the equation becomes that to a spheroid generated by revolution about Ox; similarly, if any other two of the semiaxes become equal, the ellipsoid becomes a spheroid generated by revolution about the remaining axis. 59. To find the equation to the surface of a hyper- boloid of one sheet. This surface is generated by a variable ellipse which moves parallel to itself, with its axes in two fixed planes, and vertices on two hyperbolas in those planes having a common conjugate axis coincident with the intersecticn of the planes. 44. Let AQ, BR (fig. 18) be the given hyperbolas traced in the planes xa#, yx; OC =c their common 4 conjugate axis coinciding with the axis of xs, OA4=a, OB=b, the 4 transverse axes; QPR the generating ellipse in any posi- tion, having its plane parallel to wy, its center in OC, and its vertices in the hyperbolas 4Q, BR, so that thie ordinates NQ, NR, are its 4 axes. Also let ON = 2, NM=.2a, MP =y be the co-ordinates of any point P in the generating ellipse; then x y” NO}? NR (1) NQ? SONAR ed but =i eas Sie it ye 2 Multiply the first term of (1) by zu , the second term by a” 2 , and the second member of equation (1) by the equal b? 2 quantity as and transpose and we get a y” a iit pestered the equation to the surface. 60. To determine the form of the hyperboloid of one sheet from its equation. Since the equation will visibly admit values of a, y, z positive and negative however large, the surface is extended indefinitely on all sides of the origin, If we put #=0, we obtain 2 | & y 4 +21 © w g for the equation to the trace on wy, which is the ellipse 4B; and from the mode of generation all sections parallel to wy are similar ellipses, the dimensions of which increase indefinitely, 45 the least being that of which AB is a quadrant, and which forms the interior limit of the surface. For the sections parallel to yx, putting # = +h, we have which, as long as ha, making the second member positive, the equation is y’ h? ee at which also represents a hyperbola, but in a new position, namely, with its vertices in AQ, and conjugate axis parallel to Oy. In the same manner, the sections parallel to zx” may be shewn to be hyperbolas similar to the trace 4Q. The principal sections of this surface are the ellipse 4B, and the hyperbolas 4Q, BR, The quantities a, b, which denote the distances from the origin at which the surface cuts the axes-of w and y, are called the real semiaxes of the surface; the quantity ¢ is called the imaginary semiaxis, because if we put 2 = 0, y = 0, to find where the surface cuts the axis of x, we find v= —c, or s=+0e\/-1, so that c is the coefficient of the expression for the imaginary 4 axis of the surface. ‘The extremities of the real axes are called the vertices of the surface; two of them are 4, B, and the remaining two are at distances from O respectively equal to a and 6, in AO and BO produced. The whole surface consists of eight portions, precisely similar and equal to that represented in the figure; and since it is continuous, that is, since we can pass from one point in it to any other point in it, without quitting the surface, it is called the hyperboloid of one sheet. 46 Cor. If a= 8, all sections parallel to wy become circles, and the surface becomes a hyperboloid of revolution about the conjugate axis. 61. The hyperboloid of one sheet has an interior conical asymptote. Putting the equation under the form 3 x? y° x yy a’*b? char ae ee pas) La 5 ap a PA hace b- ad 6 ay” + bx’ and expanding the value of ‘a by the Binomial Theorem, we c see, since when a and y are very great the quantity a’ b? a*y* of b? a? is very small, that the relation among the co-ordinates of points very distant from the origin, is nearly expressed by a” 2 Poteet ds CG amos this then is the equation to a surface, whose distance from the surface of the hyperboloid, measured parallel to the axis of z, diminishes indefinitely as w and y increase; and, as its ordinate is always greater than the corresponding ordinate of the hyperboloid, it lies within the latter surface. This asymptotic surface is a right cone vertex O, and base an ellipse A’B’, whose center is in C, and 3 axes equal and parallel to 40, BO; for if x, y, be co-ordinates of any point in a generating line of this conical surface, its equations will be Ye a7 ae Ze & be . the co-ordinates of the point where it meets 4’B’ are, making Z=c, 47 and as these are co-ordinates of a point in an ellipse whose 4 axes are a and 6, “ve\? yc” Pa Yad 2 Za 3b CF 5 ae be 62. To find the equation to the surface of a hyperboloid of two sheets. This surface is generated by a variable ellipse which moves parallel to itself, with its axes in two fixed planes, and vertices in two hyperbolas in those planes having a common transverse axis, coincident with the intersection of the planes. Let AQ, AR (fig. 19) be the given hyperbolas traced in the planes xv, wy; OA =a, their common 4 transverse axis coinciding with the axis of 2, OB=b, OC =e, the 4 con- jugate axes; QPR the generating ellipse in any position, having its plane parallel to yz, its center in Ow, and its vertices in AQ, AR, so that the ordinates QN, RN are its semiaxes. Let ON=a2, NM=y, MP==2 be the co-ordi- nates of any point P in the ellipse, then 3° y° Qn? + RNeT ? M) pay ahs RN* x but Q —— Out 9 9 aay 2 1 2 a b- a 2 multiply the first term of equation (1) by —, the second Cc 2 RN° term by aprrat and the second member of equation (1) by o - ES ao the equal quantity — — 1, and transpose and we get me the equation to the surface. 48 63. To determine the form of the hyperboloid of two sheets from its equation. The equation shews that all values of v, between + a and —a, are inadmissible, therefore no part of the surface can be situated between two planes parallel to yx through A and A’; but the equation can be satisfied by values of wv, y, % in- definitely great, therefore there is no limit to the distance to which the surface may extend from O. If we put # =0, we have 2 Dae ; therefore the principal section by yz is imaginary; but all sections parallel to yz, and at a distance from it greater than a, are similar ellipses as appears from the mode of generation. For the sections parallel to wy, putting x= +/, we have 2 I? — eb Te? which represents a hyperbola similar to the principal section AR with its vertices in AQ and the opposite branch, and conjugate axis parallel to Oy; and in the same way it may be shewn, that the sections parallel to xv are hyperbolas similar to the principal section AQ with vertices in AR and the opposite branch, and conjugate axes parallel to Ox. If we make at once y = 0, x = 0, to find where the surface cuts the axis of w, we have 2 = a? or a = £a3 2a is the real axis of the surface, and its extremities the vertices of the surface of which there are but two; the quantities 6b, ¢ are the imaginary semiaxes of the surface, because it does not cut either the axis of y or x. ‘The whole surface consists of two sheets perfectly similar and equal, and indefinitely extended in opposite directions, but separated by an interval in which exists no point of the surface ; it is therefore called the hyperboloid of two sheets; each sheet consists of four portions precisely similar and equal to that represented in the figure. 49 Cor. If 6 =c, or the two imaginary axes become equal, this surface becomes a hyperboloid of revolution about the transverse axis. 64. The hyperboloid of two sheets has an exterior conical asymptote. Putting the equation under the form x y° sig 3° : (4 =) ( b? Cc ) = 0 as tree Bad ede reer Witeustna y Can i ON bans Ce ey? + b?2*)” and expanding the value of — by the Binomial Theorem, we a have, for points very distant from the origin, the relation among the co-ordinates nearly expressed by this then is the equation to a surface whose distance from the hyperboloid, measured parallel to the axis of w#, diminishes indefinitely as y and s increase; and, as its. ordinate w is — always less than the corresponding ordinate of the hyperbo- loid, the latter surface lies within it. It may be shewn, as in Art. 61, that the asymptotic surface is a right cone vertex O, and base an ellipse B’C’ whose center is A and semi-axes parallel and equal to OB, OC. In this case, as in the former, we observe that the equation to the conical asymptote is ob- tained by omitting the constant term in the equation to the surface. 65. Since from the equation to the ellipsoid, the equa- tion to the hyperboloid. of one sheet results by changing ce into c\’—1; and the equation to the hyperboloid of two sheets, by changing 0b into b,\/—-1 and ¢ into CAL sevais if any result in terms of its axes be obtained for the ellipsoid, the corresponding results for the hyperboloids may be de- duced by writing GA for c; or b /—1, ae at, for 6 and c 4 50 66. To find the equation to the surface of the elliptic paraboloid. This surface is generated by a parabola which moves with its plane perpendicular to a fixed plane, and axis in that plane, and parallel to the axis of another parabola along which its vertex moves; the concavities of the parabolas being turned towards the same parts. Let OR (fig. 20) be a parabola traced in the plane of wy, vertex at the origin, and axis coinciding with the axis of «, J its. Jatus rectum; RP the generating parabola in any position with its plane parallel to zw, vertex in OR, and axis parallel to Ow, and let /’ denote its latus rectum, and ON = a, JM=y, MP =z be the co-ordinates of any point P in it; also draw RM’ parallel to Oy. Then 2 s=l'.RM =U (ON - OM’) =! (« _ =| 9 Neen aye f 7a the equation to the surface. 67. To determine the form of the elliptic paraboloid from its equation. Since only positive values of w are admissible, no part of the surface is situated to the left of the plane of yx; also since the equation can be satisfied by positive values of w, and by positive and negative values of y and x however large, the surface is extended indefinitely towards the positive direction of w If we make y=0, 3° =/w is the equation to the principal section OQ, and from the mode of generation all sections parallel to xv are parabolas equal to OQ with vertices in OR; similarly, all sections parallel to wy are parabolas equal to the other principal section OR, and having their vertices in OQ. If we make w =0, we find I'y? + /z*=0; *. y=0, x=0, or the trace on yg is a point namely the origin. If we make w =h, we have y” 3? TM 51 for the equation to a section parallel to yx, which represents an ellipse whose semiaxes are the ordinates QN, NT' which are to one another in a constant ratio \//' to \/1; therefore all sections perpendicular to the axis of the surface, are similar ellipses, and hence the name of the surface. It has only one vertex O and one indefinite axis Oa, and consists of one sheet ; and is reduced to a paraboloid of revolution when J =U’. 68. To find the equation to the surface of the hyperbolic paraboloid. This surface is generated by a parabola which moves with its plane perpendicular to a fixed plane, and axis in that plane, -and parallel to the axis of another parabola along which its vertex moves; the concavities of the parabolas being turned towards opposite parts. Let OR (fig. 21) be a parabola in the plane of wy, vertex at the origin, and axis coinciding with the axis of w; and J its latus rectum; HP the generating parabola in any position with its plane parallel to x7, vertex in OR, and axis parallel to Ox, and let /’ denote its latus rectum, and ON=a, NM=y, MP = the co-ordinates of any point P in it; draw RM parallel to Oy, then o ~ 2 =I.MR =! (OM -ON)=0 (F- w) 2 l the equation to the surface. 69. To determine the form of the hyperbolic paraboloid from its equation. The surface cuts the co-ordinate axes only at the origin, and since the equation will visibly admit values of aw, y, % positive and negative however large, the surface is extended indefinitely from the origin. If we make y=0, we have s* = —I'x for the equation to the trace on xa, which repre- 4 —2 52 sents the parabola OQ, with its concavity turned towards the negative direction of a, since its equation shews that w must be taken negatively; and from the mode of generation, all sections parallel to s# are parabolas equal to OQ, with their vertices in OR. The other principal section of the surface in the plane of wy, is the parabola OR, forming the interior limit of the figure, and all sections parallel to vy are parabolas equal to OR, with their vertices in OQ. For the trace on ya, putting wv = 0, we have EVA sabat AWA which represents two straight lines passing through the origin; and for sections parallel to yz, making x = h, we have which represents a hyperbola with its vertices in OR, and conjugate axis parallel to Oz. If we make h negative, the equation to the section becomes » 9 8" Ji Uh th | which also represents a hyperbola, but in a new position, viz. with its vertices in OQ, and conjugate axis parallel to Oy. The surface consists of one sheet, and has only one vertex O, and one indefinite axis Ow; and all sections per- pendicular to its axis are similar hyperbolas, whence its name. It does not become a surface of revolution when 7 = 7’, nor in any other case. b) 79. The hyperbolic paraboloid has plane asymptotes. Putting the equation under the form \é , U Lx v=-y—lea =F (2 - =) ; and expanding the value of x by the Binomial Theorem, we have the relation, among the co-ordinates of points very distant from the origin, nearly expressed by 53 is the equation to two planes through the origin perpendicular to the plane of yx, the distance between which and the para- boloid continually diminishes. These planes contain the asymptotes to all the hyperbolic sections parallel to yz, which, as we have seen, have their centers in the axis of #, and axes in the ratio of Vs to / le 71. The equation to the hyperbolic differs from that to the elliptic paraboloid, in having — 7’ instead of J’; this re- lation will enable us to modify all results obtained for one surface, so as to be true for the other. 72. The elliptic and hyperbolic paraboloids are par- ticular cases of the ellipsoid, and hyperboloid of one sheet respectively ; viz. when the centers of these surfaces are removed to an infinite distance. In the equation write vw —a instead of wx, then the resulting equation 2 (w-a)y ¥? ON ety ceemeese e ee AT, gree oe fe O a hous ct ei Ge i Ciden On Gi da is reckoned from the vertex of the surface. Let p and p’ denote the distances of the foci of the principal sections in wy and xa, from the vertex, . P=ad—(a-p)=2ap-p, e=2ap sp’; hence, by substitution, a” 2a y - re Fe 9 , 7 = 0, a a 2ap —p 2ap) += Pp a y x or ——-2@+ = 5 m= OS MP AY 1 lara Ay UN ae a a therefore, making @ infinite, 7,e. supposing the center of the surface to remove to an infinite distance from the vertex, 54 whilst the distances of the foci of the principal sections from their common vertex remain finite, we have 2 2 Ee — ,—~2xH=0, 2p 2p which coincides with the equations to the paraboloidal surfaces. Cor. Hence if any result be obtained for the ellipsoid or hyperboloid, it will be adapted to the paraboloids, by the modification above indicated; viz. by transferring the origin to the extremity of an axis, and making that axis infinite. Also both families may be represented by the equation Ax’ + By? + Cz? = 242, the origin being at a vertex, and A = 0 when the surface has not a center. feectilinear generating lines of surfaces of the second order. Surfaces of the second order admit of another division, viz. into those which can be generated by the motion of a straight line, and into those which cannot. This property, which we have seen to belong to the cylinder and cone, we shall now shew to be possessed by the hyperboloid of one sheet and the hyperbolic paraboloid; it is easily foreseen that it cannot belong to the remaining surfaces of the second order, as their forms manifestly preclude their having a straight line applied to them throughout its indefinite length; this will also appear from our results. 73. The Hyperboloid of one sheet can have an infinite number of straight lines entirely coinciding with its surface. Since the equation to the surface may be written 2 2 v 2 y a b 2 Ee) w it is evident that either of the following systems (each con- sisting» of the equations to two planes and therefore represent- ing a straight line) will satisfy it, viz. v& R 4 or ~ = Fay (1 +4) @*- Cc b w being an arbitrary constant; since then the co-ordinates of every point in either of the lines (1) or (2) will satisfy the equation to the surface by causing its two members to become identical, it follows that for every value of mu there are two straight lines that lie entirely in the surface. 74. ‘To determine the equations to the projections of the generating lines of a hyperboloid of one sheet. Let the equations to the surface and to a line be respectively x” y 3 erg rhe oh Ta eb 6 : vw=ms+h, y=nsr+k; if the line coincide with the surface in all its points, the equation (ms+h)? (nz+hkh) GOLgey es Bo peer 3 a” b Cc will be true for all values of x; therefore, equating to nothing the coefficients of z* and z, we have m oi 1 ae =O a” ieee Ca : mh nk ta Mi ; riggs sae ee ie eet ae b- > ef BP Eliminate & between the two latter equations, and reduce by means of the former ; h? mb h\? RENS 2 aa 4 — +e = te or ( = 1% a~ na a VUaAC b Dis ie BEG eat Me mhe > 56 therefore the two lines whose equations are respectively nae NaC C= MZ + — &=>M2Z — — £ b (1) += -Hh2)> mbe mbe y= NZ —- — Y = ns + — a a will coincide with the surface of the hyperboloid in all their points, m and m being any quantities which satisfy the equation m 1 =r 5 Cc” and if m be made to assume all values, and m be always 6 ; ae taken = — \/a? = m2c?, we shall determine two infinite sys- ac tems of lines represented by the equations (1) and (2), having the aforesaid property. Cor. No two lines in the same system intersect; for if we take two lines in the first system represented by the equations NAC fe NIE L=MSs + — C= MS + ? , mbe f m be y= ns — —— Y=NZ— a J a we shail find (Art. 22) that they cannot intersect unless (m—-m’)?=0, that is, unless they become identical. But any line in the first system intersects every line in the second system; for let the equations to two lines, one in each system, be nac Welin ae = MZ +—— L=EMZ— b b ? , 2 mbe ; mbe Y= 2S = a Y= 22. -.—— a 57 24 fo ry; ? 2 2 . ‘ : At Li: then (Art. 22) these lines intersect if — call Ghia a a ay 1 which is always the case because each equals —. c Hence if we take any three lines in the second system, and suppose them fixed, and make a line intersect them, it must be a line of the first system, and by assuming all possible positions will generate the surface of the hyperboloid; and since consecutive positions of the generating line do not intersect one another, the surface is what is called a twisted surface. The surface evidently admits a second mode of generation, in which a line belonging in every position to the second system, moves along three fixed lines of the first system. 75. The projections of the generating lines of the hyper- boloid upon the principal planes, are tangents to the traces of the surface on those planes. Suppose w = mz + h to be the equation to a line touching the principal section on sv whose equation is 9 2 ri fa & then cial SAAN as We must give two equal values for x, and Pi ms +h)? 5 ae ME Ok: +1 must be a perfect square ; 1 mM? h? m? h? . b Peps es Spaete ort FET = 4 , or h*=a° — mc’; cb a* a a e w=me+/ae—m ec, which coincides with the equations to the projections of the generating lines on zw. Similarly, the projections on the 58 other principal planes may be shewn to be tangents to the traces on those planes. Cor. If through the origin we draw a line parallel to the generating lines, its equations will be v=ms, y=nzx3 and the equation to the surface generated by it, eliminating m and » by means of the equation which connects those quantities, will be which represents the conical asymptote; hence any line drawn through the center parallel to a generating line of the hyper- boloid lies in the surface of the conical asymptote. 76. The hyperbolic paraboloid can have an_ infinite number of straight lines entirely coinciding with its surface. It is evident that either of the following systems (each consisting of the equations to two planes, and therefore repre- senting a straight line,) will satisfy the equation to the surface; viz. oo. ¥ % y x VLR tte an ae Ore Ae UD Pike keer eae es (2) DRT ARE 5 Ao A a Since then the co-ordinates of every point in either of the lines (1) and (2) will satisfy the equation to the surface by causing its two members to become identical, it follows that for every value of the arbitrary constant p, there are two straight lines that lie entirely in the surface. 77. ‘To determine the equations to the projections of the generating lines of a hyperbolic paraboloid. Let the equations to the surface and to a straight line be respectively 59 e=met+h, y=nsr+k; then, if the line coincide with the surface in all its points, the equation (nz+k)y eo eae tae ms+-h= ti i j is satisfied for all values of x; are pa} i Qnk ; toe =a TFT =U N= = <5 aoe Tieaee ttre t 2 caper Ty fe py 2n 4, will coincide with the surface of the paraboloid in all their . 1 points, m being any quantity, and ” = NES and if m be made to assume all values, we shall determine two infinite systems of lines represented by equations (1) and (2), having the aforesaid property. Cor. In the same manner as for the hyperboloid, it may be shewn that two lines in the same system never inter- sect, and that two lines in different systems always intersect. Hence if we suppose three lines in either system to become fixed, the surface may be generated by making a line move so as always to intersect them. Or, since the generating lines in the two systems are respectively parallel to the fixed planes yY=Nn2%, y= —nz, the paraboloid may be also generated in two ways by a straight line which moves so as always to Intersect two fixed lines, and to be parallel to a fixed plane. 60 In this case also the projections of the generating lines are tangents to the principal sections of the surface. Thus (changing the sign of w in both equations) —x=mz+h will bea tangent to x* =/'a, if 2° + Imzx+Uh=0 bea perfect square, ay! or 40h =1?m’®, orh= aa (+7) . -—-& =m - — fe is the equation to a tangent; which, measuring # in the positive direction, coincides with the equation to the projection of the generating lines on zw. 78. We shall terminate this Section with demonstrating the following general and important property of surfaces of the second order. If two surfaces of the second order have a plane section in common, their other curve of intersection, if it exist, will also be a plane curve. Let the equations to the two surfaces be Aa’ + By? +C2?° 4+ 2A y2z4+2B' s24+2Cay+2A"v+2B’y +20°2+ D=0, au + by +cx? + 2a'yx 4+ 2b’ sax + Qcuxy 42a" v4 2b'y + 2c"24d=0, and suppose them to have a common section in the plane of wy; then making x= 0, the curves represented by the equations Ax’? + By? +2Cay+2A"v+2B"y+D=0, ax+by+2ceary 42a a+ 2b y+d=0, 61 are identical; if therefore m be a constant multiplier, A=ma, B=mb, C=me', A” =ma", B'’=mb", D=md. But in order to determine the complete intersection of the surfaces, we must combine their equations. ‘Therefore, mul- tiplying the latter by m and subtracting it from the former and having regard to the above relations among the co- efficients, we find (C—mc)2°+2(A'— ma) yz +2(B’— mb’) z7+2(C”’—me")z=0; the equation to a surface which contains all the points com- mon to the two proposed surfaces. But this may be decom- posed into %=0, (C—mc)x +2(A’— ma’)y+2(B'—mb')a +2(C”—mc")=0; the first belongs to the assumed curve of intersection in the plane of wy; the second is the equation to a plane, and cannot therefore, when combined with either surface, give any thing except a plane curve of the second order for the other curve of intersection. SECTION III. ON THE PROJECTIONS OF LINES AND PLANE SURFACES, AND ON THE TRANSFORMATION OF CO-ORDINATES. Tur meaning of the projection of a point, and of a line, upon any plane has already been explained, (Arts. 2, and 13.) Moreover if from the extremities of a limited line we drop perpendiculars upon any indefinite line either in the same plane with it or not, the part of the latter line intercepted between the feet of the perpendiculars, is called the projection of the former line upon the latter. Also, if the sides of any plane surface be projected upon a plane, the figure bounded by these projections is called the projection of the given surface upon that plane. Between the length of a finite line and the length of its projection upon any plane or line, and also between the area of any plane surface and the area of its projection upon a plane, a remarkable relation exists, which we shall now exhibit. 79. The length of the projection of a limited line upon a plane, is equal to the length of the line multiplied by the cosine of the acute angle which it forms with the plane. Let CD (fig. 25) be the line produced to meet Gd K, the plane on which it is to be projected, in H. Let DHd be the projecting plane, intersecting GdK in Hd; and in the projecting plane draw Cc, Dd, perpendicular to Hd, and CD’ parallel to it; then ed is the projection of CD, andz DHd =i is the inclination of CD to the plane GdK; also CD’ = CD xX COS 23 « ed= CD = CD cosi. 63 80. The length of the projection of a limited line upon any other line, is equal to the length of the line multiplied by the cosine of the acute angle which the two lines form with one another. Let CD (fig. 26) be the line, 4B the line upon which it is to be projected, Cc, Dd perpendiculars upon AB, then ed is the projection of CD upon 4B. Let Ne, Md be planes through C, D, perpendicular to AB, and of course containing the lines Cc, Dd; draw CD" parallel to AB, meeting the plane Md in D’, and join DD’; then the triangle CDD’ is right-angled at D’, and DCD’ =i is equal to the angle formed by the two lines, .. CD’ = CDcosi, but ced = CD’, each being the perpendicular distance of parallel planes ; *, ed = CD cosi. 81. ‘The area of the projection of any plane surface upon a plane, is equal to the area of the surface multiplied by the cosine of the acute angle which the planes form with one another. Let ACB (fig. 27) be a triangular area traced in a plane which intersects the plane Gd, upon which the projection is to be made, in GK. Through the angular points draw planes AGa, BKb, CHc perpendicular to GA; then these planes will contain the perpendiculars Aa, Bb, Cc let fall from the angular points upon the plane of projection, and their inter- sections with each of the planes 4H B, a Hb, will be perpen- dicular to GK; also the angle CHe will equal the inclination of the planes =7. Join the feet of the perpendiculars; then acb is the projection of ACB; also join Dd, which is parallel to Ce. Then the triangles acd, ACD, since they have a common altitude GH, are to one another as their bases ed, CD; simi- larly the triangles bed, BCD, having a common altitude HK, are to one another as cd to CD; ed CD? = triangular area ABC, cos?, " triangular area abe = triangular area ABC. 64 Next suppose the figure to be projected is a polygon; then it can be divided into triangles, each of which will be to its projection as 1 to cosi; and therefore the sum of the triangles will have to the sum of the projections the same ratio, or area of projection = area of polygon . cosi ; and as this is true however much the number of sides of the polygon be increased, it is also true when the figure to be projected is bounded by a curve ; .". area of projection of any plane surface = area of surface. cos i. 82. The square of the area of any plane surface is equal to the sum of the squares of the areas of its projections on three co-ordinate rectangular planes. Let the given area be denoted by 4, and its projections on the planes of wy, yx, xu by A,, A,, A,, respectively. Also let a, (3, yy denote the angles which a perpendicular to the plane of the given area from the origin, makes with the axes of w, y, 8. ‘Then vy (Art. 27) is the inclination of the plane of the given area to the plane of wy, and therefore A,= Acosy; similarly A, = dA cosa, A,= A cos DB; “. 42 + Ai + At = AP $ (cos yy)? + (cosa)’ + (cos B)?} = A?. Cor. Hence in fig. 9, since the triangles 4OB, BOC, COA, are the three projections of the triangular area ABC, if OA, OB, OC be denoted by a, 8, c, (AABC)? = ($.4b)° + (f bc) + (hea)? ; ». DABC =}/ (aby + (ao) + OO Oblique co-ordinates. 83. To express the distance of a point from the origin in terms of its oblique co-ordinates. fig 2, AL the angles yOx, Ow, wOy, be denoted by A; wy v3; and let OA, AN parallel to Oy, and NM parallel to Ox, be the co-ordinates w, y, x of the point M. From 65 the points 4 and N drop perpendiculars 4m, Nn upon Ox; then mv is the projection of 4N upon Oz, and = AN cos), since AW is parallel to Oy (Art. 80); and Om = O4 cosy. Now from the triangle ONM, since MN is parallel to Ox, we have OM? = ON? + NM? +2MN.NO cos s ON. But ON? =a? + y° +2vy cos v, from the triangle OAN, and ON cosz ON = On = Om +mn = 4 cosun+Ycosr; . P= a +y’ +2 + 2rycosy + 2a cose + 2yX COSA. Cor. Since d is the diagonal of the parallelopiped of which a, y, s are three conterminous edges, the above formula gives the diagonal of any parallelopiped in terms of its edges and the angles which they make with one another. It is also the equation to the surface of a sphere whose center is at the origin and radius = d. 84. To find the distance between any two points in , terms of their oblique co-ordinates. Let vw, y, x; @, y’, x’, denote the co-ordinates of two points referred to oblique axes; then as in Art. 6, if through these points we draw six planes parallel to the co-ordinate planes, we shall form a parallelopiped with its edges parallel to the axes, their lengths being a —a, y’—y, and 2 —3; and the distance of the points is the diagonal of this parallelopiped ; “ @=(v — 2) + (y’ —y) + (2 - 2)? +2(a' —x)(y’— y) cosy +2(a’ —x)(x'—2)cosu+2(y’—y) (2-2) cosa. Cor. Hence we have the equation to the surface of a sphere , whose radius is d, and co-ordinates of its center, a’, y’, 3’. 85. To find the angle between two lines, whose equations are given referred to oblique axes. Piet L=Mz) vx =mMsZ ; , » be the equations to two lines y=ns) yYEns parallel to the proposed ones, through the origin. 4 66 Then, as in Art. 30, if we take a point in each of these lines, at a distance = 1 from the origin, and call their co-ordi- nates v, y, 3 wv, y’, 2’, respectively; also their distance d, and the angle between the lines 0, we have 2—2cos@ = d* = (a — x)? +(y'-y)* +(x’ —2)?+2(a'— x) (y'—y) cosp +2 (« — x) (2-2) cosn + 2(y’— y) (2-2) cosa, or, since | l= a+ y? +2" + 2evy cosy +222 cosp + 2YX COSA; 1 wo? 4 y? + 2 4 20'y' cosy + 24a'2' cosm + 2y'X COSA, cos 0 = wa! + yy + x2" + (ay + vy’) cosy + (a's + w2’) cosp + (yz + 2’y) cosa. (1) But since w=ms, y=ne, & =m’'s’, y =n'", 1=3°(1 +m’? + n° +2mn cosy +2mcosu+2nCcoSd), 1=2?(1 +m? 4? +2m'n' cosy +2m cosu + 2 cosa); * cos@ = | 1+mm’+nn'+(m'n+mn’)cosv+(m+m’')cosn+(n+n’)cosr rie i 0 é a/ 1+m?-+n®+2mn cos +2 m Cos 1+2n cosry/1+m”?+n?+2 m’n’cosv+2m’cospr+2 n’Cosr Cor. Hence the condition that the lines may be at right angles to one another, is cos @ = 0, or 1+mm' +nn'+ (m'n + mn’) cosv + (m +m’) cos u +(n +n’) cosr = 0. 86. To find the conditions in order that a straight line and a plane, referred to oblique axes, may be perpendicular to one another. et a line and plane be drawn through the origin, re- spectively parallel to the proposed ones, and let their equa- tions be \, and s= Aw + By. Y= NB 67 Now this line, since it is perpendicular to the plane, is per- pendicular to any line situated in the plane, and consequently to the trace of the plane on wz, whose equations are ae eel 9=—8, y=O; .*. (Art. 85) since m’=—, n =0, A ri ll i ( 7) + ent —- —— COS MW —— We COS 7’ COS = Fyne 4 vt+ uy, fs : or A(1+mcosu+ncosr) + (m+ ncosy + cosp) = 0. Similarly, B(1+mcospu + 2 cosr) + (2 + mcosy + CosA) = 0. Cor. Hence, we can find the angle between two planes referred to oblique axes; for the above conditions enable us to find the equations to two lines through the origin respectively perpendicular to them; and knowing the equations to the lines, we can compute the angle between them, that is, the _angle between the planes, by Art. 85. 87. If we employ the symmetrical forms of the equations to a straight line we obtain immediately, for the angle 9 between these lines, from equation (1), Art. 85, denoting by wy the angle between the axes of x and y, and similarly of the others, cos @ = Il’ + mm’ + nn' + (lm' + I'm) cos vy + (ln +1'n) cos vz + (mn' + m’n) cos yx. If we now make the former line to coincide successively with the axes of a, y, x, and call the inclinations of the latter line to those axes a, (, vy, we get 5—2 68 , cosa =l1' 4+ m'cosavy + n' cos az, , , U cos 3 = m' + I’ cosvy + n' cosy®, cosy =n + I cos ax + m' cos ys; -- cos 0 =l cosa + m cos 3 + n cosy, where a, 3, y are the inclinations of one line to the co-ordinate axes, and J, m, n the projecting ratios of the other. Hence we obtain for the condition of two lines being parallel, lcosa + mcos 3 + n cosy = 13 and for the condition of their being perpendicular, Icosa +m cos + ncosry = 0. Likewise these equations express respectively the conditions for a line whose equations are OP PS Lome ne being perpendicular, and parallel, to a plane whose equation is vw cosa + ycosP + % cosy = p. Transformation of Co-ordinates. The discussion of the nature and properties of surfaces is much facilitated, when their equations are reduced to the most simple form they are capable of, without being deprived of any of their generality. This simplification is effected by giving a suitable position to the origin, and suitable direc- tions to the axes of the co-ordinates. We shall therefore next proceed to investigate the chief formule for transformation of co-ordinates. 88. To change the origin of the co-ordinates without altering the directions of the axes. Let a, y, x, be the co-ordinates of the point M (fig. 3) referred to the origin O, and axes Ow, Oy, Oz; also let M' 69 be the new origin, OA’=h, A’N’=k, N’M’=1, its co-ordinates, and a’, y’, s, the co-ordinates of M referred to M’ as origin and to axes parallel to the original ones; then _ e= OA= MH + OA =a' +h, similarly, y=Y th, sax 41; and substituting these values of w, y, x in the equation to the surface, we shall obtain the equation referred to the new origin and to axes parallel to the original ones. 89. To pass from one system of co-ordinates to another having the same origin, supposing the first rectangular, and the second oblique. Let Ow, Oy, Ox (fig. 28) be the rectangular, and O2’, Oy’, Oz’ the oblique axes, having a common origin O. From any point P draw PM, PM’, respectively parallel to Ox, Oz’, meeting the planes of wy, a'y’ in M and M’, and draw MN, M'N’ parallel to Oy, Oy’. Then the co-ordinates of P in the two systems are ON = aN Moen yy ALB i= 2, ON’ =a, NM =y', MP=-2' and our object is to express each of the first set in terms of the latter. From N’, M’, drop perpendiculars V’n, M’m upon Ow, and join PN which is also perpendicular to Ow; then On, nm, mWN are the projections of ON’, N’M’, M’P upon the axis of #; therefore (Art. 80) , On =x cosv# a, nm=y'cosya, mN =2' cosz'a, , denoting, by wx, the ZO contained by the axes of «” and #2 produced in the positive directions, and similarly of the others. . @=ON = Ont+nm+mN = a cosa’« + y' cosy’a# + 2’ cosx’a@3 (1) that is, each primitive co-ordinate is equal to the sum of the projections of the three new co-ordinates upon its avis. 70 Hence y= a'cosa’y + y cosy’'y +2 cosx’y, / , , , , re =a cosvs +y CosyX + COS8R; or, if we denote by (m, m, 7) the cosines of the angles which the axes of a’, y’, x’, make with the axis of w; and by (m', n’, 17’), (m", n”, vr”) similar quantities relative to the axes of y and z, we have v=me+ny + re’, y sama +n'y + 7's". Pee Lad 2 ‘Seth ML+NY +7TR, Of the nine angles involved in these formule, six only are independent, there being three equations of condition; for since wv, wy, a's, are the angles which a straight line, viz. the axis of w’, makes with the three rectangular axes of Uy Y, &; .*. (cos aa)? + (cos ay)” + (cos az)? = 1; and similarly with respect to the angles which the axes of y and x’ make with the primitive axes; therefore m +m? +m”? = n+n'? +n’ 1, BE AREDE ye 2? 4 7’? = 1, Cor. In the figure, the axes of a’, y’, x’ are supposed to make acute angles with the axis of w; if this were not the case, and if for instance one of them Oy’, made an obtuse angle with Ow, m would fall to the left of », and we should have e®= On-nm+mN, with which the general expression (1) still agrees, because the term y'cosy'# is negative in the case supposed, but. numerically equal to the projection of y’. If again, the axes being as in the figure, y’ were negative, the point m would fall to the left of , and we should have e=On-nm+mNn. Hence, we collect that the formula (1) is applicable to alk wil cases, provided we pay attention to the signs of the co- ordinates, and of the cosines; the angles, as was before observed, being those formed by the axes produced in the positive directions. 90. To pass from one system of rectangular co-ordi- nates to another also rectangular. Here it would be sufficient to join to the expressions, and equations of condition of the preceding Art., three new equations of condition expressing that the axes Oa’, Oy’, O3' are at right angles to one another. _- We may, however, arrive at the expressions for x, y, %, briefly as follows, Let d be the distance of P from the origin ; then referring the lines OP, Ow to the rectangular axes Oa’, Oy’, Ox’, we have for the angle between them , , cos Px = cos Px’ cos x2’ + cos Py’ cos vy’ + cos PX cos #2; hence, multiplying by d, and observing that dcos Pw = x, dcos Pav’ = «’, &c., we have , ? , , , , V=H COSA’ + Y COSYXR4% Cosza, and similarly for y and x; hence denoting the cosines as before, L=me +ny + 72, y=ma + n'y + 1'2',70.(3)- s= malt n'y +72". In this case there are only three independent angles. For since the axes of a and y’ are at right angles, cos wa cos y’#@ + cos vy cos y’y + cos ax cos y/’% = 0; similarly, expressing that the axes of w’, x’, and the axes of y', %, are at right angles, and substituting for the cosines their values, we have three new equations of condition to be joined to those of the preceding Article, viz. mn +m'n' + mn” = 0, mr +m’ + my" = 0,>)...(4). Ld a A nr +-n7r +n’r”’ =0. 72 91. Sometimes, in, the case of two rectangular systems, it is required to find a’, y’, x in terms of a, y, x. This may be effected by regarding a’, y’, 2’ as the primitive co- ordinates, and recollecting that each is equal to the sum of the projections of the co-ordinates 2, y, upon its axis; “. @ = @ cosaxu’ + y cosya’ + 2% cosza’; and similarly for y’ and x’; hence v=maxe+my +m’, ne + n'y + n's,>...(5). v= rat ryt v's. I These expressions might also have been obtained from (3) by adding them together after having multiplied them respectively istly by m, m’, m”’; 2ndly by n, n’, n”; 3rdly by 7, 7’, 7; reducing in each case by means of the equations of condition (2) and (4). Also the equations of condition, when «x, y, x are re- garded as the new co-ordinates, and consequently their axes referred to the axes of a’, y’, x, will be m+n + 7. =1, m”? +n”? + 7? =1,>...(6), m+. 9? 4 9? = 1, mm +-nn +r 0, mm’ +nn" + rr” = 0,}...(7), m'm" + n'n" + rr" = 0. which are entirely equivalent to the relations between the same constants obtained before, and may in every case replace them. Indeed there is no difficulty in shewing that they are a neces- sary consequence of (2) and (4). For since P is at the same distance from the origin in either system, (3) , Pe id eP+y t+ Pax ty? t+ 2°75 and putting for «’, y’, x’ their values given in (5), and equating coefficients on both sides, we obtain the equations of condition (6) and (7). 73 92. In the preceding Articles, the means of passing from one rectangular system of co-ordinates to another, have been given in simple and symmetrical formule; they have however the inconvenience of involving nine constants, six of which must be eliminated by means of six equations of condition in order to make the determination of the new axes relative to the primitive ones depend upon three quan- tities. This led Euler to invent a mode of expressing each of the nine constants in terms of three others, which are the inclination of the planes of xy and x’y', and the angles which their line of intersection forms with the axes of x and , x ; as described in the following proposition. 93. To pass from the rectangular system of co-ordinates &, y, % to another rectangular system a’, y’, x’; having given the angle @ at which the planes ay, «’y’, are inclined to one another, and the angles @, Wy, which their line of intersection makes with the axes of w and ’ respectively. This is effected by passing successively through three rectangular systems, each having one axis in common with the preceding, and employing the formule relative to the transformation of co-ordinates in one plane. Thus to pass from the system of axes Ox, Oy, Oz, (fig. 29) to the system Ow,, Oy,, Oz, of which Ov, is the trace of the plane vw’ Oy’ upon wy, and Oy, is perpendicular to Ow, in the plane of wy, we must put @ = @,cos@ — y; sin a (1) y= a, singd+ y, cosh without altering x. Next to pass from the system Ow,, Oy,, Ox, to the system Ow,, Oy,, Ox’, of which Oy, lies in the plane a’y’, and Oz’ is perpendicular to Oy, in the plane y,Oz, we must put Y,=y, cos 9 — 2’ sin :t @), % = y, sin@ + 2’ cos@ without altering 2, . 44 Lastly, to pass from the system Ov,, Oy,, Ox’ to the system Ow’, Oy’, Ox’, of which Oa’, Oy’, are at right angles to one another in the plane x, Oy,, we must put @, = # cosy — y' sin) ’ - , apyd te} Y2=@ sn + y cosy without altering x’. But we may arrive at the result of these three successive substitutions by a single substitution, the formule for which will be formed by eliminating 2, y,, and y, between the systems of equations (1), (2) and (3); we shall then obtain for a, y, x in terms of a’, y’, 2’ and the three angles, the following expressions : w= sind sin @ + 2’ (cos@ cos yy — sin @ sin yy cos 8) — y (cos d sin + sin d cos yp cos 6). y= — x cos sind + 2 (cosy sind + sin yy cos d cos 8) — y (sin ¢ sin yy — cos @ cos xf cos 8). =x cos0 +a siny sin@ + y cosy sin 0. 94. If we dispense with the third transformation, we shall have Ow,, one of the new axes, in the plane of xy. The system Ow,, Oy,, Ox’, in which the new plane of wy is inclined at an Z @ to the primitive one, and their inter- section is the new axis of # inclined at an Z®@ to the pri- mitive axis of w, is for most purposes sufficiently general. The formule for it, combining the equations (1) and (2), and calling the new co-ordinates w’, y’, x’, are ! w= x cos — (y'cos@ — x’ sin @) sing, y = sind + (y’ cos 0 — x’ sin 8) cos , z=y sin@ +2’ cos@; which evidently agree with the formule of Art. 93, when we put WW =0. 95. In the preceding transformations we have supposed the origin to remain unaltered ; if, however, the origin is to be 45 changed, as well as the directions of the axes, we must employ the formule e=v' th, yay th, s=2" 4+), where h, k, ¢ are the co-ordinates of the new origin parallel to the primitive axes, and a”, y”, =” denote the values of xv, y, = found in each of the preceding cases. 96. It is important to observe that, in employing any of the preceding formule to refer an equation F(a, y, x) =0 to new axes, the transformed equation F'(a’, y’, x’) =0 will always be of the same degree as the primitive equation ; understanding by the degree of an equation the sum of the indices of the three variables in that term where it is greatest. For let this term be Aa? y! z'; then it becomes by substitution, A (ma’ + ny + 73")? (ma! + n'y! + 9's')1 (ma + ny! + 7" 2’); now these factors cannot furnish terms whose dimensions exceed p, gq, t, respectively; therefore if A’a'?’y’Vs'" be the term of highest dimension, p’ + q + ¢’ cannot exceed p+q+t; ie. the dimension of the transformed equation cannot be greater than that of the primitive. Neither can it be less; for if it could, then, as transformation of co-ordi- nates can never raise the degree of an equation, we could not return from F(a’, y’, s')=0 to F(a, y, %)=0, which is absurd. 5 97. To pass from one system of oblique co-ordinates to another also oblique. Let OM=2, MQ=y, QP=2x, (fig. 30) be the co-ordi- nates of a point P parallel to the axes Ow, Oy, Ox; and OM'=.2', M'Q’=y, QP=-2' the co-ordinates of the same point parallel to the axes Ow’, Oy, Oz’. Through the origin draw a normal ON” to the plane of wy, and on the same side of it as the positive direction of the axis of x; and from P, Q', M’ drop perpendiculars Pp, Q’n, M'm, upon ON"; "6 then PQcos N”Ox = Op = Om+mn+np = OM’ cos N" 2’ + M’Q’ cos N"y' + Q'P cos N"’2’, or xcosN” 2 = 2’ cos N”’ a2’ + y’ cos. N"y' + x cos N’’2’. Similarly, if we draw normals ON, ON’ to the planes of yz, xx, and on the same side of them as the positive directions of the axes of w and y, we shall have «cos Na = a cos Na' + y' cos Ny’ + 2’ cos N2’, y cos N’y = x cos N’2’ + y' cos Ny’ +’ cos N's Since cos VW" x = sin (x, wy), &c. these formule may be expressed without the aid of the auxiliary normals; but they are more convenient in their present shape, as the employment of negative angles is thereby avoided. 98. The position of a point P’ in space may be fixed by the following three variables; (1) the radius vector OP’=r (fig. 16); (2) the 2 COP’ = @ which the radius vector makes with the axis of x; (3) the 2 AOA’ = q which the meridian plane COP’ makes with the fixed plane COA. Of these angles the former varies from 0 to 180°, and the latter from 0 to 360°, in order that the radius vector may pass through all points of space. If #, y, x be the co-ordinates of P’, they can be expressed by means of r, 0, x2 for P’M = r cos 0, OM =r sin 0; . w= OMcosh =r sinOcosd, y=rsinOsing, and x =rcos8; and if these values be substituted for #, y, x in the rectangular equation to any surface, we shall obtain the Polar equation to the surface. Plane sections of surfaces. 99. In the discussion of surfaces, it is useful to know the nature and magnitude of the curves in which they are intersected by any planes. To do this, it is not sufficient to combine the equation to the surface F’ (a, y, ) = 0, with the equation to the cutting plane z = dv + By +c, so as to 77 eliminate one of the variables, x for instance; for the result f(a, y,) = 0 would represent only the projection of the curve required, which is not generally of the same nature with the section in space, nor sufficient to determine it. But if we transform the co-ordinates so that the cutting plane may be that of w’y’, and then put x’ = 0 in the resulting equation, we shall determine the trace of the surface on a’y’, i.e. the curve in which it is intersected by the proposed plane. And it may be here observed that, since the degree of an equation is never altered by the transformation of co-ordinates, if the equation to a surface be of the m' degree, the curve in which it is intersected by a plane cannot be of a higher order than the m™; but it may be of a lower if, by putting x’ = 0, we cause all the terms of the highest dimension to disappear from its equation. If we employ Euler’s formule for trans- formation of co-ordinates (Art. 93), since x’ is to vanish in the final result, we are at liberty to make z’ = 0 in the values of x, y, x before we substitute them, and so we may obtain the proper substitutions; these however, without being de- duced from the general case, may be readily obtained by the following independent method. 100. To determine the nature of the section of a surface made by any plane passing through the origin. The most convenient data for fixing the position of the cutting plane are (1) the 20 at which it is inclined to the plane of wy, and (2) the 2@ which its trace on that plane makes with the axis of #; these are readily obtained if we suppose the equation to the plane given; for-if the equa- 1 tion be s = Aw + By, then cos @ = tA, nad a Be (Art. 29), A Tht : and tan @ = — BR? Since Ax+ By =0 is the equation to the trace. Let «’ Oy’ (fig. 31) be the given plane, cutting the surface in the curve 4M, and the plane of vy in the line Ow’, which take for the axis of a’; and let Oy’, a line perpendicular to 78 Ow’ in the given plane, be the axis of y’, and OR=2’, RM=y’' the co-ordinates of any point M in the section re- ferred to the axes Oa’, Oy’; also let OQ=a, QP=y, PM = be the co-ordinates of the same point referred to the axes Oa, Oy, Ox. Join RP, then 2 MRP =O the inclination of the cutting plane to the plane of wy, and £wOza' = @ the angle formed by its trace on wy with the axis of &. Then PR =y' cos0, PM =~y'sin@; OQ = OR cos g + RPsingd, QP= OR sin ® -hPcos@; .e=y sind, v= cosp+y'cos6@ sin d, y =w sind — y' cos 0 cos d; and if these values be substituted in the equation F(a, y, s) =0 to the surface, we shall obtain a relation between a’ and y’ which is the equation to the curve 4M. 101. If the cutting plane pass through one of the co- ordinate axes, the formulee are simplified, and in many cases are sufficiently general. Let «’ Oy (fig. 32) be the cutting plane, passing through the axis of y; Oa’ in the plane of za the axis of «’, PM=2', OM =y, the co-ordinates of any point P in the section, ON=a, NQ=y, QP==s the co-ordinates of the same point. Join MQ, then 2PMQ= 90, and MQ =~2' cos@, PQ =. sing; . v= cos0, y=y, x= a' sind; which shew that to determine the section made by a plane through the axis of y and inclined at an 2@ to wy, we have only to write w cos @ and 2’ sin@ for w and x in the equation to the surface, without altering y; and the resulting relation between w’ and y is the equation to the curve. Similarly, if the cutting plane pass through the axes of s or w. 102. If the cutting plane does not pass through the origin of the primitive co-ordinates, or if we wish to take a 19 point in the cutting plane different from O for the origin of the new co-ordinates, we shall only have to add to the second members of the preceding formule the co-ordinates h, k, 1 of the new origin reckoned parallel to the primitive axes. 103. To determine the nature of the curve formed by the intersection of any plane with a surface of the second order. Let Aa’ + By? + Cx* = 2A'w be the equation, which may represent all surfaces of the second order; therefore, sub- stituting for w, y, x the values found in Art. 100, x= cosp+ y cos @sin @, y = v' sin pd — y cos 0 cos , s=y' sin 0, the result developed and arranged will be w (A cos’ p + Bsin’ p) + 2a'y' (A — B) cos Asin dh cos + y” (A sin® d + B cos’ P) cos’ 6 + Csin’ 6} = 24'u' cosh + 24’y' cos Osin d, the equation to a curve of the second order, which will be an ellipse, parabola, or hyperbola, according as the quantity (A — B)’ (cos @ sin d cos p)? — (A cos’ @ + B sin? ) x {(A sin’ + B cos’) cos’ 6 + C sin? 9}, or — AB (cos 0)? — AC (cos ¢ sin 6)” — BC (sin @ sin 6)” is negative, nothing, or positive. Hence every section of an ellipsoid is an ellipse, because all the quantities 4, B, C are positive. For a hyperboloid of one or two sheets, in which cases one or two of the quantities 4, B, C are negative, the section may be an ellipse, parabola, or hyperbola. For paraboloids 4 =0; therefore for the elliptic, in which case B and C have the same sign, the section is an ellipse; except when 6=0, or @=0, in which cases it is a parabola. For the hyperbolic paraboloid, 80 since Band C are of contrary signs, the section is a hyper- bola; except as before when 6 =0, or @=0, when it is a parabola. Cor. Since the section is referred to rectangular axes, it can never be a circle unless the coefficient of #’y’ vanishes, or (A — B) cos @ sin f cos h = 0; Tw Tv ; hence we must have ees or, pes or ¢=0; which shew that for a circular section, the cutting plane must be perpendicular to one of the principal planes of the surface. In the next article we shall see that this property is confined to one only of the principal planes of each surface. 104. Every surface of the second order, except the hyperbolic paraboloid, may be generated in two ways by the motion of a variable circle parallel to itself, the center of the circle moving along a diameter of the surface. Let the surface have a center, and let its equation be Au’ + By’ + C2’ = D. Since every circular section must be perpendicular to a principal plane, let the cutting plane be perpendicular to va, and inclined to wy at an Z @; and as the center of the circle must be in the plane of za, let x=h, x= be its co- ordinates ; then the equation to the section reckoned from that point as origin will be A(w cos0 +h)? + By’ + C(a’ sin 8 + 1)° = D, (Art. 101) which represents a circle if A cos’@ + Csin’@= B, or A + C tan’@ = B(1 + tan’), B-A i GER? or tan? @ and the relation between h and /, since the origin is the center of the circle and therefore the coefficient of wv’ = 0, is Acos@h4+ CsinOl=0; 81 therefore the locus of the centers of the circular sections is a straight line and a diameter of the surface. ; : D Ah?+Cr Also the radius of the section = Lot post BE Dh ae B B We must now examine, for each of the surfaces, which axis it is that coincides with the axis of y to which the cutting plane is parallel. 1 1 1. For the ellipsoid, A=, B==, C=5, tan @=+-— an @ 7 v= -. b lies between a and c, or the axis of the surface to which the cutting plane is parallel is its mean axis. 2. For the hyperboloid of one sheet, since we cannot ; 1 ] have B negative, we must put A=-, Bowe Gia aaa i fe pes a t Q = ck a CVE 3 an ayy 73 -.b>a, or the cutting plane is parallel to the greater of the real axes. 3. For the hyperboloid of two sheets, since we cannot have A and C negative, we must put 1 ] A=, Bora, Coss c G4 b* anes NAS 9? a 67 — Cc ..b>c, or the cutting plane is parallel to the greater of the imaginary axes. Since tan@ has two values, the cutting plane may be 6 82 inclined at an z@ or 180°—@ to the plane of wy, and hence the surface may be generated by a variable circle in two different ways; but it will be observed that in every case, if the surface become one of revolution, the two positions coincide in one which is parallel to the two equal axes. 105. Next, let the surface not have a center, and let its equation be By + Gz? =2A'x. Then as before, if h, 7, denote the co-ordinates of the center of the circular section, its equation reckoned from that point as origin will be By? + C (a’sin 0 + 1)? = 24'(a' cos + h), with the conditions B=Csin?@, Csin@.1= A’ cos@; ‘sing = + JZ. and 1 =~ cot 8. Hence the locus of the centers of the circular sections is a straight line parallel to the axis of the surface, 7. e. it is a diameter of the surface; also B and C have the same sign and B 6 ap wee ae J 1 a” +h b” ( a + b’ ( ) Now let P = perpendicular on ABC from O, OP=r, OK =71', p = perpendicular on PF’, p’ = perpendicular on GH; then ] 1 1 1 1 =" > ary ea ees hae Is ar , ee abide. if a bila 1 1 1 1 cos sin @\ ” ay oe “2 ae Ww a a “+ (=e - ®) 3 eee OF RY ce a b 1 1 1 1 - @Q) Now suppose r= f(@, 9) to be the polar equation to the surface. Then, considering @ as constant, r =f (qd, @) is the equation to the plane curve PM, and lst) dine! ( sa) p” gf °F yt dQ) ~ 94 Again, considering @ as constant, 2 = sin Of (@, 0) is the equation to the plane curve AL, and | 1 1 1 aa 1 1 A 1 eal peat ee ~ sin? @ |r? ot dd ; Hence, substituting in equation (2), we finally get 1 tv ft /dF\*> ‘cosec’’ Oo fare RM avi 68 Peet r iy ao r da an expression involving the partial differential coefficients of r = f(@, 0), where @ and @ are independent of one another. 117. The above expression may of course be obtained from the value of P given in Cor. 2, Art. 107, by changing the independent variables. 118. In some cases it is convenient to express the dif- ferential coefficient of the volume of a solid by the polar co-ordinates explained in Art. 98. Let ACP (fig. 39) be a curve surface; AP, A’P’, its intersections with two conical surfaces described about OC with semivertical angles 9 and 6+60; CP, CP’ its inter-— sections with two planes through CO inclined to sa at angles @ and @ + od; then it may be shewn as before, if V = vol. ACPO, that pyramid OPP = dV See Se as S00 dod therefore, taking the limit of both sides by making ods O80, vanish, in which case we may consider the base of the pyramid as coincident with the surface of a sphere, center O and radius OP, and therefore the area of the base will =Pp.P'p=rsin0dd.r00, and the vol. of the pyramid &e. ; will =—.r?sin@d60, we have 3 da’ V 1 dOdp 8 which may be also deduced from the result of Art. 112, by changing the independent variables. r sin @; 95 119. If in the above expression we substitute for r its value in terms of @ and @, and integrate with respect to 0, considering @ as constant, between the limits 9 =COG=F(¢), 0= COG = F,(), (these values being obtained from the given equations to the bounding conical surfaces BOD, B’OD’) we shall obtain a quantity, a function of @ only, which when multiplied by op is the ultimate value of the volume of the wedge GOH’ ; and this, integrated between any two values of @ will give the volume OBD' contained between two planes inclined at those _ angles to xv, and between the given conical surfaces and the curve surface. If instead of the vertex O, we suppose the figure to be bounded by another curve surface whose equation is x’ = f,(, 9), then the equation to be integrated will be dv 120. ‘To express the differential coefficient of the area of a curve surface by polar co-ordinates. Proceeding as in Art. 118, it may be shewn that if S' be the area of the surface APC (fig. bes we shall have ultimately area of surface PP’ = a consequently, if p be the length of the perpendicular dropped from O on the tangent plane at P, vol. of pyramid OPP’ = 4p (er . ultimately, d0do also = 4” sin @. od 60, ultimately ; iS sind ny .' : ‘ A 6); Dod =r A/ rsin'o (5) sint+ (— aa rt.116); Pp which may be also deduced from the result of Art. 113, by changing the independent variables. SECTION V. ON TANGENTS, AND NORMAL AND OSCULATING PLANES, TO CURVES, AND THEIR LENGTHS. 121. ‘To shew how a curve in space may be represented by equations. We have seen (Art. 44) that the equation F' (a, y, x) =0 represents a surface, and if to this we join another equation F’ (@, y, %) = 0 representing a second surface, and suppose the variables to receive only such values as satisfy both equations at the same time, we shall determine a series of points situated in each of the surfaces, that is, in the curve of their intersection. Conversely, as we have no other means of determining a curve in space than by assigning two surfaces each of which contains it, we cannot represent it analytically except by two simultaneous equations among the variables Vv, Ys 2. > 122, Among the various surfaces which may pass through a curve and so determine it, we employ for the sake of sim- plicity the cylindrical surfaces which are parallel to the co- ordinate axes, as their equations will contain only two of the variables. (Art. 43). Let QPR (fig. 43) be a curve in space, and through all its points draw perpendiculars Pp, Rr, &c. to the plane of vy (or, if the axes be oblique, parallel to 4z); the assemblage of these lines will form a cylindrical surface, called a projecting cylinder of QR, and meeting the plane of xy in the curve qpr, which is called the projection of QR on vy. Similarly, if we drop perpendiculars from all the points of QR on yz and za, we shall have two other projecting cylinders, and two other projections ; and the curve will evidently be determined if any two of its projections are given, for in that case two cylindrical surfaces will be given of which it is the intersection. Now 97 the projections qpr, Q'P’R’ are determined by equations of the form gi (av, y) = 0 ory=@ (2), Vi (8 &) = 0 or ¥ = Wy («), the complete signification of which, as we know, is the cy- lindrical surface erected upon each as its base. Therefore the curve QPR will be determined by the system of simul- taneous equations y= (x), x=W(a). In these, only one of the variables, w for example, is arbitrary ; and any assumed value of w joined to the corresponding values of y and x derived from them, will belong to a point in the curve. The equation to the projection of QPR on the plane of yz, is deduced from the other two by eliminating 2. Hence it appears that every curve in space may be con- sidered as formed by the intersection of two cylindrical surfaces erected on its projections as their bases, perpendicular to the co-ordinate planes. | 123. To find the equations to the line which touches a given curve at a proposed point. Let PT’ (fig. 43) be a tangent to the curve QPR at a point P; also let qpr be the projection of the curve on the plane of wy, and pé a tangent to the projection at p; we must first prove that p¢ is the projection of PT. Let R be a point in the curve near to P, r its projection; draw the lines PR, pr; then pr is the projection of PR, and continues so, however near & approaches to P, and consequently r (which is always in a line through # parallel to Pp) to p. ‘Therefore in the ultimate positions of the lines, when R co- incides with P and r with p, and they become tangents at P and p, pt is the projection of P7'; that is, the projection of the tangent at any point coincides with the line touching the projection of the curve at the corresponding point. The same is of course true relative to the other co-ordinate planes; if therefore y = d (vw), s = (a) be the equations to the pro- 98 jections qr, Q’R’, the equations to the lines pt, PT’, or to the line P7' of which they are the projections, will be dz d s (w’ —#), x -—s= oT v— x); y -y= v, y, % being the co-ordinates of the point of contact, and x, y’, x those of any point in the tangent, and the co- ordinates being either rectangular or oblique. 124. The two equations to the curve may evidently be supposed to arise from the elimination of ¢ between equations of the form « = f(t), y=fi (4), 2 = fo (t); consequently, when the co-ordinates of a point in the curve are all regarded as functions of the same variable ¢, the equations to the tangent line will take the symmetrical forms dz dy dg dt dt dt Or if the equations to the curve, instead of having the explicit forms assumed in Art, 123, should be Mie VAG YS) = 0, We, et, (@, Ye) 0, then considering y and x each as a function of wv, we have du Hi dudy dwdz _ dw dydzx A dzduv du, du,dy du,dz aye Ce a d ’ : from which =; pe may be obtained and substituted in the v dx equations to the tangent line. b) e 9 125. If the tangent to a curve makes angles a, (3, y with the axes of aw, y, #, since its equations, taking the pro- jections on the planes of x” and yx, are 99 dy , 1 , , da i eae: Fa ©) Vom Ue (Fae ®)s dx dx dy 1 dav cosa LS eee Vise (Ge) Me) +() a dx} ' \dwx i dx +(= dx COS? = 1+ (2) +(Z) 4 dx dx 126. To find the equation to the normal plane to a curve at a proposed point. A curve can have only one tangent at a proposed point, but it may have an infinite number of normals, that is, of lines perpendicular to the tangent through the point of con- tact ; these all lie in one plane called the normal plane. Let x, y, be the co-ordinates of the proposed point of the curve, ‘then since the normal plane passes through that point, its equation will be vs’ —-2=A(e'-a)+ By -y); and since it is perpendicular to the tangent whose equa- tions are dy 1 , , d. x am (s — 2), Yonsself amar (Ham 8)s da dx dy i dz A+=—==0 =ume = 0) ; Pr 4 HER ees da dx 100 hence, substituting for 4 and B these values, the equation to the normal plane at a point wy%, is : , dy ? dz v—at+y FU) et TAS sn ti Baim Cor. If a, y, x be each regarded as a function of the same variable ¢, as in Art. 124, the equation to the normal plane takes the symmetrical form dx dy dz a aki by inks Vee N iseadeeeiye Dre) fay gl ee ey pinto Pa) tay 127. To find the equation to the osculating plane to a curve at a proposed point. The osculating plane at any point is that which has a closer contact with the curve than any other plane passing through the same point. Let «#, y, x be the co-ordinates of the point of the curve ; then the equation to a plane passing through it will be ACK Saye BY Syl Ze oF =) let Oo be the length of a perpendicular let fall on this plane from a contiguous point in the curve whose co-ordinates are wth, y+kh, x4+1; Ah Bk ail \/ A + B 4 C dx dy ‘dz ad’ x CY 6a pelea ty a cS) +}3( 4 Bey cS) ae ( Ai cdi) \ al li, Ge en tp (Cor. Art. 37). 3 if we suppose the points consecutive, and substitute for h, k, l their developements in series ascending by powers of 7; + being the increment of ¢ the variable of which the three co-ordinates are assumed to be given functions. Now if we determine the constants so that the coefficients of + and 7* 101 may vanish in the numerator of 6, the portion of the curve immediately contiguous to the point wysz will coincide more nearly with the plane so determined than with any other plane that can be drawn through that point; and as only the ratios of A, B, C to one another are required, this can be effected by establishing the two conditions dx dy dz dx Ei d’ — a —— =0; 4A — =| Bri aeat) satin’ ae dae ede eae whence eliminating A, B, C between these and equation (1), 2) 2 &e., by a’, 2”, &c., we get di and denoting a EE (y/'2"—2!y")(X—a) + (w/a""—a'e”)(Y-y) +(a'y"-y'a")(Z—2) =0, the equation to the osculating plane at a point wyz. Cor. If we assume y and g to be functions of a, in the ordinary way, then d & d & Ah+B/( ella ct 1) 40(4= + 4o + Be) s_ dx dx dx dx” (2) / A+ B+ C dy d’y ad? z oA — t= Oy ——- — = eer cS aE ere et and the equation to the osculating plane becomes d’y Gydzs dzdy oa rere ora which evidently agrees with the above, when & = ¢. as — a)+ agit? - Y), 128. In order that the contact may be of the third order, the coefficient of h* in the numerator of 6 in equation (2) d? zx d’y must also vanish, or Cis + Lister =e ae eT SE A eT dx’ da da’ da’ : the condition which must be satisfied at any point of the curve where the contact is of the third order. When the contact is only of the second order, 1 We Bs _ dy bee ty" ily Ro ge J A? + B+ C {3 (Cas ss ia) ate 4 } a quantity which changes its sign at the same time that h does ; consequently the curve generally cuts its osculating plane. On the contrary, when the contact is of the third order, the sign of 6 does not change with that of h; and the curve is said in such cases to have a point of inflexion. The condition then of there being such a point is asdy ay ds eS ea ee SSS — _— da’ dx*® da? dx’ 129. When a curve in space is a plane curve, if w, y, = be the co-ordinates of any point in it, they will always satisfy the equation to a plane v= Aw + By’ +ce...(1), so thats = d~e+By+e; therefore the differential coefficients of = and y will be such as to satisfy the equations dz iy dy ds d°y a dv dx dx dx®’ A and B denoting constant quantities ; ad’ 2 d [ dx’ Tyds asd'y ed a\ ai) Esai a Me eae ee dx* which must be rendered identically true by the equations y=@(v), x= \(v), when they represent a plane curve; 103 otherwise, the curve is of double curvature. This condition being satisfied, if we substitute for 4A, B, ¢ their values in (1), we find the equation to the plane in which the curve is situated the same as that to the osculating plane, as might have been foreseen; for when a curve in space is a plane curve, the plane in which it is situated is the osculating plane at every point. Hence when the above condition is satisfied, the equation to the plane in which the curve is situated may be obtained by writing for the differential coefficients their values in the equation to the osculating plane. If we assume each of the co-ordinates vw, y, x to be a function of the same variable ¢, a symmetrical expression for the condition (2) may be easily obtained by eliminating the constants from the three derived equations of lw + my+nzx=c. 130. The osculating plane at any point of a curve in space is perpendicular to the line of intersection of con- secutive normal planes at that point. The equation to the normal plane at a point wyz is dy sy ds ve — Ut CRE mae 28D) Seye & E : ; dy dz and at the contiguous point 7+h, y+ he +&c., 2 the + &e. it 1s ’ dy dy d*y wa -he(y -y- hg te.) (+ + hot 4 &e.) d dz d® + (¥ ~# ~h— — &e.] (= + a + Be.) = 05 or, combining it with the former and dividing by 4, it is ; d’y dy d’s dz\? s Bet ET ge (3) + apn (ag) ttm h, h*, &c. = O. 104 Therefore, making 4 = 0, the equations to the line of inter- section of consecutive normal planes are OY dz w-—at+y a dean ne my eye 2 2 2 2 oF peo Cees) be (=) 5 (=) hee Os dx” d x” dx dx (the latter equation being evidently that which results from differentiating the former with respect to #, considering «’, y’, %’ as constant) and if we deduce the equations to the pro- jections of this line on the planes of yz and ga, it will be seen that the conditions of being perpendicular to the os- culating plane (Art. 25) are fulfilled. 131. To find the differential coefficient of the length of the arc of a curve in space. Let y= (2), * =wW («) be the equations to the curve, 2, y, % the co-ordinates of any point P in it (fig. 43), at+h,y+hk, 2+, the co-ordinates of a contiguous point R, s =length of arc QP, Q being a given fixed point in the curve, and &R = chord PR; then R?=7? +447 | dy hd’ dz hh dx ans (A 4 = <7 4 &e V4 + (Aa at &e.] da rd dx da 2 dx dx’ dy\? dz\* = h? {1 a : yes + (2) + oy fan + &c DSN He... R a/ dy. * dz\? , — = | fete — —— ae ay Imit o h 1 + Gy ae (= Cor. Hence the formule for the angles a, 3, yy, which | a tangent at the point wys forms with the axes of a, y, 2, may be written dy dz dax da cosa=—, cosB=——, cosy =—; dw dv dn or, supposing s to be the independent variable, dxv B dy dz a oe a eS a) co = —— COAG = ——s COs rF sty 132. If we employ polar co-ordinates so that x =r cos 0, # =rsinOcosd@, y =17 sin @sin @, then d ; dr\* : d d\? ie / + (=) + r’sin® @ (52) : ds r Also, —- = ————, p being the perpendicular on the dr re p tangent line at the extremity of r. SECTION VI. ON THE DISCUSSION OF THE GENERAL EQUATION OF THE . SECOND ORDER. 133. To find the position of the center of any surface. The center of a surface is a point O (fig. 22) such that any chord of the surface PP’, drawn through it, is bisected in it. (It must be observed, however, that if PP’ cut the surface in more points than two, it would be sufficient that these points combined in a certain order should be, two and two, equally distant from OQ). If the surface be referred to any three axes originating in O, and PM, P’M’ be the ordinates parallel to Ox of the extremities of a chord, we see from the equal triangles POM, P’OM’, that these ordinates are equal and of contrary signs ; the same thing would be true for the other co-ordinates of P and P’, as well as for every other chord passing through O. If therefore f (xv, y, x) =0 be the equation to the surface, and if it be satisfied by 7 =a, y= 6b, x =c, it must also be satisfied by # = —a, y= —b, s = —¢; that is, it must be such as not to alter when the signs of the three variables are changed ; and, conversely, if it have this property, the origin is the center of the surface. When f(a, y, %) = 0 is algebraic, it cannot have the above property unless the dimension of every term be even in an equation of an even degree, and the dimension of every term be odd in an equation of an odd degree; for in the former case the equation is not at all altered by replacing vw, y, s by — a, — y, — #3; and in the latter case (in which the equation cannot have a constant term) the sign of every term will be altered, and therefore the whole equation unaltered. 107 Hence, to find whether a proposed surface admits of a center, we must refer it to parallel axes, through a new origin having co-ordinates h, k, 1, by putting eaa't+th, yay +k, x= +], and equate to nothing the coefficients of all the terms which are of a dimension different (as far as regards odd and even) from the degree of the equation; if these conditions can all be satisfied by real and finite values of h, k, 1, the surface has a center, and h, k, / are its co-ordinates; in the contrary case the surface has no center. 134. To find the co-ordinates of the center of a surface of the second order represented by the general equation of the second degree. The general equation of the second degree is FS (&, Ys 8) = an? + by? +02? + 2a yx +20 su+ 2c uy + 2a" e2+2b"y + 2c"s+d=0. Here we must make v=a’ +h, y=y +k, s=2'+1, and equate to nothing the coefficients of terms of odd dimensions, that is, those involving a’, y’, 3’. The result of these sub- stitutions is S(@ +h, y +k, 2 +0, which when expanded will consist of three parts, (1) terms of two dimensions which must be the same as those of FT (2, y’, #’) which are of two dimensions; (2) terms of one dimension which are va f(y i). df (h, bl). dfeecan CMMNM ied aa) as Stas each of which must disappear; and (3) a constant term f(h, k, 1); therefore the result will be aw” + by” +62" 420'y'x' + 2b sv + 2c uy +f (h, kh, D =% 108 df (hs k; !) dh with the conditions =0, &c., or ah+bl+ck+a"=0 bk + al+céh +h’ =0 wl)» cl+ak+bh+c’ =0 for determining h, k, 7. Multiplying the two latter equations respectively by indeterminate coefficients ¢, uw, and adding them to the former, we have (a+te +ub)h+a"+tb’ + uc’ =0, provided c +tb+ua'=0, 6 +ta'+uc=0; these two latter equations give ¢ and w, and then substituting in the first we find h = = where D=aa’ + bb” 4+ cc” —abe -2a'l'e," N=a’ (bc — a”) +b" (vv — cc) +0" (ac — bb); aN Wve similarly, & = —, / = _. y> D’ D Hence, provided D is not = 0, these values of h, k, 1, which are always real, are finite; and the surface will have a single center of which they are the co-ordinates, and its equation reckoned from the center as origin will be, suppressing accents, ax’ + by? +cex +2a ys + 2b ea4+2ay+ah+b'k aa cl+d= 0% for, multiplying equations (1) respectively by h, k, 7, and adding, we have | fi, ky Dl) = ah WK +014 d. 135. If the constant term f(h, k,l) disappears, the surface represented is a cone, since if we combine its equation with the equation to a plane through the origin s = 4a + By, 109 & the result will be of the form y = a (p + /q) indicating two straight lines through the origin; unless the radical be impossible for all values of 4d and B, in which case the pro- posed equation represents a point. If the coefficients of the given equation be such that D=0, and the three numerators are not all =0 at the same time, then one at least of the co-ordinates of the center will be infinite, which signifies that the surface has no center. If at the same time that D =0 the three numerators vanish, then the surface admits of an infinite number of centers; for in that case the three equations (1) are reduced to one, or to two realiy distinct equations, as is shewn in most treatises on Algebra, and may therefore be satisfied by an infinite number of values of wv, y, x. If they are reduced to two, that is, if the values of h and k deduced from the two first for instance, satisfy the third whatever / be, then there will be an infinite number of centers situated in a straight line which is the locus of the two independent equations; the surface will therefore be a cylinder on an elliptic or hyperbolic base. If the three equations (1) are reduced to a single equation, that is, if the value of # deduced from the first, for instance, satisfies the other two whatever k& and 7 be, there will be an infinite number of centers situated in a plane which is the locus of the single independent equation, and the proposed surface will be a system of two planes parallel and equidistant from that plane. In this latter case the proposed equation must be capable of being resolved into two rational factors of the first degree. 136. The locus of the middle points of a system of parallel chords of any proposed surface is called its dia- metral surface. ‘This surface will have several sheets, if each of the chords has more than two points in common with the proposed surface; if, for instance, the proposed surface be of the m™ order, the points of intersection with 110 its chords, real or imaginary, will be in number m, and their combination on the same indefinite line will form in(n-—1) different chords, and as many middle points; and therefore the diametral surface, since it may be met by an indefinite line in $2 (2-1) points, will have an equation of the degree $(m—1). For surfaces of the second order where n= 2, the diametral surfaces can only be planes. When any surface admits of a diametral plane, if we make it the plane of wy, and take the axis of x parallel to the chords which it bisects, the equation to the surface, for every pair of values vw =a, y = 6, must furnish for x values which, taken two and two, are equal and of contrary signs; and therefore the equation, supposed algebraic, can only involve even powers of x. And, conversely, whenever an equation contains only even powers of one of the variables, x for instance, the plane of wy is a diametral plane, and is said to be conjugate to the chords parallel to the axis of zg. Also if a diametral plane be perpendicular to the chords which it bisects, it is called a principal plane, and the chords principal chords. Moreover the intersection of any two diametral planes is called a diameter of the surface. 137. To find the equation to a diametral plane of a surface of the second order. Let v=mz, y=ns, be the given equations to a line through the origin to which the proposed system of chords is parallel, and f(a, y, 3) =0, the general equation of the second degree, the equation to the surface. Let h, k, 1, be the co-ordinates of the middle point of any chord, and let the surface be referred to axes parallel to the former ones passing through it; then the equation will become S(a@ +h, oy +k, x +20) =0, and the equations to the chord itself will be a = mz’, y’ =n; therefore the values of x’ belonging to the points 111 of the surface where the chord meets it, are given by the equation f(mx' +h, nz’ +k, x +1) =0......(1), which is of the form Rz? + S2'+T=0; and since the values of x’ are equal and of opposite signs, S=0. But Sis the coefficient of x’ in equation (1) ; mdf (hy ky 1). df (hy yD af (hy hy 2) dh eens S/%) Mark Soni lee Whe uae or mM(ah+Ulock+a")+n(bk+al+ch +b") tel+avk+h +c" =0...... (2); or (am+cen+b)h+(bn+em+a)k+(c+0m+an)l +am+b’n+c" =0, the relation among the co-ordinates of the middle point of any chord, or the equation to a diametral plane. Cor. As the coefficients of h, k, J are possible, there will be a diametral plane for all values of the constants m and n, unless the three coefficients should all become nothing at the same time, when the diametral plane will be situated at an infinite distance. But the equations am+cn+b =0, bn+cém+a=0, c+bm+an=0, containing only two unknown quantities, have an equation of condition which is the same as D = 0 (Art. 134); in this case therefore the surface has not a center. When the surface has a center, every diametral plane passes through it, or through the locus of the centers; for equation (2) is visibly satisfied by the co-ordinates of the center furnished by equations (1), (Art. 134). This also follows from the definition. 112 138. Any diametral plane of a surface having a center is parallel to the tangent plane applied at the extremity of the diameter to which it is conjugate. Taking the center for origin, the equation to the surface will be ax’ + by? + cx? +20 yx 4+ 202742 xy+d=0; therefore the equation to the tangent plane at a point vyz is (Cor. 1, Art. 107), (av + bx +c'y) (# -— 2) + (byt+ax4+ec'x) (y’-y) + (cx +a'y4+0'2) (x — 2) =0, or (av +Uxtcy)a+(bytaxstea)y+(extay+d'x)2 +d=0; and if it be applied at the extremity of the diameter whose equations are v = mz, y = nz, the equation becomes / ] , / 4 / , d (am+cen+0')a'+(bnt+em+a')y + (c+ bm +a‘n)z'+-=0, R and therefore (Art. 18) represents a plane parallel to the diametral plane which is conjugate to the diameter x = mz, y= nz, the equation to which is (putting a” = 6b” =c”’ =0, in the equation Art. 137) (am+cn+)a + (bn+emta)y+(ce+ Um+a'n)x'=0. This result might have been foreseen; because the straight lines in which a diametral plane and a tangent plane at the extremity of the conjugate diameter are cut by any plane through that diameter, must, by the nature of lines of the second order, be parallel to one another. 139. We shall now proceed to the reduction of the general equation of the second degree ax? +by+ex°+2a'yxt2bear2cuvy+2ax42b"y4 2c x+d=0, f 113 where we suppose the co-ordinates rectangular; for if they were oblique, by transforming them to rectangular co-ordinates we should obtain an equation of the same degree as the above (Art. 96), and which could not therefore be more general than the one which we have assumed. We shall prove, as affirmed at Art. 53, that this equation, after being simplified as much as possible, will always assume one or other of the forms Ax’ + By? + Cz’ =D, By? + C2? =2A'e, the co-ordinates being rectangular; and therefore the general equation of the second degree can never represent any other surface than one of those discussed in Section 2. 140. Every surface of the second order has at least one diametral plane which is perpendicular to the chords bisected by it. Let the equation to the surface be ax’ + by? +c2°4+2a'ys+2b'za4+2cvyt+2a vt+2b" y+2c'%+d=0, and v=mz, y= nsx, the equations to the line to which a system of chords is parallel; then the equation to the plane which bisects the chords is (Art. 137) (am+en+b)xut (bnt+em+a)yt+(c+hm+a'n)s ed Meh +o = 0. and our object is to shew that real values can be assigned to mand nm, such that this plane shall be perpendicular to the chords. The conditions for this are (Art. 25) am+en+b bn+em+t+a ee ee OE wat te , 9 c+bm+an c+bm+an’ from which, by eliminating one of the unknown quantities m or n, we shall obtain a cubic equation which will always give a real value for the other; and the direction of the S~ 114 system of principal chords will be determined. But we shall obtain a more symmetrical result by assuming for the un- known quantity c+bm+an=s, ors—-c=bm+an, then am+cen+b'=ms, or m(s—a)=cn+b e+ (1): bn+cm+a=ns, orn(s—b)=cm+a Hence, determining m and 7 from the two latter equations, m {(s—a)(s—b)-c*} =U (s—b)+ ae nm {(s —b) (s—a) —c”} =a'(s —a) + yap And substituting for m and m in the former, we have (s —a)(s— b)(s —c)—a*(s —a) —b?(s — b) —c?(s—c) -2a'b'c'=0, or 8 —(a+b+c)s°+ (ab+ ac+be-a’—b’—c")s | — (abe —aa*— bb”? — cc? + 2a'U'c’) = 0. This equation, being of an odd degree, will always have one real root which substituted in (2) will give real values for m and»; and therefore in every surface of the second order there is at least one principal plane, or, which is the saine thing, one system of principal chords. Also there can- not be more than three, unless the particular form of the proposed equation of the second order should render any two of the equations (1) identical, in which case m and nm would be indeterminate, and the number of principal planes would be infinite. Cor. ‘That the cubic has all its roots real, may be shewn by putting it under the form (s—c) }(s—a)(s—b)—c} — $a"(s—a)+b°(s—b) +2a'b'c? =0, and substituting for s, a and ;3 the roots of (s—a)(s—b)-—c?=0. The results of these substitutions, since (a ~ a) (a — b) = oF. (a = B)(b- B) =e, are we fa'\/a —-axt ba — 6}? and + fa'\/a — B + b'\/b — B}*, 115 for upon solving the equation (s — a) (s — b) = c”, it is easily seen that one of its roots a is greater than both a and 8, and the other 8 is less, Therefore since +, a, 3, — 0 when substituted for s give results +, —, +, —, there is one root greater than a, another between a and #3, and a third less than p. 141. Every variety of surfaces of the second order referred to rectangular co-ordinates is comprehended, without exception, in the equation Av’? + By + Cx? +24e4+2By4+2Cxe+D=0. Let the surface be represented by the general equation of the second degree f(a, y, s) =0, and let it be referred to three new rectangular axes Ow’, Oy’, Oz’, by the substitutions of Art. 90, and let the transformed equation be Ax’ it By”? i Cz” + 2 A'y’s! ate 2 RB’ =x’ ae 2 C'a'y’ 4 2 A" a +2B"y' +2C"s'+D=0; also let the equations to a line parallel to a system of principal chords (the existence of which in every case is certain) referred to the new axes, be a’ = m3’, y’=nz’, then m and n satisfy the conditions Am+Cn+ B=m(C+ Bm+ An) Bn+Cm+A=n(C+Bm+ An). Suppose now one of the new axes, that of s' for instance, to be parallel to the direction of the principal chords; therefore m=0, n=0; consequently we must have A’=0, B’=0. Hence whenever one of the rectangular axes is parallel to the direction of a system of principal chords, the general equation is freed from two of the rectangles, and takes the form Aw? + By? + Cx? +2C'a'y'+2 A"a' +2 B’y' +2C 2’4+D=0...(2). This equation comprehends all surfaces of the second order without exception, and it may be still further reduced if, without altering the axis of x’, we turn the axes of a’ and y/ in their own plane through an angle @, so as to make the term involving ay’ disappear, by the substitutions (Art. 94.) 8—2 116 a’ = x, cosg — y, sin ®, y =a, sind + y, cos d; this gives — 2cos@ sin (4 — B) + 2C’ Scos* ph - sin? hb} = 0, 2C" tan2@ =——_., or tan2@ Gap which value, being real and always admissible even when A = B, shews that equation (2) may be reduced to the form (suppressing accents) Aa’ + By + C+ 2Ae+2By+2C2+ D=0; which (the co-ordinates being rectangular) comprehends all surfaces of the second order. 142. It is at this point of the reduction of the general equation of the second order, that the separation of the sur- faces represented by it into two classes takes place. 1. If none of the squares of the variables are wanting in the equation Aa’? + By +024+2Ae+2By+2C2e+D=0, i.e. if A, B, C are all different from zero, by writing eth, y+k, =+1 for x, y, x and determining h, k, 1 by the conditions Ah+A=0, Bk+B=0, Cl+C'=0, (which will give finite values for h, k, 1) the origin will be changed, but the directions of the axes unaltered, and the equation will be reduced to the form Av’ + By? + Cz? =D... (1) which represents the first class of surfaces of the second order. 2. If one only of the squares is wanting, A for instance, and the corresponding coefficient A’ is different from zero, then the term involving w cannot be made to disappear, for that would give / infinite; but instead of that we may deter- 117 ‘mine f# so as to exterminate the constant term, and determiné k and 7 by the same conditions as before, and the equation will be reduced to the form Bye OS = 2A (2) which represents the second class of surfaces of the second order, ‘The figures and properties of these two classes of surfaces have already been discussed, (Art. 53). 143. There are still two particular cases to be examined, belonging, as we shall see, to the two sorts of cylindrical surfaces of the second order. First, suppose that the coefficient of one of the squares, A for instance, vanishes, and at the same time the correspond- ing coefficient A’=0, without either of the other squares dis- appearing; the equation is then By’ + C2*4+2By4+2C2+D=0, which, containing only two of the variables, represents a cylinder perpendicular to the plane of yz on an elliptic or hyperbolic base; and if the center of this curve be taken for the origin, the equation will be reduced to the form By iG ei which is deducible from equation (1) by making 4 = 0. Secondly, suppose only one of the squares to remain in the equation, x? for instance; then writing »+/ and a+h for s and #, and determining # and 7 so as to exterminate the term containing x, and the constant term, the equation becomes C#+2Aa7+2B y=0; now, without altering z, turn the axes of w and y in their own plane through an angle @ such that 4’ sin p = B' cos , then the equation will be reduced to the form 118 which represents a cylinder perpendicular to the plane of wy on a parabolic base, and is deducible from equation (2) by putting B= 0. It is unnecessary to examine the case where the three squares disappear, as no equation of the second order could ever by transformation of co-ordinates be reduced to that form; the cases in which it represents a cone, or two parallel planes, have been noticed. 144. Hence, we conclude, that all surfaces of the second order with all their varieties, are comprised in the two classes represented by the equations Ax’? + By + C2 =D, By? + C2? =2Aa, the co-ordinates being rectangular. In the first class, each of the co-ordinate planes is a principal plane; therefore there are three systems of principal chords, and consequently the three roots of the cubic in (Art. 140) are all real. In the second class, only the planes of s# and wy are principal planes; therefore there are at least two systems of principal chords; the third system, determined with the others by the cubic equation just referred to, must therefore be also real, but the corresponding principal plane is situated at an infinite distance. Hence it results from this discussion, as well as from the form of the equation (as proved in Cor. Art. 140), that the three roots of the cubic equation which determines the positions of the principal chords are always real. 145. By means of the preceding results, we are able to determine the species and form of the surface represented by any proposed equation of the second order, without recurring to the laborious process of transformation of co-ordinates. We must first ascertain whether the surface has a center by the method of Art. 134, and if it has, find the co-ordinates of the center h, hk, 1 from the equations which result, by equating to nothing the three derived equations of the first ——— 119 order, w, y, # being replaced by h, k, 1; then taking that point as origin, the equation is reduced to aa’ + by? +c3x?4+2ayx4+2bsa+26ay4+ah+b"k+c'l+d=0, from which the species of the surface and the values of the real or imaginary axes can be readily determined, as will be seen in the next Art. In the succeeding Articles, the charac- teristics of the different sorts of surfaces that have not a center will be deduced. 146. The equation au’ + by? + cz? + 2a ys + 2b ea + 2uvy =d, (d being a positive quantity) belongs to an ellipsoid, hyper- boloid of one sheet, or hyperboloid of two sheets, according as the cubic equation (s —a)(s—b) (s —c) —a?(s — a) —b?(s—b)-c?(s—c) -20'b'c'=0, gives for s three positive values, two, or one. Let w=mz, y=nsz, be the equations to a line through the origin, which will be a diameter of the surface, since the origin is the center, and let the length of the diameter be denoted by 27; then the chords parallel to it will be bisected by a plane, of which the equation is am+cen+b)a+(bn+emt+ta)y+(c+dm+an)s=0; Yy and if this plane be perpendicular to the chords, or if 27 be a principal diameter, making c+ bm+an=s, am+en+b=ms we must have }..0) bn+cecm+a'=ns, from which eliminating m and 7, as in Art. 140, we find (s—a)(s —b)(s—c) — a? (s— a) —b?(s—b)—c? (s—c) -2a'b'c'=0; 120 also multiplying the two latter equations by m and m respec- tively, and adding them to the former, we find s(1 +m? + n*) = am? + bn? 404 2a'n + 2b'm + 2Cmn. But if w, y, x be the co-ordinates of the extremity of the diameter, 9 ~ ra eP+ypPrs= (1+ m +n’) 2; also since wv, y, % are co-ordinates of a pcint in the surface, 3 (am? + bn? +c0+2an+2bm+2cemn) =d; d d 2? Pim Ss tw) ae y = or r = 3° x —.sl+m+n')= s ee Ss Now d is a positive quantity, therefore according as s has three positive values, two, or one, r has three real values, two or one, or the surface has three real principal diameters, two, or one, and is therefore an ellipsoid, hyperboloid of one sheet, or hyperboloid of two sheets. 147. Hence, when we have an equation of the second order reckoned from the center, and the constant term forms one member of the equation and is positive, if with the numerical coefficients we form the above cubic, and observe the signs of its terms, we shall determine the nature of the surface represented by the proposed equation. For since all the roots of the cubic are real, there will be three positive roots, two, or one, according as there are in it supposed com- plete, three changes of signs, two, or one. If the cubic offers no change of signs, but only continuations, its three roots are all negative, and the three axes of the surface are imaginary, and consequently the surface itself imaginary. Cor. If we solve the cubic, we may then easily calculate the lengths of the axes, and fix the position of each by means of the values of m and n, corresponding to each value of s, given by equations (1). Ex. 1. 2a°+5y°+32°42yx%-420-20y+2u+8y-624+8 =0, 121 Forming the three derived equations and replacing a, y, z by h, k, J the co-ordinates of the center, we get Diy ra FEA a Te Bo as: 10k +21 —2h+8=0,> which give = —1, 614+2k-—4h—6=0, b= 2. Then the equation, referred to the center, and the cubic for determining the axes, become respectively 20° + by? + 32° + Qyz—430 —-Qwy=—-14+446-8=], s§—10s°+ 78+ 388 =0; and as this complete equation has all its roots real and has two changes and one continuation of sign, it has two positive roots and one negative root, and the surface is a hyperboloid of one sheet. Ex. 2. v0 +2y? 4+ 3274+ 2yx4+22u+4+2uyt+ut+y+s=1. The co-ordinates of the center will be found to be h= — i, b=0, 1=.0; and the equation referred to the center becomes a+ 2y + 32° 4+ 2ye + 2Q2u + 2aHy =F. Hence the cubic becomes 8 —6s?+8s—2=0, having three real roots; and as it is.complete and has three changes of sign, it has three positive roots, and therefore represents an ellipsoid. Ex.3. ystxuv+ay=a’. Here the cubic is s* — 3s — 2 =0, which has one positive root and cannot have more; therefore the surface is a hyper- boloid of two sheets. | Ex. 4. «4+ 2y°+3ys-2ey-6x+7y+624+7=0. 122 The co-ordinates of the center will be found to be h=1, k=—2, t=, and the equation referred to that point uv +2y°? + 3y% —2xy =0, wanting the constant term; and as it is satisfied by w =0, y =0, it contains the axis of x, and represents a cone and not a point. 148. When the proposed equation axv+by+ex 42a yxet+2b su42Cay+2a vVt+2b y+2c' zx +d=0, is such that abe — aa® — bb”? — cc” + 2ab'c = 0, the surface represented either has not a center, or it has a central line or plane (Art. 135), and admits of four varieties, viz., paraboloids, parabolic cylinders, elliptic or hyperbolic cylinders, and a system of two parallel planes; for each one of which, certain conditions must be satisfied which are not all true for the others. Hence we can determine the dis- tinctive characteristics of each of these surfaces. 1. For paraboloids, since the co-ordinate planes cannot be all parallel to the axis of the surface, and since it is only such planes which intersect the surface in a parabola (Art. 103), one of the quantities ab, (1) (which determine the nature of the traces on the co-ordinate planes) must be different from zero, and negative in the case of the elliptic, and positive in that of the hyperbolic para- boloid ; also one at least of the co-ordinates of the center will be infinite, and therefore one of the equations for determining those co-ordinates imaginary. a*—be, b?-—ac, c 2. For parabolic cylinders, since all sections are either straight lines or parabolas, each of the quantities (1) vanishes, 123 and one of the equations for determining the co-ordinates of the center will be imaginary. 8. For elliptic or hyperbolic cylinders, the equations for determining the co-ordinates of the center are reduced to two distinct equations, and one at least of the quantities (1) is different from zero, and is negative in the former, and positive in the latter case. Also, by taking sections parallel to the co-ordinate planes, it will be necessary further to examine whether the cylinder is wholly imaginary, or reduced to a line, or to a system of two planes not parallel. 4. For a system of two parallel planes, the equations for determining the co-ordinates of the center must be reduced to a single distinct equation, and the three quantities (1) must each vanish. It will be further necessary, by taking sections, to examine whether the two planes are confounded in one, or are imaginary. Ex.1. a’ -—2y’?-— 3y2 4+ 32a + ay +42 =0. The equations for determining the co-ordinates of the center are 20+3%3+Yy=0, —4y—-32+4+07=0, —3y+ 3%+4=0; and since the last subtracted from the sum of the two former gives 4=0, this impossible equation shews that the surface has not a center. Also, c? — ab = (4) + 2, which is different from zero and positive; therefore the surface is a hyperbolic paraboloid. Ex.2. a? +y? +92" + 6y2 — 62a —2ay + 2u—4z% =0, The three derived equations are ] 2e —-62—-2y+2=0, 2Qy + 6% —2e@ = 0, 18x + 6y — 6% —4=0, 124 the two former of which give 2=0, therefore the surface has no center; and the traces on the co-ordinate planes cannot be ellipses or hyperbolas, therefore the surface is a parabolic cylinder. Ex.3. #® -y-24+2yx+e+y—-x =a. The derived equations are 2%+1 =u), —-2y+22+1=0, which are reduced to two distinct equations; also the traces on wy and ws are hyperbolas; therefore the surface is a hyperbolic cylinder; except when a@=0, when it represents two planes intersecting in the axis of the cylinder whose equations are 2v+1=0, 2x -—2y+1=0; for when a=0, the equation is resolvable into (w + y — 2) (w@-y+2+41)=0. Similarly, wv + 3y? + 42° -—6yz —2%x” =a represents an elliptic cylinder; except when a =0, when it represents the straight line « = %, y = #, forming the axis of the cylinder; for it may then be written (@ —%)? + 3(y—x)?=0. If a be negative, the surface is imaginary. Ex. 4. w+ 4y°+ 2 4+4y2-22a -40y43au—-6y—32=0. The derived equations are 20-23 —-4y+3=0, 8y +42 —-4v—-6=0, 23+ 4y —-24%-—-3=0, which are equivalent to only one equation; therefore, the proposed equation can only represent two parallel planes; and putting x = 0, the trace on wy has for its equation w+ 4y? —4vy + 3x —-6y = 0, or (w — 2y)’? +3 (x — 2y) = 0, representing two real straight lines; consequently the proposed equation represents two separate parallel planes, and, as we see, it may be written (v7 — 2y — x) (vw- 2y—%+4 3) =0. 125 149. To find the relations among the coefficients of the general equation of the second order when it represents a surface of revolution. Let av? + by? + c2°+2a yx 4+2b'ax 4+ 2c ary + 2a ex +2b°y4+2c'x+d=0, (1) be the general equation to the surface; and let a, 8, y be the co-ordinates of a point in the axis of revolution. “Then a sphere whose equation is (v7 —a)’+(y—-B)y+ @-y)- R=0, (2) will intersect the surface in two parallel planes i pircaih. eit len +my+nz =p. Now if \ be an arbitrary multiplier, (1) +A (2) =0 gives the intersections of the surface with any sphere such as that above mentioned ; hence this equation must be identical with O=(la+my+nz—p)(le+my+nzx—p’); .O0O=PH +m yt ns? +2mnyx4+ 2lnae + 2lmay + &e. is identical with O=(a+A)a’?+(b+A)y?+ (C+A) +420 YX 42h az42Cry+&e.; -P=at+aA, mM=b+r, W=c+”, mn =a’, In =0’, lm=c b'e! ; ae’ ih pel=sat+nr, —-=m=b+)d, —-=Wect+d; a b C bo! a’e a’b’ ° ee eer b = —— —"¢, a b are the required conditions. To this must be added that each of these equated quantities, being equal to — A, must be finite. 126 Cor. Hence the plane through the origin to which the circular sections are parallel has for its equation Veutacy+abz =0, consequently all chords of the surface parallel to this plane are principal chords. Also if we substitute for a, b, c the values furnished by these equations in the cubic (Art. 140), it will take the form bic, 20 0 we ean (+r {srn-“ ~~ hao, Hence, when the above conditions are satisfied, the cubic for determining the directions of the principal chords has two roots equal to — A, so that of the three systems of principal chords two may be drawn in any manner parallel to the plane represented by the above equation. Moreover when the sur- face has a center the equal values of s lead immediately to the value of its equatoreal axis (Art. 146), and the remaining value of s to that of the polar axis, 150. To find the equations to the axis of revolution. By comparing the terms of the first order in the two identical equations of the preceding article we get 2a"—-2r\a=—-I(pt+p’), 2b"°-2\B =—m(p+p), 2c" —- 2rAy=—n(p+p'); but mn=a’, In=b*, Im=c’, (3) *. a (a —da) = 5 (b" -AP) =e (c" —Ay); or, replacing a, 3, y by a, y, s, and X by its values, the relations amongst the co-ordinates of the center of the sphere that always intersects the given surface of the second order in two parallel planes, become i A a’a”’ u( : bp”! ( ant TE Ls emerges garry Pi SB re rrr mh eae recs nae be Leh DN & +o ay) which represent a straight line forming the axis of revolution of the surface; and, as appears from equations (3), perpen- dicular to the parallel planes. Although the results of this 127 and the preceding Article may be so modified as to embrace all particular cases, yet when a proposed equation wants several terms, it will generally be better to apply the direct investigation to discover whether or no it represents a surface of revolution. 151. Three diametral planes are said to be conjugate to one another, when each bisects the chords which are parallel to the intersection of the two others; and the inter- sections in that case are called conjugate diameters. When a surface admits of three planes of this sort and they are taken for the co-ordinate planes, its equation supposed algebraic can only contain even powers of the three variables a, y, x Hence the principal planes of a surface of the second order having a center, are conjugate to one another, since the equation referred to them is Av? + By? +C2? =D; and this, as we have seen, is the only rectangular system of co-ordinate planes which can give the equation of this form, or which can be conjugate to one another. 152. We shall now shew that there is an infinite number of systems of diametral planes oblique to one another which, taken for the co-ordinate planes, will make the general equation of the second order to be of the above form, that is, free from the terms involving za, wy, yx, and which are therefore con- jugate to one another, Let w=mz, y=nz be the equations to any diameter Ox’ (fig. 50), then the equation to the diametral plane «’ Oy’ conjugate to Oz’, is (Art. 137), df (x, Y, 2) ii AAC Y, &) = df (x, Ys %) as x ae O iy: dy dx ; or, mAv+nBy+Cz=0. Since the diameter O2’ is parallel to the chords which 2 Oy’ bisects it must according to the definition be the intersection of the two planes which are conjugate to wv Oy’; and those planes will cut the plane #’Oy’ in two lines Oa’, Oy’, such that if 128 together with Ox’ they be taken for a system of oblique axes, the equation to the surface will only contain even powers of the three variables, and therefore be of the form Ae? + Byy? + Ciz" = D,. But if we make x’ = 0, the result A,a”’ + B,y? = D, is the equation to the section a Oy’, and by its form shews that the axes Ow’, Oy’ are.conjugate diameters of that section ; hence it follows, that any diameter Ox’ being proposed, if in the section made by the diametral plane which is conjugate to it we draw any two conjugate diameters Oa’, Oy’, at pleasure, we shall determine three planes z’Oa’, x’ Oy’, y' Ox’ which are conjugate to one another; the number of such systems is therefore unlimited ; and the equation to the surface when referred to a system of conjugate diameters as co-ordinate axes, may be put under the form a’, b’, c’, being the distances, real or imaginary, from the center at which the surface cuts them. Cor. If Oz’ be in a principal plane, the tangent plane at the extremity of Ox’ and consequently the diametral plane conjugate to Ox’ will be perpendicular to that plane, but not perpendicular to Oz’. Also if Oz’, Oy’, be axes of the section w’ Oy’, they will be at right angles to one another; or if Oz" coincide with an axis of the surface, it will be perpendicular both to Ow’ and Oy which will lie in a principal plane; but in no case will three conjugate diameters be mutually at right angles, unless they coincide with the axes of the surface. 153. We have seen (Art. 138) that any diametral plane is parallel to the tangent plane, applied at the extremity of the diameter to which it is conjugate. Hence, in a system of conjugate diameters, the tangent plane at the extremity of each is parallel to the plane of the two others. Also, if through the extremities of each of a system of conjugate diameters we draw planes parallel to the plane of the two others, we shall form a parallelopiped, which is said to be 129 constructed upon the diameters, and which in the case of the ellipsoid will be circumscribed about the surface. We shall now shew that the known properties of conjugate diameters of curves of the second order may be extended to surfaces of the second order. 154. Ina surface of the second order, that has a center, the sum of the squares of any system.of conjugate diameters is equal to the sum of the squares of the axes; also the volume, and the sum of the squares of the faces, of the parallelopiped constructed on any system of conjugate dia- meters, are respectively equal to the volume, and the sum of the squares of the faces, of the rectangular parallelopiped constructed on the axes. 2 y 2 , Let wtiatat 1 be the equation to a surface of the second order, referred to a system of conjugate diameters, the inclinations of which are Zyz=A, Z2v=p, Ley =r; and let r=mz, y=nz, be the equations to any diameter ; then the equation to the diametral plane conjugate to this diameter is me ny z TE DER LES Fae topo de a ae a b c and if it be perpendicular to the diameter, we have (Art. 86) mec*(1 + mcosu + ncosr) = a?(m + CoS v + COS 1), ne*(1 + mcosu +n cos)) = 6" (nm + m cos v + cos dX). Let r =c"(1+mcosp + 2COSA) 3; “mr = a" (m +n Cos vp + COS p), nr° = b?(m + mcos v + 0s A) 3 Les 10. va Ne id Sate & +—+ =] =1+m’+n?+2mncosyv+2mMcosu+2N Cosas; 1 a’ *. (as in Art. 146) ris the length of the semidiameter whose a 130 equations are a= mz, y=nzx. Also if we determine m and nm from the two latter equations, and substitute them in the former, the result is (7° —a’*) (7? —b”) (r? —c’*) —a’*b? (1? —c”) (cosy)? —a’*c?(r* —b”) (cosu)? — be? (r* — a”) (cos d)* — 2ab”c? cos p cos vy cosA = 0, or 7° — 7" (a? +b" + 6") 47°f (a'U'siny)’?+(a'c'sinu)?+(b'c'sind)”} — (a'b’c’)*$1 + 2cosd cos u cosy — (cosd)* — (cos)? — (cos v)*} =0, But if a, b, c be the three semiaxes of the surface, then a*, b*, c are the values of r? in the above equation; there- fore, by the theory of equations, we have a? +b7° 4+ Pra V+ +0’, (a’b’ sin v)?+ (ac sinu)?+ (0'e' sind)? = (ab)? + (ac) +(bc)*.... (2); (a’b’c’)* $1 + 2 cos X Cos pm Cos v — (cos \)? — (cos nx)” — (cos v)*} = (abc)’.....(3), which are the three required results, the first member of the third being the well-known expression for the square of the volume of a parallelopiped in terms of its three edges, and their inclinations to one another. Cor. When one or two of the axes are imaginary, there will be an equal number of the conjugate diameters imaginary; and it will be necessary to change the signs of the squares of these axes and these diameters in the above equations. 155. The preceding results may also readily be arrived at by the following method. Let Ow, Oy, Ox (fig. 29) be the semiaxes of the sur- face a, b, c; Ow’, Oy’, Oz’ any system of semiconjugate diameters a’, 6’, c’; let the plane of a’y’ intersect that of avy in the 4 diameter Ow,=a,; also let Oy,=b, be the semidiameter of the curve #,x’, which is conjugate to Oa; a+b, =a? +b” 131 Now the plane aw Oy' is by supposition conjugate to 0, therefore Ox’, Oy,, Ox,, form a system of conjugate diameters, or Ow, is conjugate to the plane x’ Oy. Let the plane x’ Oy, intersect wy in the semidiameter Oy, = b,, then this plane must contain Ox; for being con- jugate to Ow, in a principal plane, it must be perpendicular to that plane (Cor. Art. 152); hence Ox,, Oy,, Ox form a system of semiconjugate diameters, (for Oz, Oy, are the 4 axes of the section y,y,%) and any two are semiconjugate diameters of the plane section in which they are situated ; U0; Ho =0, ey a+b =a + bi; therefore, adding these equations to the former, a’ “fs b? += q? ib? 6. Again, employing the same auxiliary systems of diameters as above, and denoting each parallelopiped by its three edges, we have vol. (a’, b’, c’) = vol. (a, be, ¢’), for these figures have the same altitude, viz. the perpendi- cular from ’ on the plane of «’y’, and equal bases, viz. the parallelograms constructed on the two systems of 4 conjugate diameters a’, 6’, a,, b,, belonging to the same curve in the plane of ay’. Similarly, | vol. (a, b,, c’) = vol. (a), 0, ¢), vol. (a), bj, c) = vol. (a, 6, c) ; .. vol. (a, b’, c) = vol. (a, 8, c). Cor. If the conjugate diameters 2a’, 2b’, 2c’ are each equal to 2R, then 3R*? =a’? +b’?+c’; also since there are only two equations, viz. (2) and (3) Art. 154, to determine the angles of inclination A, uw, v of the conjugate diameters, there may be an infinite number of systems of equal conjugate diameters, their extremities all lying in the intersection of the surface and a concentric sphere radius = R. 9—2 132 And of all systems of conjugate diameters of an ellipsoid the principal diameters have their sum a minimum and the equal diameters their sum a maximum. For in any proposed system, if there were two not perpendicular to one another, by substituting for them the axes of the section of which they are diameters, we should obtain a system whose sum is less than that of the proposed system. Again, if there were two not equal to one another, by substituting for them the equal diameters of the section to which they belong, we should determine a system whose sum is greater than that of the proposed one. And in this manner we could shew that no oblique system could have their sum a minimum, and no unequal system their sum a maximum. 156. In surfaces of the second order not having a center, represented by the equation to rectangular co-ordinates By + Cz2*°=2/4' a, the planes of xa and vy only are diametral and principal, each bisecting perpendicularly the chords parallel to the intersection of the other with the plane of yz, which is the tangent plane at the vertex. In this case no three diametral planes can be conjugate to one another; for, taking a system of chords parallel to a line whose equations are v = mz, y = 72; the equation to the diametral plane conjugate to them is Therefore all the diametral planes are parallel to the axis of a, and consequently their intersections, which are the diameters of the surface, are parallel to that axis, and cannot therefore form a system of co-ordinate axes. But we can find an infinite number of oblique co- ordinate planes related to one another in the same manner as the three rectangular planes mentioned above are, viz. so that two, ’# and ay’, shall be diametral planes, and each of them conjugate to the chords parallel to the intersection of the other with y's’; the latter being a tangent plane to the surface at the extremity of the diameter in which 3’ a’ 133 and a’y’ intersect; and the equation to the surface when referred to them will therefore preserve the same form. For let a’ O'y’ (fig. 51) be the diametral plane represented by equation (1), and which, being parallel to the axis of the surface, cuts it in a parabola AO’B. In AO'B take any point O’, and draw O’’ parallel to the chords to which the diametral plane is conjugate. Then the two co-ordinate planes which are to go along with this diametral plane will pass through O's’, and cut the diametral plane in two lines O'v’, O’y’, such that being taken for the co-ordinate axes they shall give the equation to the surface under the form Byy® + Cis- = SA it. Therefore, making ’ = 0, the equation to AO’B is By” = 2 Aa’. which by its form shews that it is referred to a diameter of the parabola and a tangent to the parabola at the extremity of that diameter. Hence the position of the second diametral plane x’ O'x’, and also of the third co-ordinate plane x’ O'y’, is determined, which latter, since it passes through two tangent lines to the surface O'y’ and O'’, is a tangent plane to the surface at O'; and as not only the plane 4O’B is arbitrary, but also the position of the point O’, the number of systems of oblique co-ordinate planes similar to the above is unlimited. 157. When the general equation of the second order represents a paraboloid, to find the position and magnitude of the surface. Let aa’ + by? + c2x* + 2a'yxz4 2b ax4+2Cuy+2a' a + 2b’y4+2c's+d=0, be the equation, then D = aa’ + bb”? + cc? — abe — 2a'b'e' =0; consequently the cubic which determines the directions of the principal chords has one root s=0, and the corresponding values of m and 7 are (Art. 140), m(ab—c”) =a'c — bb, n(ab—c”*)=)c'- ad’. 134 Since the plane bisecting the chords for which s =0 is at an infinite distance, these chords are in the direction of the axis of the paraboloid; and therefore the equations to a line through the origin parallel to this axis are A Y Z MLS PA ea a OL) ac—-bb be-—ad ab-c® Also if w,y,% denote the co-ordinates of the vertex, expressing that the normal at that point must be parallel to (1) we get the equations 1 du 1 du 1 du ab—c? dz ac’—bb' dz wc'-—aa’ dy’ which, together with the equation to the surface wu = 0, will determine a, y, x the co-ordinates of the vertex. Next to find the values of the parameters of the principal sections, let ’ , M aa + by? + cx? + 2ays + 2az + 2Cuy + Dies where M=a"N+b’N'+c"N"4+dD (Art. 134), be the equation to a surface of the second order referred to its center ; then the roots of the equation D D? 4 s+(a+b+0c) By peat as Lae teat ad a oe) ps _- 0, PD D? D' “or gf ln Feit ae - ays 7 0 (Att 146) are the squares of the reciprocals of the semi-axes a, 3, +. Now suppose D very small, then one value of s will be very 1 1 1 small, let this be =; and the remaining values of s, ead a p yy” will be nearly equal to the roots of PD QD s 4+ ——'s + ——_ = 0; uM** MM? aye 1 PD\? Pet eal er ae I 135 D' 1 QD? (aByy MP’ (Bye MP’ we war tt Q Q’ ak \ tea. C)* [oe eed ~ Consequently, the squares of the reciprocals of the 4 para- meters of the ue ag sections of the paraboloid, being the ] 2 Q . and’ @* = Dr” since values af poh — eo 4.9 p= ("= 2Q) =. p+ when D = 0, are the roots of the quadratic, Q' im If 1,7, be those 4 parameters, we have 1 ut (Jay Jakes Cle pat Cpe ogy! 2S pt pn ' Var Wr aw et We Es - sa ll Q ? which shews that paraboloids represented by the general equation of the second order, have the parameters of their principal sections in the same ratio, or are similar to one 2 another, when has the same value. Also the surface will be an elliptic or hyperbolic paraboloid according as Q is posi- tive or negative; for as a is a real semi-axis and therefore a’ positive, Q and M must have the same sign in order that any surface may exist. Cor. Similarly we may find the conditions that two central surfaces of the second order may have their axes proportional, or be similar to one. another. As above, the cubic equation . 1 e e e e will have roots —, Let this equation in which we a 1 1 136 may suppose M = 1 without altering the relations of its roots to one another, be transformed into another whose roots are 1 it ; (Seen tee) male Be 4 \B Wal ko” Wa pede the result will be found to be PQ jie aig 8S es ) Ey) | Slap +3)@-—|——4+1] =0. ale a) 16\D? D* D* as . ae ah eee Now the values of ¢ depend only on the ratios =, — and —; SRO fi 8 in consequently these ratios will be the same, provided > and el 5 hy ae es =p have the same values; these therefore are the con- ditions for central surfaces, when represented by the general equation of the second order, having the ratios of their axes the same, or being similar to one another. SECTION VII. ON CYLINDRICAL, CONICAL, AND CONOIDAL SURFACES, AND ON SURFACES OF REVOLUTION. 158. Brrore proceeding with the proper subject of this section it may be useful first to introduce the following remarks relative to the number of constants in the equations to surfaces, and to the intersection of surfaces. If the general equations of the n™, (m — 1)" &c. orders between two variables x and y be formed, and multiplied respectively by the quantities 1, z, 2’, &c. %", we shall by adding all such products together form the general equation of the 2" order between three variables a, y, x; and the number of constants which this equation will contain will be E(m+1)(n+2)+4n(n+1) +h (m—-1)n+...4+$1.2 =1(m +1) (n+ 2) (m + 3). Consequently, diminishing this by unity we shall have the number of independent constants in the general equation of the 7" order between three variables expressed by N=1n(n?+6n+ 11); which is also the number of points through which a surface of the nm order can be made to pass, for the co-ordinates of every one of these points being substituted in the equation to the surface would furnish N linear equations for the determination of the constants. Let w=0, v =0, be the equations to two surfaces that intersect; then w+XAv=0, where XA is an indeterminate constant different from zero, represents a surface passing through their curve of intersection, since it is satisfied by all values of wv, y, and x that simultaneously satisfy the equations w= 0, v = 0. 138 Also if w= 0, v =0, be the equations to two surfaces of the mn order passing through N—1 given points, then uw+2Xv=0, since it contains an additional constant ), in- cludes all the surfaces of the m' order that can pass through the given points, which shews that all those surfaces will pass through the intersection of any two of them; therefore all surfaces of the 2“ order that pass through NV — 1 given points, have a common curve of intersection. And if a surface of the m™ order be made to pass through a number of fixed points two less than the number sufficient to completely determine it, the surface will also pass through an additional number of fixed points such that added to the former it makes up m* the entire number of points in which three surfaces of the n“ order can intersect one another. Since an infinite number of surfaces of the m™ order can be described through the N —2 given points, consider any three of them whose equations are wu =0, v=0, w =03 then the equation w~+Av+uw=O0 (1) where A and pw are in- determinate constants, will include all the surfaces of the nm order that can pass through the given points, because the equation of every such surface could involve only two un- determined constants. But equation (1) will be satisfied by every system of values of x, y, and x that simultaneously satisfy u=0,v=0,w=0; that is, all the surfaces will pass through the points of intersection of any three of them; therefore all the points of intersection must be fixed points, and the N —2 given points will determine the n?— N + 2 remaining fixed points of intersection. | Again, if the system of equations u,=0, v,=0 taken together represent the curve of intersection of the surfaces U, =0, v,=0; and w,=0, v,=0 taken together represent. the curve of intersection of the surfaces of w,=0, v,=0; then will 2, + AVv,v; = 0 represent another surface which passes through both these curves of intersection, and is of the order indicated by the greater of the quantities ” or r +s. Hence if v=0, v =0, be the equations to two planes intersecting a surface w=0; then «+)vv'=0 is the equa- 139 tion to a surface passing through the two plane curves of intersection; and if the planes become coincident, w+? = 0 represents a surface touching w= 0 along the curve in which the plane v = 0 cuts it. A surface of the m‘" order may in general have n (n — ii tangent planes applied to it passing through a fixed straight line. Let V=0 be the equation to the surface, then if its tangent plane at wysz contain the straight line A=mZ+a, Y=nZ +0, we must have dV dV adaV — 0 Ce Tae dV dV dV — vw) —+(b-y)—-2—=0 (1). (@- 0) T+ 0-5-8 5-9 O But if V=w+v=0, so that w is the sum of all the terms of m dimensions in V, we have by a property of homogeneous functions oes, Whe, du pros By pad pote z—_= = = da "dy dz therefore equation (1) is reduced to Puls Cael dv aay, ———w | Sant av —- ——._- ———_—_- _ a se ad = de dev dy - dy dz d which is of the (z — 1)" degree; and the other two equations from which together with (1) the co-ordinates of the points of contact av, y, x are to be determined by elimination, are of the n™ and (n —1)™ degrees; therefore the degree of the final equation will generally be m(—1)*, which expresses the number of the points of contact, or of tangent planes that can be drawn through the fixed line. 159. In the preceding sections we have considered several instances of surfaces generated by a line, straight or curved, which so moves and changes its position or form, as constantly to pass through one or more fixed curves or directrices. We shall now extend the same considerations 140 to the general case of a generating curve represented by the equations SCAR PA EN aah Pi Oy Opec eyed Ra 6 containing two variable parameters, a, $8, and subject to pass through one fixed curve or directrix. The species of the generating curve is determined, because the func- tions f, f,, are supposed to be known, but its position and dimensions will change corresponding to the different values of a and 3; and it will generate a solid if the parameters a and £ vary independently of one another. Thus the circle represented by +(y-By=ce-a’, v-a=0, gives, by the elimination of a, the equation S++ (y- Blac’, which represents a sphere, radius c, and center in the axis of y; and if £6 receive all values from 0 to + o, the equation will belong to all the points of a solid cylinder, radius c, and axis coinciding with the axis of y. But if we suppose the generating line (1) in every position to have a point in common with a fixed curve or directrix represented by the equations y= (#); s=vV(a), then the equations to the generating line and directrix must be simultaneously satisfied by the same system of values for xv, y, * belonging to that common point; if therefore we eliminate #, y, x between them, we shall have 3 = F(a), the relation between the parameters, in order that the gene- rating line may meet the directrix. Hence, the generating line in any position will be represented by the system of equations S(® Ys % a, B)=0, fi(*, y, % a; 6B) =0, B= F(a). If therefore we eliminate a and 8 between them, that is, if we determine a and 3 from the two former in terms of Xv, Y, %, so that B =u, a=v, and substitute in the latter, 141 we shall have w= F'(v) for the equation to the surface; and we observe that w and v do not change for surfaces of the same family, that is, for those which admit the same gene- .rating line, but that F', which depends upon ¢ and y, changes for each Pcichion! surface of the family. These considerations we shall now apply to several of the more common cases, where the generating line is a straight line, or a circle. 160. ‘To find the general equation to cylindrical sur- faces. A cylindrical surface is generated by a straight line which moves parallel to itself, and always passes through a given curve. Let the equations to the generating line in any position be v=mz+a, y=nzx +f, a and B being variable quantities depending upon that position, and m and m constant quantities since the line is always parallel to itself. Also let y= P (a), #% =) (w), be the equations to the directrix or curve through which the generating line always passes, and a’, y’, x’ the co-ordinates of the point in which they meet; then a’, y’, 2 must satisfy both the equations to the directrix, and gene- rating line ; .a@ems ta, y=ne +B, y=O(v), x=’); now by means of these four equations, we may eliminate a’, y’, s, and there will remain 8 = F(a); but B=y- nz, a=LX—-Ms; . y—-nv=F(#- ms), a relation among the co-ordinates of any point in the gene- rating line, and therefore the equation to the surface which it describes. In this case the quantities w and v are y — nz, and « — mz, which remain the same for all cylinders, whilst the function F' will alter with the different directrices employed. Cor. If for the purpose of eliminating the arbitrary function, we differentiate the equation y—nz = F'\(# — mz) successively with respect to # and y, we find —np=F'(4@—mz)(1-—mp), 1—-nq =F" (a — mz) (— mq) + 142 therefore, dividing one result by the other, np 1— mp dz dz = > or m—+n— =I, 1—nq mq dx dy which is the differential equation to cylindrical surfaces; we may however obtain it more easily by the consideration of the tangent plane, as follows. 161. To find the differential equation to cylindrical sur- faces. One of the distinguishing properties of cylindrical sur- faces is, that the tangent plane, since it always contains a generating line, (Cor. 2 Art. 108) is always parallel to a fixed straight line. Let 2 = ms’, y’=nz' be the equations to a line through the origin, to which the generating line is always parallel; then this line is parallel to the tangent plane whose equation is s—s=p(e#—x#)+q(y'-y); dz d . (Art. 23) mp+nq=1, or mn Ere 1, as before. Cor. This equation may be employed to discover whether the surface represented by a proposed equation w = 0, is cylin- drical or not. For, obtaining the values of p and q, as in Cor. Art. 107, and substituting, we have du du du WP 9 0, x z which must be satisfied for all points of the surface, that is, for all values of w, y, x, if the surface be cylindrical; hence we must equate to zero the coefficients of the different powers of the co-ordinates in it, and examine whether the resulting conditions can be satisfied by real values of m and n. 162. Having given the direction of the generating line, to find the equation to the cylindrical surface which envelopes a given curve surface. 143 We have seen that whatever be the nature of the directrix, the equation mp+nq=1, must subsist between the differential coefficients p, g, derived from the equation to a cylindrical surface. But at the points where the cylinder touches the surface, the values of p, g, are the same for both ; therefore at those points the above relation subsists between the differential coefficients derived from the equation to the surface; that is, the co-ordinates of the points of contact are such that the equation mp + nq =1 is satisfied. Hence if we differentiate the given equation to the surface «=0, and write the values thence obtained of p, g, in the above equation, the result together with the equation to the surface, will be the equations to the directrix ; and we know the direction of the generating line, and therefore can find the equation to the required surface by Art. 160, 163. To find the general equation to conical surfaces. A. conical surface is generated by a straight line which passes through a given point, and always meets a given curve. Let the co-ordinates of the given point or vertex be as Cs @—-a=a(s—-c), y-b=B(s-0), are the equations to the generating line in any position, a and f being variable quantities depending upon that position. Also let y = ¢ (#), % = W (@#), be the equations to the directrix or curve through which the generating line always passes, and a’, y’, x, the co-ordinates of the point in which they meet; then a’, y’, x must satisfy both the equations to the generating line and directrix, wv —-a=a(%—-c),y -b=B(e'-0¢), y= (#), X= (e’). Now by means of these four equations we may eliminate we, y’, x, and there will remain 8 = F(a); =U ioe but 6 =~ sat ease ip ? yw — 6 w= 14.4 J Y= (2) Reb EG a relation among the co-ordinates of any point in the generating line, and therefore the equation to the surface which it de- scribes. When the vertex is situated in the origin, the equation is reduced to Pay (<) , which expresses that the @ Pe equation is homogeneous in w, y,%. It is easily seen that if the directrix be the curve of intersection of two surfaces of the m™ and 2" orders, the equation to the cone will be of the mn" order. Cor. If from the above equation we eliminate the arbitrary function by differentiation, as in Cor. Art. 160, we find tee d zen 7 (@—a)t 7 yd), which is the differential equation to conical surfaces; but which may be obtained more easily by the consideration of the tangent plane, as in the following Article. 164. To find the differential equation to conical surfaces, The distinguishing property of conical surfaces is, that the tangent plane, since it always contains one of the generating lines (Cor. 2 Art. 108), always passes through the vertex. Let a, b, c denote the co-ordinates of the vertex, then they must satisfy the equation to the tangent plane at any point wyz; that is, x —s=p(a —«x)+q(y'—y) must be satisfied by # =a, y =b, x'=c; therefore, whatever be the nature of the directrix, the differential coefficients p, g, derived from the equation to the surface, must be such as to satisfy the equation x-c=p(w—a)+q(y—4), which is the differential equation to conical surfaces. Cor. As in cylindrical surfaces (Art. 161), this equation, when put under the form du du du eae \ eee ny EMRE Cy ee 145 may be employed to discover whether a proposed equation u = 0 represents a conical surface or not; a, 6, ¢ being the unknown quantities to which must be applied what was there said relative to m and n. 165. Having given the position of the vertex, to find the equation to the conical surface which envelopes a given curve surface. Whatever be the nature of the directrix, we have seen that the differential coefficients derived from the equation to a conical surface must satisfy the equation s—-ce=p(w-a)+qty—); (1) but at the points where the cone touches the surface, the values of p,q, are the same for both; therefore at those points the above relation subsists between the differential coefficients derived from the equation to the surface, that is, the co-ordinates of the points of contact are such, that the above equation is satisfied. Hence if we obtain values of p,q, from the given equation to the surface, and sub- stitute them in the above equation, the result, together with the equation to the surface, will be the equations to the curve of contact of the two surfaces, or to the directrix ; and we know the co-ordinates of the vertex, and can therefore find the equation to the required conical surface by Art. 163 ; and it will be of the m(m —1)™ order, if the given surface be of the v" order, because (1) will become of the (7 — 1)" order, in the same way as equation (1), p. 139. Cor. 1. If the vertex of the cone be considered as a luminous point, the curve of contact whose equations we have just found, is that which on the surface separates the illumined and obscure parts; if it be considered as the place of the eye, the curve of contact is the line of the apparent contour of the surface. Cor. 2. Let w= 0 (1) be the equation to a surface of the second order; then the curve of contact of the enveloping cone is easily shewn to be a plane curve determined by an equation of the first order, v =0 suppose. Hence w +2 .v* =0(2) is another surface of the second order whose tangent cone 10 146 from vertex (a, b, c) is identical with that to surface wu =0 from the same vertex, whatever be the value of the constant ) 3; for from Art. 158 the surfaces (1) and (2) touch one another along the curve in which they are intersected by v = 0, and therefore each is touched by the same enveloping cone along that curve. Now no surface of the second order can pass through the vertex of its enveloping cone; for if it did, that cone would become the tangent plane; therefore if > be determined so as to make (2) pass through the vertex of the cone, the equation w + Av” = 0, will represent the enveloping cone of the surface wu = 0. Similarly, if v =0 be the plane of contact of the envelop- ing cylinder having its axis in a given direction; and 2d be determined so as to render infinite that radius vector of the surface % + \v* = 0 which is parallel to the axis of the cylinder, we shall obtain the equation to the cylinder enveloping a surface of the second order. 166. To find the general equation to conoidal surfaces. A conoidal surface is generated by a straight line which moves parallel to a given plane, and always meets a given fixed straight line, and ‘a given curve. Let the given plane, called the directing plane, be taken for that of wy, and the point in which the straight directrix OA (fig. 54) meets it, for the origin; then the equations to OA will be r=mz, y=nz. Also let the equations to the curvilinear directrix BC be y=@(@), s =W(a); and let s=B, y=ax+- be the equations to the generating line DC in any position, since it is always parallel to the plane of vy; then since the generating line always meets OA, we must have nbB=amB+y; therefore, eliminating y, the equations to the generating line become v= B, y-nB=a(w—mB); we must next express that it meets the curve BC, whose 147 equations are y=@(w#), x= (wv); therefore, eliminating x,y, %, we have (3 = F(a); hence the generating line in any position will be represented by the system of equations e=B, y-nB=a(e-mp), B= F(a); therefore, eliminating a and , the equation to the surface generated is g- (Zo). C— MZ Cor. 1. This surface is twisted, that is, no two consecu- tive generating lines are in the same plane; for the line DC in passing to the consecutive position D’C’ may be supposed to glide along the tangent C7’; therefore, in order that DC and D’C” may be in the same plane, OA and CT' must be in the same plane, which cannot happen for all the tangents at least, unless the curve BC is entirely in the same plane with AO, in which case the surface generated will be a plane, If the straight directrix be perpendicular to the directing plane, m=0, m =0, and the equation to the surface, which is then called a right conoid, becomes z = r(2); at which we may readily arrive by a direct investigation, as the equations to the generating line will be s =, y=aa. Also it is manifest that the equation to the oblique conoid will a ; : assume the form z= r(2) if we refer it to a system of & oblique co-ordinate axes, of which OA is one, and any two lines drawn through O in the directing plane are the others. Cor, 2. If from the equation z = P(t) we elimi- L—-Ms nate the arbitrary function by differentiation, as in the former cases, we find p (w@—ms) + q(y— nz) =0, the differential equation to conoidal surfaces; at which we may arrive more easily as follows. LO 148 167. To find the differential equation to conoidal surfaces. Their characteristic property is that the tangent plane at any point contains the generating line which passes through that point, (for it contains the tangent lines to all curves traced on the surface through that point, and a straight line is its own tangent,) that is, a line parallel to the plane of wy, and intersecting another line whose equations are a =m, y’=nz'. Now the equation to the tangent plane at a point vyz is x —s=p(a-x)+q(y'-y), and if we intersect it by a plane parallel to ry at a dis- tance x from it, their line of intersection will have for its equations s'=s, p(w—x)+q(y-y) =0, and since this is coincident with the generating line, it will intersect the rectilinear directrix of which the equations are wv =m, y=ne'; p(w-—msz)+q(y—nz)=0, which is the relation that must be satisfied by the co-ordinates of every point in the surface. For a right conoid, or for an oblique conoid referred to the rectilinear directrix as one of the axes, the differential equation is p“e+qy=d0. 168. Having given the rectilinear directrix, to find the equation to the conoidal surface which envelopes a given curve surface. As in the former cases, if %=0 be the equation to the given surface, the curve of contact which forms the curvi- linear directrix, will be given by the equations du du 0, Cl atisted eg 88) ED arte 149 which must be employed instead of the equations y = ¢ (2), x =wW(w#); the remainder of the process will be the same as that explained in Art 166. 169. To find the general equation to surfaces of re- volution. A surface of revolution is generated by a curve revolving about a fixed axis. If the curve be situated entirely in a plane passing through the axis, then every section of the surface through the axis, which is called a meridian, will reproduce the generating curve; otherwise, the section will be different from the generating curve. Also every section perpendicular to the axis, which is called a parallel of the surface, will be a circle whose center is in the axis and plane perpendicular to it, and whose perimeter passes through the generating curve; hence a surface of revolution may be supposed to be generated by a circle whose center moves along a fixed straight line to which its plane is perpendicular, and whose perimeter always passes through a given curve. The advantages of the second mode of generation are, that the moveable circle is a generating curve of an invariable nature and common to all surfaces of the class, whilst the curve through which it always passes is a variable directrix which changes for each particular surface, agreeably to the mode of generation occurring in every instance in this section. Let v-—-a=m(z-c), y-b=n(s—- Cc), be the equations to the axis or fixed straight line along which the center of the generating circle moves, and which we suppose to be drawn through a given point (a, 6, c) in a known direction. Then every one of the parallels of the surface may be re- garded as the intersection of a plane perpendicular to this axis, with a sphere whose center is any fixed point in the axis, the point (a, b, c), for instance, and whose radius varies so as to give the section of the proper magnitude; conse- quently the equations to the generating circle will be Neate () (vw — a)? + (y — b)? + (ew — cp whis,0 ate ; ll 7 150 Let y = ¢ (x), = = (2), be the equations to the directrix or curve which the perimeter of the generating circle always meets; then these four equations will be satisfied by the same system of values of #, y, x, which are the co-ordinates of the point of intersection; hence, eliminating a, y, x from them, we find B = F(a) the condition to be joined to the equations (1) in order that the circle which they represent may really be a parallel of the surface; hence, eliminating a and 3 from the equations to the parallel, we find s+me+ny =F }(e—a) + (yb)? + (z- 6), a relation among the co-ordinates of the generating circle in any position, and therefore the equation to the surface which it describes. Cor. 1. If the axis of revolution coincide with the axis of zg, and we take the center of the sphere for origin, we have m=n=0, a=b=c=0, and z= F(a#+y¥ + 2°), or x= f(a" +y’), as we have already seen, Art. 52. In this case it is more convenient to regard each parallel as the intersection of a right cylinder with a plane perpendicular to it, that is, instead of equations (1), to use = B, a +y’ =a, which joined to the usual relation 6 = F(a), give imme- diately s= F(a’ + y’). Cor. 2. If we differentiate the above equation with respect to w and y, and divide one result by the other, so as to eliminate the arbitrary function, we find m+p «w-a+(x—c)p m+q y-b+(s—e)q’ or pjy—b—n(z—c) —q}w-a—m(z-c)} =n(w—a)—m(y-b), the differential equation to surfaces of revolution, at which we may also arrive as follows. 151 170. To find the differential equation to surfaces of revolution. Their distinguishing property is, that the normal always meets the axis of revolution. This will be readily seen if we consider that the tangent plane at any point necessarily contains the line touching the parallel at that point, and is therefore perpendicular to the meridian plane passing through that point; consequently the normal will lie in the meridian plane, and therefore meet the axis. The equations to the normal at a point wyxz are eo —-ve+p(2'-x)=0, y-y+q(e' -2) =0; let, as above, the equations to the axis be wo —-a—m(x —c)=0, y —b-—n(s'-c)=0, then if these two lines intersect, their equations must be satisfied by the same values of a’, y’, 2’, which will be the co-ordinates of their point of intersection. Hence if we eliminate a’, y’, 2’, the result will be the differential equation to the surface. Subtracting, we find m (x'—c)+p(2’-2) -@ +a=0, n(x’ —c) +q(%—2)-yt+b=0, or, (m +p) (x —2) +m(x—c) -v%+a=0, (n +q)(s —s) +n (e@-c)-yt+b=0; -, eliminating x’ — x, we find as before (m+ p) {n (# —0) —y +b} = (m4 q) {m(w—c)-w+a}...(1); which expresses that a surface is one of revolution about an axis passing through a point a, 6, c, and having a direction determined by the quantities m and 7. If the axis of revolution coincide with the axis of z, the equation becomes py—-gqr=d0. Cor. The above equation will enable us to ascertain whether a proposed equation w=0, represents a surface of revolution; fer, deducing the values of p and q, and substituting them in (1), the result ought to be true for 152 all values of w, y, x. Hence we must equate to zero the coefficients of the different powers of the co-ordinates, and we shall obtain among the unknown quantities a, b, c, m, n a certain number of equations, which must be consistent with one another, in order that the surface may be one of revolution. The process applied to the general equation of the second order will furnish the conditions already obtained (Art. 149) by a different method. 171. Having given the position of the axis, to find the equation to the surface of revolution which shall envelope a given curve surface. As in the similar cases which have preceded, we must begin by finding the equations to the curve of contact which will be the directrix of the moveable circle. Let wz=0 be the equation to the proposed surface, then at all points along the curve of contact the tangent planes of the two surfaces are coincident; and therefore the values of p and q derived from w = 0, and substituted in (1), must satisfy it for all points in the curve of contact; the result of this substitution together with w= 0, will therefore be the equa- tions to the directrix, and then the equation to the required surface of revolution may be found as before. It is manifest that we thus determine a surface which, if the proposed surface revolved about the given axis to which it was fixed, would touch and envelope the proposed surface in every position. 172. The above are the principal propositions relative to the commoner families of surfaces which admit only one directrix, or only one of a variable form; for though Conoidal surfaces have two directrices, yet one of them, namely the rectilinear one, is constant for all surfaces of the family. Consequently, for Conoidal, as well as for all the other families of surfaces considered in this section, the finite equation involves only one arbitrary function, and the dif- ferential equation cleared of the arbitrary function, is only of the first order. SECTION VIII. ON SURFACES HAVING MORE THAN ONE ARBITRARY CURVI- LINEAR DIRECTRIX, AND ON DEVELOPABLE SURFACES, AND ENVELOPES, 173. In the preceding Section we considered several surfaces where the equations to the generating curve con- tained two variable parameters; and the necessary dependence of these parameters upon one another, in order that a surface and not a solid might be generated, was established by making the generating curve in every position have a point in common with a given fixed directrix. The resulting equation to the surface consequently involved but one arbitrary function; and the differential equation, characterizing the nature of the surface independent of that directrix, was of the first order. We shall now take a more general view of the subject of the generation of surfaces, and suppose the equations to the generating curve St (@, Yo By As Bs Y &e.) = 0, fi (a, Y, Bs a, B; ie &c.) = 0, to contain any number 7 of variable parameters a, 9, y, &c. This curve will not, by passing through all the various positions and forms corresponding to the various values of the parameters, generate a determinate surface, unless we subject it to such conditions as will leave only one of the parameters, a for instance, arbitrary ; and this may be done by making it, in every position, pass through m —1 curves represented by the equations y=P(%) F=W(*%)s Y= hilt) x= VWn(a); &e. For in order that it may have a point in common with the first directrix, its two equations and the equations y = ¢ («) z= (a) must be satisfied by the same system of values of w, y, x, which are the co-ordinates of that point; there- fore, eliminating w, y, * between these four equations, we 154 find F(a, 3, y, &c.)=0, which expresses the necessary dependence among the parameters in order that the gene- rating curve may pass through the first directrix. In the same manner, each directrix will furnish a fresh relation between a, 3, yy, &c.; so that to represent completely the generating curve passing through the 2 — 1 directrices, we must join to its two equations, the 2 —1 equations of con- dition F' (a, B, y, &c.) = 0, F, (a, B, y, &c.) = 0, &c. And if from these latter we suppose $B, y, &c. to be all de- termined in terms of a, and substitute their values in the equations to the generating curve, those equations will then involve only one parameter; and if we finally eliminate that parameter between them, we shall arrive at an equation con- taining no arbitrary quantity and representing a determinate surface generated by the moveable curve. Hence the final equation will contain as many arbitrary functions as there are directrices, and therefore the differential equation characterizing the surface independent of the di- rectrices and consequently not involving the functions, will be of an elevated order generally exceeding the number — 1 of functions, and never less than 22—3. ‘These principles we shall now illustrate, confining ourselves, however, to the cases where the generating curve is a straight line. 174. Surfaces generated by a straight line are divisible into two classes, each of which has distinct properties; viz. into twisted surfaces, where two consecutive positions of the generating line are never in the same plane; and into de- velopable surfaces, where two consecutive positions of the generating line are always in the same plane. One remarkable difference in the properties of these two classes has reference to their contact with the tangent plane ; in both, the tangent plane at any point (since it contains the tangent lines to all curves traced on the surface through that point, and a straight line is its own tangent) must contain the generating line passing through that point; but in twisted surfaces, the tangent planes along a generating line are all distinct and each touches the surface only in 155 a point; whereas in developable surfaces, it is one and the same plane which touches the surface at all points along a generating line. For let DC, DC’ (fig. 54) be consecutive positions of the generating line of a twisted surface; BC, ED sections of the surface through any points D and C, and TC, AD tangent lines to those sections. Then the tangent planes at all points along DC contain DC; and at D and C the tangent planes are ADC, T'CD, which cannot be coincident unless AD, T'C are in the same plane; but in that case D’C’ (since between the positions DC, D’'C’, the generating line may be supposed to glide along 4D, CT’) would be in the same plane with DC, which is impossible since the surface is twisted. But if the surface be developable, then since D'C’ is in the same plane with DC, the tangents 4D, TC are in the same plane, and therefore the tangent planes at D and C, as well as at every other point along DC, are coincident. This property has already been proved for cylinders and cones, which are evidently particular cases of developable surfaces. 175. We shall first consider twisted surfaces; and among the various ways in which the motion of a straight line may be governed so as to generate a twisted surface, we shall select the two following: viz. when the moveable line constantly passes through two fixed curves and remains parallel to a fixed plane, and when it constantly passes through three fixed curves. 176. To find the equation to the surface generated by a straight line which constantly passes through two given curves, and remains parallel to a fixed plane. The motion of the generating line is here, manifestly, completely determined, as we have only to cut the curves by planes parallel to the fixed plane, and to join the points of section by straight lines, to obtain as many positions of the generating line as we please; and as the tangents to the 156 fixed curves at those points of section will not usually be in the same plane, the surface will generally be twisted. Taking the directing plane for that of ay, let the equa- tions to the generating line in any position be S=a, y=PBat+y, and let the equations to the two directrices be y= oie) #=\,(0)3 y= dua) = yA (0); then successively combining the equations to the moveable line with those to each of the directrices, so as to eliminate VY, Y, %, we find F, (a, B, y) = 0, 1 Gd (rey decry Lal Oe which we may suppose reduced to the forms B = p(a), y = W (a). Hence, eliminating a, 3, y between these and the equations to the generating line, we have for the equation to the surface y = vp (zx) +(e). Cor. ‘To find the differential equation, we have, differen- tiating successively with respect to w and y, and dividing one result by the other, 0= (2) + 2p) p+ ¥'()p, 1=0$' (2) 9+ V4 2 oh ea (=~ 9G) therefore, differentiating again, and dividing one result by the other, employing the usual notation, qr—ps yee qs — pt j g =- (2) p, ny =— (2) q; “ Fr—2pqst+ pt =0, the equation common to all surfaces of this class. 157 177. To find the equation to the surface generated by a straight line subject to constantly pass through three given fixed curves. 3 It is easily seen that the motion of the generating line will be completely determined. For conceive any point in one of the fixed curves to be the common vertex of two conical surfaces, having the other two fixed curves for their directrices; then these surfaces will intersect one another in a finite number of straight lines, each of which passes through the three fixed curves and is therefore a position of the gene- rating line; and as we take fresh points for the vertex, the successive generating lines belonging to the same sheet will pass through points contiguous to one another on the three directrices. Also the surface will generally be twisted, because the tangents to the three directrices at the points where a generating line meets them, will not, except in very particular cases, be in the same plane. Let the equations to the generating line be w=axsty, y=Bz +o, containing four variable parameters, a, (6, y, Oo; and, as explained in Art. 173, let the three relations among the parameters necessary for the line passing through the three given directrices, be obtained and reduced to the form B=9(2), y=) d=7(a)s therefore, substituting for 8, y, 0 in the equations to the generating line, we have w=azt+W(a), y=xh(a) +7 (a); and it remains to eliminate a between these equations, in order to get the equation to the surface ; but this cannot be done without particularizing the functions, or the direc- trices on which they depend. ‘Therefore we must retain the above system of equations to represent this family of surfaces ; regarding a as an indeterminate quantity to be eliminated, after the forms of the functions in each individual case have been determined. 158 It may be observed that the various, other modes of gene- rating this sort of surface, as for instance when we replace one or more of the directrices by a surface which the moveable line is always to touch, are reducible to this method; for in every case we may take for the three directrices, any three sections of the generated surface made at pleasure. Cor. To obtain the differential equation independent of the directrices, we must, as before, obtain by successive differentiations of the above system a sufficient number of equations to eliminate a, and the functions @, vy, mw and their derived functions; the result is a complicated equation of the third order. When in the two preceding cases, the directrices all become straight lines, the surfaces generated become re- spectively the hyperbolic paraboloid, and the hyperboloid of one sheet, as is seen in the Appendix; they are the only twisted surfaces whose equations do not rise above the second degree. Developable Surfaces. 178. We next come to the consideration of the second class of surfaces which admit of being generated by a straight line, the characteristic law of the motion being that two consecutive positions are always in the same plane. Before proceeding to point out the various modes in which this condition may be satisfied, we shall shew that surfaces ge- nerated in this manner are developable; that is, supposing them flexible but inextensible, they may without rumpling or tearing be made to coincide with a plane in all their points. Let fig. 57 represent a surface of this sort, and let AN, A'N’, A”N”, &c. be positions of the generating line inde- finitely near to one another; then from the definition of the surface, 4N will be intersected by A’N’ in some point m, A'N’ by A’ N” in m’, and the latter by the next generating line in m", and so on; so that these successive points of 159 intersection will form a polygon mm’m’”..., or rather a con- tinuous curve to which all the generating lines are tangents. Also AN and A’N’, and similarly every pair of consecutive generating lines, will include a sectorial area Am A’ of inde- finite length, but infinitely small angle, which may be re- garded as a plane element of the surface. If now the first of these elements be turned about its line of intersection with the second, till they are in the same plane; and then the system formed by these two be turned about the line of intersection of the second and third till they are in the same plane; and if this operation be continued through all the plane elements, they will all thus be brought into one plane, and the given surface will be developed without rumpling or tearing. 179. We have already seen (Art. 174) that the plane which touches a developable surface in any point M’ is the tangent plane at every point in the generating line passing through JZ’; and that to construct the tangent plane to any point 1’, we have only to draw a tangent line M’T' to any curve on the surface passing through that point, then TM'N’' is the tangent plane required. 180. As the generating lines are all tangents to the curve mm'm’,.. formed by their perpetual intersection, the surface may be supposed to be generated by a moveable straight line which is always a tangent to a fixed curve; the curve must of course be of double curvature, otherwise the surface generated would be a plane. Hence it is suf- ficient to assign one fixed directrix (to which the generating line must be always a tangent) to completely determine a developable surface. If the equations to the fixed curve be v= (2%) y=), the equation to the surface will result from eliminating a from the equations to the line touching the curve at a point for which s =.a, which (Art. 123) are 2 $@)=$@MC-@, ¥-V¥O=V@(e- 45 160 the elimination cannot be effected without assigning the forms of the functions; but every developable surface may be represented by the above system of equations, regarding a in the first as a function of y and x to be determined from the second. 181. The curve mm'm”... is called the edge of re- gression of the surface which its tangent generates; the reason of which is that the two portions of any tangent, produced both ways from the point of contact, generate two distinct sheets of the surface which are united in the curve; so that if we wished to pass from a point in one sheet to a point in the other without quitting the surface, our path would have a point of regression at the curve, provided the two points were not in the same generating line. In conical surfaces the edge of regression is reduced to a point, in cylindrical surfaces it 1s removed to an infinite distance. 182. We may also generate a developable surface, by assigning two fixed directrices through which the moveable line is always to pass; in that case the points in which the moveable line meets them must be constantly so chosen that the tangent lines to the directrices at those points are in the same plane; then, as the moveable line in passing into its consecutive position may be supposed to glide along those tangents, every two consecutive positions will be in the same plane. Consequently, after having obtained two relations among the parameters which enter into the equa- tions to the moveable line by subjecting it to meet the two directrices, we shall obtain a third by expressing that the tangents to the directrices at the points where the moveable line meets them, are in the same plane; hence only one arbitrary parameter will remain in the equations, by eliminating which we shall obtain the equation to the surface. 183." There is yet another way of expressing the gene- ration of developable surfaces; for since every two consecutive generating lines include between them a plane sectorial area 161 of infinite length, we may regard these elements as forming parts of the successive positions of a plane subject to move according to a given law. This method will usually be found the most convenient in practical applications; and it leads easily to the general differential equation to developable surfaces, and to other results, which we shall now obtain by means of it. 184, To find the general equation to developable sur- faces, considered as generated by the consecutive intersections of the positions assumed by a plane subject to move after a given law. The law according to which the plane moves will vary with each particular surface; but in order that the motion may be completely determined, and that a single surface and not an infinite number of surfaces may be generated, it must leave only one arbitrary parameter in the equation to the moveable plane. The successive positions assumed by the plane by virtue of the infinitely small variations of the parameter, will cut one another consecutively in straight lines, which taken two and two are in the same plane; these lines will therefore form a developable surface touched by all the planes. Hence the law of succession of the planes which generate a developable surface requires that two of the three arbitrary constants which enter into the equation to a plane should be functions of the third, or, which is the same thing, that they should all be functions of the same parameter a; let therefore s=ap(a) + yf(a) + (a) be the equation to the generating plane in one of its positions depending upon the parameter a; then the equation to the plane, which differs insensibly from it in position, will be ce an) Sat &et + yf f(a) +f (a). da + &c.} | +W(a) + W (a). Sa + &e.; and the co-ordinates of the points in which they intersect LL 162 must satify these two equations; that is, they must satisfy z= 0(a) +yf (a) + (a), 0O= w$ (a) + &e.t + y sf’ (a) + &e.$ + W' (a) + &e. 5 or if the planes be consecutive, making da = 0, the equations to their line of intersection are s=adp(a)+yf(a) + (a)...... Gly. O0=ad'(a)+yf (a)+ (a)... (2); and the general equation to developable surfaces will result from eliminating a between these equations; but as the elimination cannot be effected without fixing the forms of the functions ¢, f, .y, that is, without particularizing the surface, we must retain the above system to represent this family of surfaces, regarding a in the former as a function of w and y to be determined by the latter. Cor. Hence the differential equation to developable surfaces can be obtained; for, differentiating (1) successively with respect to w and y, regarding a as a function of # and y to be determined from (2), we have p=$(a) + feg'(a) uf) +O} S. vis prin’ el ibsidetta q=f(a) + {2p'(@) + of (a) +V(a@)} ar which by virtue of equation (2) are reduced to p= (a), g=f(a); and since p and q are functions of the same quantity, they are functions of one another, “p= 7 (q); which is the differential equation of the first order to de- velopable surfaces, containing one arbitrary function; this may be made to disappear by differentiating the equation 163 p = m(q) successively with respect to # and y, and dividing one result by the other; for we have =a (q).s, 8=(q)-é; .rt—s=0, the differential equation of the second order common to all developable surfaces. 185. To find the equations to the edge of regression of a developable surface. The edge of regression is the locus of the intersections of the successive generating lines; and to find its equations we must combine the equations to any generating line (1) and (2), in which a is an arbitrary constant, with the equations to the consecutive generating line resulting from (1) and (2), by changing a into a + 0a; we shall thus obtain four equations which by reduction are equivalent to the three z= ad(a)+ yf (a) + W (a), 0=ag (a) + yf (a) + ¥ (a), 0= ag" (a) +f (a) + W' (a); these, for any value of a, will furnish the three co-ordinates x,y, % of the point where the generating line corresponding to that value of a, is intersected by the consecutive generating line; so that in eliminating a from the above system, we shall have the two equations to the locus of all such points of intersection; that is, to the edge of regression of the surface, by the tangent line to which the surface may be considered as generated. 186. ‘To find the equation to the developable surface which touches at the same time two given curve surfaces. Let the equations to the two surfaces be F, (a, Yis %1) ee Ag ee, (#5, Y25 Bo) = 0, dz dz fie dase tt dy 11—2 and let p,4i, P2q2, denote the values of for the 164 first and second surface respectively. Then the moveable plane, since it touches the first surface, will have for its equation B—-R= pi (©-X)+UnY-Y%); and in order that it may touch the second surface, that is, in order that it may coincide with the tangent plane at the unknown point «,y,%,, we must join the conditions Pi = Por Nh = Joo F1 — PiG — NY = Fe — P22 — G2Yr2- If therefore from these three equations, and the two equations to the surfaces, we determine the five unknown quantities Yis Zip Voy Yoo Vy in terms of &,, the equation to the moveable plane will take the form x =ad(a) + yf(a) + (a); and by eliminating the arbitrary quantity #, between this equation and the derived equation with respect to v,, we shall obtain the equation to the required developable surface. If we suppose an opaque body to be illuminated by a luminous one, the surfaces which circumscribe the umbra and penumbra occasioned by the interposition of the opaque body, are two sheets of the developable surface which touches both the bodies; and its lines of contact with the surfaces of the bodies are, one, the curve on the opaque body which separates the illumined and obscure parts; and the other, the curve on the luminous body which separates the illumi- nating part from that which can send no rays to the other body. Consequently this problem is the general problem of umbras and penum bras. Similarly, we may find the developable surface generated by the intersections of the tangent planes to a surface along an assigned curve traced on the surface. 187. We may arrive at the differential equation to de- velopable surfaces, and at the equation to the generating line passing through any point, by the following method. Since a tangent plane applied to the surface at any point, touches it in a series of points the projection of which on any 165 of the co-ordinate planes is a straight line, and since the equation to the tangent plane at a point wysx is = pa +qy +8 —pxr—qy, if y’ =ma' +n be the equation to the projection of the ge- nerating line which passes through the point wy, and in the values of p, g, and s — paw —qy, we change y into ma + n, the above equation must remain the same for all values of xv whatever, and therefore the quantities p, q, 7 — px —qy, which become functions of # only, must have their differential coefficients with respect to w equal to nothing, dp dy _— = —~=0 dw 4 u ane: ; d d q =s+f y = 0, dav dav d 1733 4 BS ( + FMS aiN a WP Gar HP Oe te sg : Hence, eliminating as rt —s* =0, the equation required ; x : . s ; and y -—y=— 5 (# —uv)y=-— - (x — x) is the equation to the projection of the generating line which passes through the point xy%. 188. When a developable surface is made plane, the absolute lengths of any determinate portions of the gene- rating lines, as well as of any curve traced on the surface, are not altered; and the angle which the tangent line to the curve at any point forms with the generating line through that point also remains unaltered ; but the curvature at any point is changed. For let AN, A’N’, &c. (fig. 57) be consecutive positions of the generating line of a developable surface, and let PR, P’R, &c. be the chords produced of a curve traced on this surface, and meeting the generating lines in the points P, P’, &c.; and for the surface let us substitute the series of plane sectorial elements Am JA’, A’m’ A”, &c., and for the curve the 166 polygon PP’P” &c. Then in turning the first plane element about A’m to bring it into the same plane with the second, and in turning the system formed by the two about A”m’ to bring it into the same plane with the third, and so on, it is evident that neither the lengths of PN, P’N’, &c., nor the angles which they form with PP’, P’P’, &c., nor the lengths of the latter, are altered, but that the angles RP’R’, RP’ R", &c. will be changed; and as this continues true, however small we take the plane elements, the properties announced above for the surface and the curve traced upon it, are established. 189. The shortest line on a developable surface has the property that its osculating plane at every point is perpen- dicular to the plane touching the surface at that point, or contains the normal to the surface at that point. If the polygon be such that, upon bringing all the plane elements into the same plane, it becomes a straight line, then two consecutive sides must always make the same angle with the intermediate generating line; that is, for every point we must have L4mPR=em PR; for when this condition is fulfilled, in the developed surface every two consecutive sides will be the prolongment of one another. Also since RP’, R’P’, make equal angles with P'N', they may be regarded as two generating lines of a conical surface whose axis is P’N’, and therefore the plane RP'R’ will ultimately be the tangent plane to this surface along P’R, and therefore perpendicular to RP’N’, which is a meridian plane of the cone containing P’R; hence, passing to the curve, the plane RP’R’ is its osculating plane at P’, and RP'N’ is the tangent plane to the surface at the same point, and these planes are perpendicular to one another; also since, when the surface is developed, the curve becomes a straight line, it is the shortest line which can connect two points on the surface through which it passes, and it has the property announced at every point. 167 190. That the same property is true for the shortest line traced on any surface whatever, appears thus. If on any proposed surface we conceive the shortest line between any two points to be traced, we may describe a developable surface which shall touch the surface according to that curve; consequently the curve of contact will be the shortest line between the same points on the developable surface, and therefore its osculating plane at every point must contain the normal to the developable surface, i. e. the normal to the proposed surface at that point. Hence if y = d (x), x = yy (x) be the unknown equations to the shortest ; d line on a surface for which = 7, - =q; expressing that the osculating plane at any point contains the normal to the surface at that point, we find dy (1 =) d’sf dy ); dx? beers = Sag -4 d d which together with e =p+t+ 7 will determine the curve. v c If we take s the length of the arc for independent variable, the equations to the curve will assume the symmetrical forms ax d’z d’y d’z = : a ds” Leryn ?- ds? os Tae 0. 191. From the property that the lengths of the ge- nerating lines of a developable surface, and of any curve traced upon it, as also the angles which the generating lines form with the curve, are not altered by the development of the surface, we can find the nature of the curve when the surface is made plane; or, conversely, a curve being traced upon a plane, we can find its nature when the plane is applied to a given developable surface. This is shewn in the case of cylindrical and conical surfaces, in the appendix. Envelopes. 192. Developable surfaces, considered as generated by the successive intersections of planes drawn after a given law, 168 are a particular case of a general family of surfaces to which we shall in the next place direct our attention. 193. To find the equation to a surface which envelopes a series of surfaces described after a given law. Let w=f(x, y, %, a)=0 be the equation to a surface containing, besides other constants, a parameter a; then if we give a particular value to a, the form and position of the surface in space will be completely determined; and if we give to it all possible consecutive values, we shall obtain an infinite number of corresponding surfaces, usually inter- secting one another two and two. The surface formed by these successive intersections, and having with each one of the surfaces the infinitely narrow zone in common which is contained between the curves in which that individual surface is intersected by the preceding and succeeding ones, (and which consequently envelopes each one of the first series of surfaces, supposing them all to exist together, and touches it according to a curve of intersection) has been named by Monge the Envelope. Also the curve in which any two consecutive surfaces intersect, he has named the Characteristic of the Envelope, because it indicates the mode of its genera- tion; thus the characteristic of developable surfaces which are generated by the intersections of planes, is a straight line ; and that of surfaces generated by the intersections of spheres, a circle, : To find the equation to the envelope, we observe that if after having given the parameter a determined value a in u = 0, we give it a new value a + oa differing insensibly from the former, we obtain the equation to a second surface differing in form and position insensibly from the first, and intersecting it in a series of potnts the co-ordinates of which satisfy the equations du . @u (da)® = 0, U-+ ——.0a + ——. da d a’ 24 du or u=0, apes ee a 169 or if the surfaces be consecutive, w= 0, — =0;3 it follows therefore that the equations to the characteristic or curve of intersection of two consecutive surfaces, corresponding to the value a of the parameter, are d S(@, Y, % a) =9, Spake Y,.% a) =0;3 and since the envelope is formed by the assemblage of the characteristics, its equation will result from eliminating a between the same equations. As this elimination, however, cannot be effected without assigning the form of f; we must retain the above system of equations to represent the envelope, regarding a in the former as a function of w, y and z to be determined from the latter. 194, Again, if in the equations to the characteristic, after having given the parameter a particular value a, which determines the position of the characteristic in space, we give it a new and insensibly increased value a+ da, the two resulting equations will be those to a second characteristic, differing insensibly in form and position from the first, and intersecting it in general in a finite number of points, the co-ordinates of which satisfy the two sets of equations du u = 0, ory du du du —. ae U) pa ——— , eee Uti da + &c. = 0, FEEL da + &e : which latter two equations, by virtue of the former, are equivalent to du & " ad’ u rere Cs = da i ‘ da’ Consequently the co-ordinates of the points in which two consecutive characteristics intersect, satisfy the three equations du d?u u =,0, wa () - =0; + &c. = 0. 170 and from these equations we may either determine a, y, and x in terms of a, that is, the co-ordinates of the point in which a given characteristic is intersected by the con- secutive; or eliminate a between them, and so find the two equations in w, y, and x, to the curve formed by all the successive points of intersection. This curve will be touched by all the characteristics, and will be the edge of regression of the envelope; for, as explained in the case of develop- able surfaces, the portions of the characteristics which are on contrary sides of their points of contact with the curve under consideration, will form two distinct sheets of the envelope, and these sheets will touch one another in that curve and have it for their common limit. 195. The propriety of the term Characteristic will be more apparent, if we generalise the equation w= 0, and suppose it to contain together with a a function of a, @ (a), so as to have the form f 2, Y, ®, ay @ (a)} =n) Then whatever form we give to @, the intersection of two consecutive surfaces represented by this equation will always be a curve of the same sort, and will therefore offer a character common to the whole family of surfaces. For this curve of intersection will be represented by the equations Smaking ¢@ (a) = B} Bes df dB Tid, ys. a, (3) = 0; at, da Saas and we see that by changing A we shall only alter the constant parts of these equations, without at all affecting the way in which a, y, # enter into them; so that for all values of B; the curve represented by them will be of the same species. 196. ‘To determine the equation to the envelope when the equation to the series of surfaces contains two independent parameters. Suppose the equation #«=0 to contain two parameters 171 a and £ independent of one another; then the equation to the surface which differs insensibly from that represented by u = 0, will be du 1 ——~ fa Sh. OB + &e. me = da and the co-ordinates of the cy in which these surfaces intersect must satisfy their two equations, that is, they must satisfy a wu = 0, ot om 0B + &c. =0; or if the surfaces be consecutive - du x du Radia aay. where A = limit of d8+da; but since A may have any value, the latter equation resolves itself into du du da ” dB these then together with w=0, are the three equations for determining the curve of intersection of two consecutive sur- faces; from whence we may obtain its two equations involving only one of the parameters; and finally by eliminating that parameter we may obtain the equation to the surface generated by the perpetual intersection of the first series of surfaces. = 0, SECTION IX. ON THE CURVATURES OF CURVES IN SPACE. 197. Preparatory to finding the radius of curvature, and evolutes of a curve in space, consider figure (59), where for the curve is substituted an equilateral polygon mm'‘m”..., and through the middle points of its sides are drawn planes respectively perpendicular to them, which intersect, two and two, in the lines-kh, k’h’, k”h”, &c. Then the plane which contains the two consecutive sides mm’, m’m”, is perpendicular to each of the planes gh, g’h’, and therefore to their common intersection kh; let kh meet this plane in the point q, then q is the center of a circle passing through the three angles m, m’,m”; and every point in the line kh is likewise equi- distant from the same three angles. The lines kh, kh’, &c. will be parallel only when the sides of the polygon mm’m”...are in the same plane; in other cases, if they be produced till each meets its consecutive, they will form a polygon hop..., the angular points of which are equidistant from four consecutive angles of the first polygon mm 'm'’...The point o for instance, since it is situated in kA, is equidistant from m, m’, m”; and again, being situated in k'h’, it is equidistant from m’, m”, m’”; that is, it is the center of a wr sphere passing through four consecutive angles, m, m’, m”, m’”. 198. The preceding results being true when the number of sides of the polygon is indefinitely increased, it follows that the normal planes of a curve generate, by their perpetual intersection, a curve surface; also, since of the lines of intersection kh, kh’, &c. every two consecutive ones are in the same plane, the surface which they generate is developable. 173 199. On the same supposition, the polygon hop...be- comes a curve, to which all the lines ko, k’p, &c. are tangents ; that is, hop...is constantly touched by the straight line which generates the developable surface, and is therefore the limit to the surface, for no part of the surface can fall within the space towards which the curve is concave. Moreover its tangent, being produced both ways from the point of contact, generates two sheets of the surface which are united and terminated in the curve; consequently, as was explained in Art. (181), the curve hop...is called the edge of regression of the developable surface, from the analogy it bears to a point of regression in a plane curve. This curve is also the locus of the points of intersection of three consecutive normal planes, or of the centers of spheres passing through four consecutive points of the first curve mm’m”..., that is, it is the locus of the center of spherical curvature. 200. The plane ggg’ passing through two consecutive tangents to the curve, and which is perpendicular to the intersection of two consecutive normal planes, is the osculating plane, whose equation was found in Art. 127; and the point qg, in which that intersection meets it, is called the center of absolute curvature. 201. To find the equation to the surface generated by the perpetual intersections of the normal planes to a curve in space. Let y= («), = = y (2), be ‘the equations to the curve, dy dy and let qn? da? &c. be denoted by 9, y”, &c.; then we have seen Art. (130), that the equation to the normal plane X—a2+(¥-y)y +(Z-%)% =0, (1) and the derived equation with respect to a, viz. (Y—-y)y + (Z—-s) a" -1-y*-x°=0, (2); are the equations to the line of intersection of two consecu- tive normal planes; XY, Y, Z, denoting the co-ordinates 174 of any point in that intersection. If therefore we eliminate w between them, we shall have the equation to the surface generated by the line of intersection in all its successive positions. On this surface lie all the evolutes of the curve, as will be shewn; it will be a cylinder if the given curve lie all in one plane, and if the given curve be traced on the surface of a sphere it will be a cone with its vertex in the center of the sphere. 202. ‘To find the radius, and co-ordinates of the center, of spherical curvature of a curve of double curvature at any point. The center of spherical curvature, or of the sphere which passes through four consecutive points of a curve, is the point of intersection of two consecutive generating lines of the sur- face considered in the last article; hence we must join to equations (1) and (2) of last article, the derived equations with respect to v, as explained Art. (185). If therefore , Y, Z be the co-ordinates of the center of spherical curvature, we have in order to determine them, the equations, X—-a+(Y-y)y + (Z—2)2 =0, (Y-y)y" + (Z—-s)2"- A +y? + 2%) =0, (P-y)y" + (Z—#)2" — 3 (y'y" + 2'2") = 0, and the radius of the sphere - a V/(X— a) + (Fy) + (Z- 2)". Moreover if from these three equations we eliminate a, there will result two equations between X, Y, Z, which will be those to the locus of the center of spherical curvature, or to the edge of regression of the developable surface generated by the intersections of the normal planes. 203. To find the radius, and co-ordinates of the center of the circle of absolute curvature. 175 The center of absolute curvature is the point where the line of intersection of two consecutive normal planes meets the osculating plane. Let s be the length of a curve from a given point to a point 7, y, x; and instead of taking two equations between &, y, x to represent the curve, let us suppose each of them to be a function of another variable ¢; and let a’, wv’, &c. denote their differential coefficients with respect to ¢; then the equation to the osculating plane at wyz is (y's — 2'y") (X - a) + (2 a” —a's")(Y —y) + (e'y" -y'2")(Z-z)=0. (1) The equation to the normal plane at the same point is we (X-—wv) +y (Y-y) +2 (Z-2)=0, (2) and this, in conjunction with the equation obtained by dif- ferentiating it . a (X-a2)+y" (F-y) +2" (Z-2)=8", (3) (since a? + y? +2" =s8", and». a’ a” +y'y" + 32/2" =8's") (4), defines the line of ultimate intersection of two consecutive normal planes (Art. 130). Hence if .Y, Y, Z, satisfy at once these three equations, they are the co-ordinates of the center of absolute curvature at wyz; and the radius R will equal V(X — «)? + (¥ -y)? + (Z - 8)? Now from (1) and (2) taking account of equations (4), we get by ordinary algebraical operations, AX —& Y-y Z-— savas’ sy —ys" sls" — 2! 8" 5 : from (3) rom (; s a wy + y”? 4 9/2 _ 32 oe! 4. yf” 4. 9/2 — gl ? which equations give the radius and co-ordinates of the center of the circle of absolute curvature. Cor. If we make w the independent variable by putting 176 , Ad ”? d a t=, so that 7 =1, 7 =0, and y'sy &c., become et “ y die da’ we get by reduction (1+ y? +s”) = ja eh Gisele yy Sy + 8? + (2"y! — xy") 204. Also if we make s, the length of the arc, the in- dependent variable instead of ¢, we get immediately 1 R ad J x se y” “ 2 i” dx where wv denotes ds?” &c.; and the expressions for the co- ordinates of the center assume the following simple forms, a ” A-a“7= a” 4/8? 43 1129 Y-y= a? 4 of 4 agit? y gl! -Z%= WN wey + 3/2 205. A curve of double curvature may either have three consecutive elements in the same plane, or two consecutive elements in the same straight line; the first is called a simple inflexion, that is, when the curve at that point becomes plane, and therefore the radius of spherical curvature infinite; and the second a double inflexion, as necessarily including the former, that is, when the curve at that point becomes a straight line, and therefore the radius of absolute curvature infinite. If in Art. 202, we actually determine the values of XY — a, Y-y, Z-—z, we find for their common denominator the expression vIn OP she Md 2 wy -YyY ® 3 hence the condition 177 expressing that the center of spherical curvature is at an infinite distance, will determine the points of simple inflexion, and agrees with Art. 128. At points of double inflexion, the radius of absolute curva- ture changes its sign, and the curve, after being concave in one direction, crosses its tangent line and becomes concave in the opposite direction; hence also its projections on the planes of sv and wy cross their tangents, and have corresponding points of contrary flexure; at such points therefore, and at cusps, d’ z | d’y ——=0, Or ©, — =0, or ©; da® div® which are the equations for determining the abscissee; and which we observe make the expression for the radius of absolute curvature infinite or evanescent, as ought to be the case. 206. The tangent lines of the given curve mm’,..(fig. 59) will generate a developable surface, whose edge of regression is the curve mm’...; and since the tangent line is the intersection of two consecutive osculating planes, this surface would also be generated by the perpetual intersection of the osculating planes. But the lines of intersection of the normal planes are perpen- dicular to the osculating planes; therefore the angle between two consecutive lines of intersection, that is, two consecutive tangents of op..., is equal to the angle between two consecutive osculating planes of mm’... Again, the angle between two consecutive tangents of mm’... is equal to the angle between its corresponding normal planes, that is, to the angle between two consecutive osculating planes of op... If therefore we call the angle between two consecutive tangents of a curve of double curvature its first flewion, and that between two con- secutive osculating planes its second flewion, we may enunciate the above properties as follows. The first flexion of a curve of double curvature is equal to the second flexion cf the edge of regression of the developable surface generated by its normal planes; and the first flexion of the latter, is equal to the second flexion of the former. 12 178 Evolutes of a curve in space. 207. Every curve, whether plane or of double curvature, has an infinite number of evolutes, all of which lie on the developable surface generated by the perpetual intersection of the normal planes. Recurring to fig. 59, since kA is perpendicular to the plane through two consecutive sides mm’, m'm”, of the polygon, any point f in it is equidistant from the middle points of the sides g and g’. If therefore we draw gf in the first normal plane, and g’ff’ in the second, the lines gf, g’f are inclined at the same angle to kh; and if with center f and radius fg a circle be described, it will touch the two consecutive sides mm’, m'm” in their middle points g and g’, and when the describing radius comes to g’ it will be confounded with g’f’. In like manner if we draw 2 f'f” in the third normal plane, the lines g’f’, gf’ are equally inclined to k’h’; and a circle described from f’ with radius f’g’ will touch the consecutive sides m’m”, mm” in their middle points, and the describing radius will be con- founded with g”f” at the point g”. By continuing this con- struction the polygon fff’... will be formed, by unwinding a thread gf from which, circular arcs, touching every two consecutive sides in their middle points, will be described. Therefore when the number of sides is indefinitely increased, the polygon fff’... will become a curve traced on the develop- able surface formed by the intersections of the normal planes ; and by unwinding a string from it, the proposed curve of double curvature will be traced out. Also since the direction of the initial radius gf is arbitrary, any other direction would have produced a curve endowed with the same properties as ff/f”...; it follows therefore that every curve has an infinite number of evolutes all contained on the surface generated by the perpetual intersections of its normal planes; in the case of a plane curve all the evolutes, except that in its own plane, are curves of double curvature, traced on the right cylinder whose base is the evolute in its own plane. 179 208. Since ze fk =e" f'k' = h'f'f’, if the plane hk’ were brought into the same plane with h’k” by being turned about kh’, ff’ would be brought into the same straight line with ff’; and since the same may be proved successively of all the other sides of the polygon fff’..., it follows that if the surface formed by the intersections of the normal planes were developed, the evolute ff'f’... would become a straight line, and therefore would be formed by stretching a thread in the direction gf and applying it freely to that surface. Hence if from any point in a curve of double curvature, any line be drawn touching the surface formed by the normal planes, and be applied freely to that surface, it will form an evolute of the proposed curve; and will be the shortest line that can be drawn between any two points of the surface, through which it passes. 209. It is to be observed that the centers of absolute curvature are not situated on an evolute of the curve. For if g’q’ be perpendicular to kh’, then it cannot pass through the point g; because it would then coincide with g"q, and therefore cut kh at right angles, which would require that kh and kh’ should be parallel; and this can never happen as long as the proposed curve is of double curvature. Hence the consecutive radii of absolute curvature gq, gq’, since they are in different planes, and do not meet one another in the common intersection of those planes, therefore they do not meet one another at all; and therefore cannot be consecutive tangents of the same curve; that is, the locus of the center of absolute curvature is not an evolute. 210. ‘To find the equations to any proposed evolute of a curve of double curvature. As we know the equation to the surface on which all the evolutes lie (Art. 201), it only remains to find the equation which particularizes a given evolute. Let X, Y, be the co- ordinates of a point in the projection of an evolute, wv, y, the co-ordinates of the corresponding point in the projection of 12—2 180 the curve; then a tangent to the projection of the evolute, must pass through the corresponding point in the projection of the curve; Sed Oey Ad Xs This joined to the two equations oe +4 Yo y) y+ (Z—2)x'=0, (VY —y)y"+(Z —2)2"- (14+ y? +2”) =0, which belong to the generating line of the developable surface containing the evolutes, will give by eliminating w the two equations to the evolute, one of them being a differential of the first order; and its integral will introduce an arbitrary constant, by means of which the evolute may be made to satisfy that condition which particularizes it; as, for instance, to pass through a given point of the developable surface on which it is traced. SECTION X. ON THE CURVATURE OF SURFACES. 211. ‘To find the requisite conditions for a contact of the first, second, &c. order, between two surfaces. If two surfaces, referred to the same origin and axes, pass through the same point, the co-ordinates of which are xv, y, %; and if we change w into w+h, and y into y +k, the equation to the first surface will give for the value of the new ordinate, st+ph+qk+4 (rh? + 2shk + tk’) + &e. and the equation to the second surface s+ Ph+Qk+h(RW4+2ShK 4 Th) + &e.; the distance of the surfaces, measured in the direction of their ordinates, will therefore be expressed by (P-p)h+(Q-gQk+3}(R-r)h?+2(S-s)hk+(T-d kh} + &e. If we suppose the equation to the second surface to contain a certain number of arbitrary constants, we may determine them so as to make the first terms of this dis- tance vanish; and it will follow that any other surface, for which these terms do not disappear, cannot be situated between the two former with reference to the points which are contiguous to their common point; at least so long as we take h and & so small, that the sum of the terms of the first order may be more considerable than that of all the terms of succeeding orders. When we have P—p=0, Q - q=0, the surfaces will have a contact of the first order ; if besides these, we have R-r=0, S—s=0, T'—-?#=0, the contact -will be of the second order, and so on. 182 212. Let therefore V=0 be an equation between three variables a’, y’, x and a certain number of arbitrary constants, and according to the usual notation, let P, Q, R, &c. denote the partial differential coefficients of ’ with respect to a’ and y 3 by the determination of the constants we may make this surface have with one completely given (the co-ordinates of which we shall represent by a, y, x, and the partial differential coefficients of z, by p, qg, &c.) a contact of an order which will depend on their number. ‘The first condition to be satisfied is that the surfaces may have a common point, that is, that by changing #’ into a, and y’ into y, in V= 0, we may find x’ = x. Besides this, for a contact of the first order, upon making the same substitutions for # and y’, we must have P= p, Q=@q;3 and for one of the second, besides all the foregoing conditions, we must have R=r7r, S=s, J'= Hence it appears that a contact of the first order requires three arbitrary constants, and one of the second, six; and in order to have a contact of the » order, since the terms of three, four, &c. dimensions in # and & in the develop- ment of the difference of the ordinates in the preceding Article are, in number, 4, 5, &c., the number of disposable . constants must be | 14+424+3+4&c.4+ (n+ 1)=4$(n +1) (7 + 2). 213. But if we confine our attention to the sections of the surfaces made by a plane containing their common ordinate #, and inclined to the plane of za at an angle whose tangent is m, then, making k=mh, the difference of the consecutive ordinates in these sections will be $P—p+(Q-9)mth+4{R-r+2(S—s)m+(T-t)m t{th’+&e. Hence if the surfaces have a common point and a common tangent plane at that point, that is, if upon making a’=a, and y= y, we have x=, P=p, Q=gq, then the first term of the above difference will disappear, and the second will also vanish if we add a fourth condition, viz. R-7r+2(S-s)m+(T'-t)m =0.— 183 Hence four disposable constants in the equation to a surface are sufficient in order to establish between it and one com- pletely given a contact of the first order at a proposed point, and also to establish a contact of the second order between the sections of the surfaces made by a See: plane containing their common ordinate. 214. Previously to applying these considerations to es- timate the curvature of any surface at a proposed point, we may observe that we cannot attempt to assimilate the curvature of a surface in general to that of a sphere; because in the latter the curvature is uniform about the same normal, whereas for surfaces in general that is far from being the case. The mode of proceeding must therefore be to imagine several planes drawn through the normal at the point under consideration, to calculate the radius of curvature of each of these sections at that point, and by comparing them to judge of the greater or smaller curvature of the surface in those directions about the point, as well as towards what parts the curvature is turned; for, as we know, certain surfaces contiguous to any point are situated partly above and partly below the tangent plane at that point. In making different planes pass through the same point of a surface, so that some contain the normal to the surface at that point, and some do not, we shall find the radii of cur- vature relative to that point both of the normal and oblique sections, bearing remarkable relations to one another, inde- pendent of the particular form of the surface. We shall begin by demonstrating the simple relation which exists be- tween the radii of curvature of a normal and an oblique section, made by planes passing through the same tangent line to a surface, first noticed by Meunier. We shall then, by the doctrine of contacts, deduce the radius of curvature of any normal section of a surface; and afterwards prove Euler’s theorems relative to the curvatures of the principal normal sections of a surface. ‘The circumstances under which the normal at any point may be intersected by the normals at immediately adjacent points will next fall under considera- 184 tion, as throwing great light on the subject and leading to many results of interest. 215. If a normal and an oblique section of a surface be made by planes passing through the same tangent line to the surface, the radius of curvature of the oblique section is equal to the projection on its plane, of the radius of curvature of the normal: section. Let the tangent plane to the surface be the plane of wy, the point of contact the origin, and the tangent line the axis of w; then the normal to the surface will be the axis of =; let OP (fig. 62) be the normal section in the plane of sv, OP the oblique section made by a plane s’Ow through Ow, and inclined to the normal section at an 22Oz=6@; NP, NF’, ordinates to the two curves corresponding to the common abscissa ON =h; also let NP=2, and let h, &k, 2’ be co-ordinates of P’, Then, (assuming that in any plane curve when the axis of the abscisse is a normal at the origin, the radius of curvature at the origin is equal to 4 limit of oe eee) ———— | we.have abscissa milidevorMbureature of Onests0 oe ion: cee FA=radius of curvature o at O= 5 Imit o WP? ON? Casita ; ; a ( ee ise ye sa ee... R= radius of curvature of OP at O=4% limit of NP” ~ NP’ x’ sec — = limit of ——= limit of - -— =sec@. limit of NP x a | & But since the plane of wy is the tangent plane at O, p=0, q=C, and s=trh?+shk+4tkh + &e. and making k = 0, ee pas trh? +t+—.h' + &e.3 185 tk k : ; because limit of re since Ow is a tangent to the pro- ) jection of OP’ on the plane of wy; . R= Rcoos8. Cor. Hence it follows that the osculating circles of all sections of a curve surface that have a common tangent line, are situated on the surface of a sphere the radius of which is the radius of curvature of the normal section pass- ing through the same tangent line; for OA’= OA cos AOA (and consequently AdA’O is a right angle) if OA be the diameter of the circle of curvature of the normal, and OA’ that of the circle of curvature of the oblique section; the latter circle is therefore situated on a sphere whose diameter is AO. 216. To find the radius of curvature of any normal section of a surface at a given point, in terms of the co- ordinates of that point. Let w, y, x be the co-ordinates of the given point P (fig. 63) of the surface, p, q, 7, &c. the partial differential coefficients of x expressed in terms of those co-ordinates ; PT the tangent line through which the normal section is to pass, which, since it lies in the tangent plane at P, will be determined by the equation to its projection on the plane of vy, viz. y—y=m(a'—«). Then if we determine a sphere having a contact of the first order with the surface at P, and whose section by the vertical plane PQT' has a contact of the second order with the section of the surface by the same plane, all planes passing through P7' will cut the sphere in circles, which are the circles of curvature to the corresponding sections of the surface, and therefore the radius of the sphere will be the radius of curvature of the normal section; and this we are enabled to do, because the four disposable constants in the equation to a sphere will enable us to satisfy the four requisite conditions, which are, 186 that upon making 2’=«a and y'=y in the equation to the sphere, we must have s=2, P=p, Q=q, R-7r+2(S-s)m4+(T-t)m’ =0. Now the equation to the sphere gives, by differentiation, (2 — a)? + (y'- By + @'- y= 8 a’—-at P(s-y)=0, y'-B+ Q(2’-y) =9, 14+-P°+R(s'-y)=0, PQ+S8(%-y)=0, 14Q°?+T7(s'-y)=0; hence, there result between the constants a, B, yy, 0, and the co-ordinates vw, y, x, the following relations (w—a)’+(y- 8)? + @-yyP =e, w-atp(s-y)=0, y-B+q(?-y) =9, 1 2 2 Mica +rgo( Pt 4) m + (= +t) m= ; e—%¥ z-¥ z The latter gives 1+p°+2pqm+(14+q)m y+2sm+tm e-y=— and then a, 2, 6 are known from the equations w-a=—p(z—-y), y-B=-9g(e-y), P=(e-y)P(+p+e); Vi1l+ pit’ {1 +m? + (p+qm)°} . the radius = ry +2sm +tm’ Hence we have determined the radius and co-ordinates of the center of the circle of curvature of the normal section at a point wy, whose intersection with the tangent plane at that point is projected into a line represented by the equation y-—y=m(a'— 2). Cor. Suppose the tangent line to make angles dA, pu, v with the axes of a, y, x, and let o be the length of the arc of the normal section, then (Art. 28 and 131), 187 COS ph m =——, and cosr do HAG (54) +m + ( + mg = (2 2 (=) poe: da +(= m4 P iy = | , hence, by substitution, we get for the radius of curvature Y1l+p+¢ ~ # (cosA)* + 28 cos dA Cos p + t (cos nu)?” 217. But if the equation to the surface be proposed under the form F'(a, y, #) = 0, the radius of curvature of a normal section may be more readily obtained by considering (Art. 200) that the center of curvature of the section is the point in which the normal meets the line of intersection of two consecutive normal planes. dk d dF lor ee eae eal — = W, so that dx dy dz Ude+ Vdy+ Wdz=0; then the equations to the normal at wy are AX-« Y- Z—s a ays a ees Q, suppose (1). Now if ds be an element of the arc of the normal section, and da, dy, dz its projections on the co-ordinate axes, the equations to the normal plane to the section will be (Cor. Puric. 120. } (X -x)dx+(Y-y)dy+(Z-—sz)dz=0; (2) and the line of intersection of this with its consecutive is given by (2), together with its differential, viz. (X — 2) @a+(Y-y) dy +(Z—2)a@x=ds*. (3) Hence if XY, Y, Z be determined so as to satisfy the above three equations, they will be the co-ordinates of the center of curvature of the normal section; and if R be the radius of curvature, we shall have from (1) CMS OL yy + (2-3 x) v= \/U? + VV? + W?.Q 188 but from (3) we get (Uda + Vay + Wd’s) Q= ds’; ds?.4/U? 4+ V2.4 W Oy are ae Udu+ Vay + Wax But since Ude + Vdy + Wdz =0, if we differentiate and call dF d’F dF Teak —-=UV, ——~=W; dye? dx" aon pains aay ihe fatdtin tie dydzs’ dads’ dudy we get UPx+ Va@y + Wd’: + uda’ + vdy + wdz + 2uwdydz +2vdedz+2wdady =0; consequently, substituting, and calling dx } dy dz Bei —=m, —=N, ds ds ds et we have \/ U? + V24 W ~— at mv + niw + 2imw' + 2lnv'+ 2mnw 3 which coincides with the former expression for R, if we suppose (wv, y, 7) =2—f(a, y) = 0. 218. Hence we can find the conditions, in order that the curvatures of the normal sections of a surface at a pro- posed point may be all in the same direction, or in opposite directions; that is, in order that the surface may be convex or non-convex about a proposed point. If in the value of # (Art. 216) we always take the radical positive, as the numerator 1 + m* + (p+mq)’ is incapable of changing its sign, the sign of J will depend upon that of its denominator ] r+ 28m RE in S(mt +s)? +rt— 8°}; 189 if therefore xv, y, x the co-ordinates of the proposed point be such that rf — s*>0, then # cannot change its sign for any value of m, and the surface contiguous to that point will be situated entirely on thé same side of the tangent plane, or will be convex at that point; and if this be true for every point, as in the ellipsoid, the surface will be entirely convex. If, on the contrary, r¢—s*<0, there are two values of m for which the denominator will vanish, and R will be infinite and will change its sign as m passes through each of these values; therefore the surface will have opposite curvatures about the point in question, or will be non-convex. Thus in the surfaces considered (Art. 176), of which the equation is gr—2pqs+pt=0, or (qr—ps)’ +p? (rt —s’) =0, rt —s° is necessarily negative, and therefore surfaces of this class are non-convex at every point. For developable surfaces, where r¢— s* = 0, the denomi- nator of # is a perfect square, and consequently retains the same sign; therefore the radius of curvature will haye the same sign for every point of the surface; only it will at every point be infinite when mt +s = 0, the direction so determined evidently coinciding with that of the generating line passing through the point. 219. 'To determine the normal sections of greatest and least curvature at a given point of a curve surface. For the same point of the surface, the expression for R varies with m, and we may find what value of m will make it a maximum or minimum by putting the differential coefficient of & with respect to m (for the quantities p, q, r, &c. are in general independent of the position of the tangent line which fixes the normal section) or rather of x —y on which R depends, equal to 0; hence putting equation (1) (Art. 216), under the form (x -—y) (v7 +28m + tm’) +14 p> + 2pqm + (1 + q’) m® = 0, we have (x -—-y) (s+ tm) + pat (1+q’)m=0; 190 therefore eliminating successively x —-y and m from these equations, we get S(14q°)s—pqt}m?+ }(1+q°)r-(1+p*)t} m— { (14+ p*)s—paqr} =0, (rt-s°)(x-y)?+{(1+p’)t-2pqs+(1 +7 )r}(s-y)+14+p°+q=0. These equations being of the second degree, it follows that in general the radius of curvature of the normal section, as the cutting plane revolves about the normal, will have only one maximum and one minimum value; which, in absolute magnitude, may be two maxima or two minima, when the curvature of the surface changes its sign about the given point. These two normal sections, one corresponding to the least, and the other to the greatest radius of curvature, are called the principal sections of the surface relative to the point at which the normal is drawn, and the corresponding radii the principal radii of curvature. Cor. Hence the former of the above equations enables us, for each point of the surface, to determine the directions of the principal sections, and we might also shew from it that these directions are at right angles to one another. To determine however the angle between the principal sec- tions in a simple manner, let us suppose the plane of vy to be coincident with the tangent plane at P, or only parallel to it, (this of course we may do without at all altering the form of the surface or the positions of the tangents to the principal sections) then we must put p=0, g=0, and the equation becomes Brg ae PGR TTD 28, the two roots of which are always possible and satisfy the condition therefore the principal sections always exist and are at right 191 angles to one another. Also the same supposition gives for the radius of curvature of any normal section (Art. 216), 1 +m? e+ 25m4+tm- Having however made the foregoing process answer its pur- pose of giving the radius of curvature of any normal section of a surface in terms of the co-ordinates of the point, we shall not deduce from it all the important results which it is capable of furnishing, but obtain them by a direct and much simpler method. 220. ‘The sum of the curvatures of any two normal sec- tions at right angles to one another, is a constant quantity at the same point of a surface. Let AO, the normal at any point A of a curve surface, be the axis of zs, and wdAy, the tangent plane at the same point, the plane of wy (fig. 64). Let AB be a section of the surface made by any plane passing through AO, AC the corresponding section of the tangent plane, which therefore touches the plane curve 4B at 4; and let y = tan@.« be the equation to AC, so that 0@=ZNAC, Then if 40 =R be the radius of curvature of the plane curve 4B at A, and BC be parallel to 40, we have by a well-known theorem A 2 R= 3 limit of = - Let dN=h, NC =k, be the co-ordinates of C; “ BCa=trh?+shk+ttkh + &e, for p=0, gq =0, since the plane of wy is the tangent plane ; also kA = tan@0.h; +i * R= limit of —— —_ —__— 2 Lrht-+shk+ 4th + &e. a): Petal Oe = limit of ; dr r+2stan@+étan?@ + met htan?@ + &e, aw 192 1 + tan?@ or 2 = — r+ 2stan0+¢tan’@’ where 7, s, £ are the values of d’z d’x d’s du?’ dady’ dy’ at the point A derived from the equation to the surface. Hence, if R’ be the radius of curvature of another section inclined at an angle = 90°+ @ to AN, we have 1 : Ke rea 7 cos’ @ + 2s cos @ sin @ + ¢ sin’ 0, ] ; : . —=rsin?@ — 2ssin 0 cos@ + ¢ cos’ @; R' which (taking, as usual, the inverse of the radius of curva- ture to measure the curvature) expresses that the sum of the curvatures of any two normal sections at right angles to one another, is a constant quantity at the same point of the surface. 221. Of all sections of a curve surface made by planes drawn through the normal at any point, to determine those of greatest and least curvature at that point; and to shew that they are at right angles to one another. We have fay dart ; R= 7 cos’ O + 28 sinO cos O + ¢ sin’ A, he r (1 + cos 20) + 28 sin 20 + ¢(1 — cos 20) =r+i#+4 (r —- ¢t)cos20 + 28 sin 20, which is to be a maximum or minimum by the variation of 0; 193 therefore putting its differential coefficient with respect to @ equal to zero, we get — (r — f) sin20 + 2s cos20 =0; 28 *, tan20 = ari tan (180° +20); (1) , 28 yr—t sin 20 = £ ——______—. , cos 20 = ok Sey J (r — ty + 48° VJ (r —t)? + 48 2 (r — ty 48° We —_ = 7 + Zt + yg de a ; a JS (r—t +48? SA (r—f) + 48 : 1 17 Sete ot or Fy es (r — t)* + 48°, which are the values of the reciprocals of the greatest and least radius of curvature; and the positions of the sections to which they belong are fixed by the angles 0 and 90° + @ obtained from equation (1); consequently the sections of greatest and least curvature at any point of a surface, called the principal sections at that point, are at right angles to one another. 222. The curvature of any normal section is equal to the sum of the curvatures of the principal sections, multiplied respectively by the squares of the cosines of the angles which its plane forms with their planes. Suppose the axes of w and y to be drawn in the principal 28 planes, then since 0=0 and tan26 = , we must have yr —t s = 0; hence 1 : 2 cet —=rcos'@ + ¢sin’@; R consequently, if we denote by p and p’ the radii of curva- ture of the principal sections corresponding to @=0 and 8 = 90°, we have 1 wi ye p p 194 therefore substituting, we have for any normal section, hp roe D daisy a p This theorem, due to Euler, joined to that of Meunier, contains the whole theory of the curvature of surfaces; for it is hence sufficient to know at any point of a surface the directions and curvatures of the principal sections, to deduce the curvature at that point of every other section, normal or oblique. 223. When the two principal radii have the same sign, the formula ha —cos’@ + oa sin® 0 La p proves that every normal section will have a radius of that sign, and therefore the surface will be entirely on the same side of the tangent plane at the proposed point; also the principal radii will be the maximum and minimum values of the radius of curvature at that point, and the form of the surface will be that represented in fig. 65, where AO, AO’ are the principal radii, and AQ is the radius of the intermediate section AP. When the principal radii are equal, as well as of the same sign, the formula gives R =p, and therefore all the normal sections have the same curvature, and may be regarded as principal sections; of this we have examples at the vertex of a paraboloid of revolution, and the poles of a spheroid. 224. When the principal radii are of contrary signs, p positive and p’ negative for instance, 1 1 e € cos® 9 — — sin’ 0, it ft aep p which vanishes, or R is infinite, when @ has such a value w 195 that tanw = + Je. hence for values of 6 between 0 and + P R is positive, or the section situated above the tangent plane; and for all other values, R is negative, or the section situated below the tangent plane; also p will be the minimum of the positive radii, and p’, numerically, the minimum of the negative radii. Hence at the proposed point, the form of the surface will be that represented in fig. 66; DBE being the intersection of the surface with a sphere center 4; AE, AF sections having respectively a positive and negative radius, and AB the section made by the limiting normal plane throu gh AO. Thus in the hyperboloid of one sheet, we know that the tangent plane at any point cuts the surface in two straight lines, limiting the normal planes which give positive and negative radii of curvature, or which separate the convex and concave parts of the surface. In surfaces of a higher order, the limiting normal planes will cut them in curves having with the corresponding sections of the tangent planes a contact at least of the second order, since the curvatures of those sections are infinite. In developable surfaces, if we take a principal plane at any point for that of sa”, then s =0, and since rf —s°=0, r=0 or ¢=0; therefore one of the principal radii is infinite. In fact, we know that the tangent plane to a developable surface does not cut it, but touches it along a generating line; and that this generating line is a principal section whose radius of curvature is a maximum and infinite, whilst the minimum radius belongs to the section made perpendicular to the generating line. Hence, the curvature of any surface at every one of its points may be assimilated to that of a surface of the second order at one of its vertices, as will be seen in the following Article, where we shall take a paraboloid, although an ellipsoid would do equally well. 225. To determine a paraboloid of the second order, which shall have at its vertex, a complete contact of the second order with a given surface at a proposed point. 13—2 196 Let A (figs. 65 and 66) be the given point of the sur- face, Az the normal, and let the principal sections meet the tangent plane in Aw, Ay, which lines take for the axes. Also, let z= os A se 2p 2° be the equation to a paraboloid, p, p being the principal radii of curvature of the proposed surface at 4. Then if AP be a section of this paraboloid through its axis inclined at an Z2@to ga, and AN=r, PN =2, be co-ordinates of P, the equation to AP, putting rcos@ for # and rsin@ for y, (Art. 101) is 1 Lhe sat ef cos’ @ + — sin? af : 2 if which represents a parabola, the reciprocal of whose semi- latus rectum, and therefore of the radius of curvature at its vertex, Hence the curvature of every section of the paraboloid is the same as that of the corresponding section of the surface, and therefore the paraboloid has a complete contact of the second order with the surface. Cor. What we here assume, viz. that if two surfaces, having a common point and common normal at that point, have the curvatures of all normal sections equal, (or, which comes to the same thing, have their principal sections coin- cident and equally curved,) they have a complete contact of the second order at that point, agrees with Art. (211); for this, from the general expression for the radius of curvature of any normal section, requires that x and its differential co- efficients p, q, 7, s, #, should have equal values in the equations to the two surfaces, when in them we substitute the co-ordinates w and y of the point under consideration. 197 226. In the equation to the paraboloid, suppose x constant, and =e, therefore x y rel oge 2 pec 2p ec 1; this is the equation to a section of the paraboloid, or to a section of the surface if ¢ be indefinitely small, Hence it appears that if the surface be cut by a plane parallel to the tangent plane at any point, and indefinitely near to it, the section is ultimately a curve of the second order whose center is in the normal, and axes in the planes of greatest and least curvature; also the square of the diameter in which the curve is intersected by any plane drawn through the normal, is proportional to the radius of curvature of the corresponding section of the surface. This curve has been called by Dupin, the Jndicatrix of the surface, because it indicates the directions of the curvatures; if at any point it be an ellipse, p and p’ must have the same sign, that is, the curvatures are in the same direction; if a hyperbola, the curvatures. are in opposite directions; and if a circle, p= p> and the curvature of every normal section is the same. For instance, it has been shewn (Art. 104) that an ellipsoid may be generated in two ways, by a circle of variable radius moving parallel to itself; consequently there are four points on its surface at which a plane, parallel and indefinitely near to the tangent plane, cuts it in a circle; that is, the indicatrix at those points is a circle, and therefore the two radii of curvature equal. These points are called wmbilicit, and are symmetrically placed in the four angles of the principal section containing the greatest and least axes. It is manifest that at these points, a sphere can have a complete contact of the second order with the surface. Intersection of Consecutive Normals of a Surface. 227. As two lines in space will not intersect unless the constants which enter into their equations satisfy a 198 certain equation of condition, so if from any point in a curve surface at which we have drawn a normal we _ pass to a contiguous point, the normal at the latter point will not intersect that at the former, however near the points are to one another, unless the second point be taken in such a direction as to satisfy the equation of condition for the intersection of the normals. ‘The consideration of consecu- tive normals throws great light on the subject of the curvature of surfaces, as will appear from the following propositions. 228. Having given a point on a curve surface, to find the directions in which we must pass to consecutive points, in order that the corresponding normals may intersect. Take the normal at the given point for the axis of x, and the tangent plane for that of wy, and let h, k, Jd be co-ordinates of a contiguous point situated in a_ section through the normal made by a plane inclined at an angle 0 to that of zw, so that K=hAtan@; then the equations to the normal at that point are wv —h+p,(2' -l) =0, y —kign(s-J) =0, where p,, g, denote certain functions of h and k (p and q d d: denoting the values of 7. and a v y fore each = 0). Then in order that this line may meet the axis of x, its two equations must agree in giving the same value for x’ when wv’ and y’ = 0, that is, at the origin, and there- / 2 aay s =—+land vs = —+], Pi Nh must be the same. But if the points be consecutive, then the ultimate values of x’ must be the same when h = 0, that is, (expanding p, and q,), ie A h 1 SY scilimit of pee cae 3 rh+sk+&c. r+stan@ 199 k tan @ sh+tk+&c. 8s +#tan9 d’°s dz d? x (7,8, ¢ being the values of qa®? bray ; ia the origin) must and x = limit of be identical ; ..s+é#tan@ =rtan@ +s tan’ 0, 2 tan @ 28 or tan 29 = ——————. = ——_ 1—tan®@ r-—f# r! an equation which being the same as that for finding the positions of the principal sections through the origin, shews that there are two, and only two, directions at right angles to one another in which we may pass from a proposed point in a curve surface to a consecutive point, so that the normals may intersect; and that these directions coincide with the principal sections through the proposed point. Also the value of 1 | ee poe oe eee sar+stanO =} frt te (1-2) + 48°} shews (Art. 221) that the points where the normal is in- tersected by consecutive normals, are the centers of curvature of the principal sections. 229. This property of consecutive normals enables us in some cases to determine at once the principal sections of a surface at a proposed point; for instance, in surfaces of revolution the plane of the generating cwrve, supposed to be a plane curve, is necessarily one of those sections, because in it consecutive normals intersect, and therefore a plane through the normal perpendicular to it is the other. The first series of consecutive normals intersect in the evolute of the generating curve, the second in the axis; hence, in a surface of revolution, the loci of the intersections of consecutive normals, or of the centers of curvature of the principal sections, are the axis of revolution and the surface formed by the revolution about that axis of the evolute of the generating curve. Hence 200 we can find immediately the radius of curvature of any section of a surface of revolution, the principal radii being the radius of curvature of the meridian, and the portion of the normal intercepted between the proposed point and the axis of revolution. Lines of Curvature. 230. A series of points on a surface determined by the condition that the normals at any two consecutive ones intersect is called a line of curvature of the surface. Every point of a surface, as appears from the above investigation, is situated on two curves of this kind, whose dircctions coincide with those of the principal sections through that point, and cut one another at right angles. Thus in start- ing from a point P of a surface (fig. 68), there are two contiguous points P’ and Q the normals at which will inter- sect the normal at P; and if of these we take only that which is in the same direction with KP, namely P’, and then advance in the same direction to the contiguous point whose normal intersects the normal at P’, and so on con- tinually, we shall obtain KPP’n a first line of curvature of the surface; the second line of curvature which passes through P will be obtained in the same manner, and will be RPQE; and as these constructions may be repeated for every point of the surface, we shall thus form two series of lines of curvature dividing the surface into curvilinear quadrilaterals whose sides in space cut one another at right angles. It must be observed that the lines of curvature which pass through any point and the principal sections through the same point, although they have common tangent lines at that point, will not usually coincide. Thus in a surface of revolution the meridian PB (fig. 67) and the section through the normal perpendicular to the meridian QPG are the principal sections at P; but only one of them is a line of curvature through P, viz. the meridian; the other line of curvature being manifestly the parallel PD, which has a common tangent with PQ at P. 201 231. To find the differential equation to the projection of the lines of curvature which pass through a given point of a curve surface. The equations to the normal at a point wysx of a curve surface, are we —ax@t+ p(s -x)=0, y —-y+q(% — 2) =0; and at a contiguous point v+h,y+hkh, x +1, the equations to the normal, are wv —xvx—-h+p,(2’-x-l=0, y-y-kig(s'-x-l=0; Pir Qs» denoting the same functions of «+h, y +k, that p and q are of w and y, as derived from the given equation to the surface x = f(v, y). If these two normals intersect, their equations must be satisfied by the same values of a’, y’, 2’, which will be the co-ordinates of their point of intersection ; eliminating therefore wv’, y’, x’, between the four preceding equations, we shall have an equation of condition expressing that the normals intersect, and which will establish a relation between A and k& or fix the direction in which we must pass from the proposed point to a consecutive point in order that the normals may intersect. Subtracting the first equations from the second, we have —h+(p,— p) (# - 2)-pil=0, -—k + (4-9) (# - 2) -Ql=9, and these equations must agree in giving the same value for 2, oS h+pl_ pi-P k+ ql ara Now let the points be consecutive; then expanding p,, q,, and /, and retaining in their developments only those terms which involve the simple powers of h and k, we must have h + (; *h +sk)(ph k ie A h+sk limit of ec ei) (2 oe ) = limit of sath : k+(q + sh+tkh) (ph + qk) sh + tk But if 6 be the angle which the tangent to the projection of a line of curvature on the plane of wy makes with the axis of a, 202 we have & = Atan@ ultimately ; hence substituting this in the above equation, and making / vanish, we have 1+ p* + pq tand r+stan@ tan0d+pqt+q'tanO s+é#tand@ But if we now consider wv and y to be the co-ordinates of the ae : d projection of the line of curvature, we have tan @ = one hence, Lv substituting and reducing, {(1 + 9°) s — pqtt (5%) airy ’ +f +g) r= (+p) = — {14 ps - par} = 0. It remains to substitute for p, q, 7, s, ¢ their values in terms of w and y derived from the equation to the surface and to integrate; the result will be the equation to the projection of the lines of curvature. e © . . e ° d ° Since the above equation is of two dimensions in — its x integral when completed by the arbitrary constant C’, will be of the form C’ + Cd (a, y) + (w, y) = 0. Suppose the line of curvature is to pass through a point for which vw =a, y = 6; therefore C? + Cq@ (a, 6) + (a, b) = 0, which will give two values of C; and by substituting them successively in the complete integral, we shall have the equations to the two lines of curvature passing through the given point. Cor. We may find the differential equation to the lines of curvature of a surface whose equation is given under the form F' (a, y, ) = 0, as follows. Let U, V, W have the same meanings as in Art. 217, then the equations to the normal at a point wyz and at a con- tiguous point are X-w Y-y Z-2z _U Ley eye + W) Ape da eae y — dy ee Gass tds Ted Wade? ewe awe 203 the second of which by expanding and neglecting small quan- tities of the second order becomes dv dU # nie = Sly be ya ee ms ss v + ( v) i +(Y- py wt (Z Aye ye Lares Eliminating 1, Y, Z between (1) and - we get for the condition that consecutive normals may intersect, U(dzdV —dydW)+V(dedW —dzdU)+W(dydU-dedV)=0, the integral of which joined to the equation F(a, y, x) = will determine the lines of curvature of the surface. If we suppose (x, y, 3) =f(a, y)-#x=0 so that V=p, V=gq, W=-1,dU=rdx+sdy,dV=sdax +tdy, dW =0, we fall upon the equation given above. 232. The points in which the normal at any point is intersected by the consecutive normals, coincide, as we have seen, with the centers of curvature of the principal sections at that point; and the portions of the normal intercepted between these points and the point of the surface at which it is drawn, (which, as we have seen, are the radii of curvature of the principal sections of the surface at that point,) have been called by Monge the two radii of curvature of the surface at that point, as being the radii of two spheres which alone can touch the surface in two consecutive points. [or if with the points O, O’ (fig. 65) as centers, and radii OA, O'A, we describe spheres, the former will touch the surface along 4G and the latter along AD, because consecutive normals to the surface in both these sections meet 4O. But if with Q, the center of curvature of any section AP, as center, and radius AQ we describe a sphere, it will not touch the surface along AP, because QP cannot be a normal to the surface, since it intersects AO; only, this sphére will have the same tangent plane as the surface at 4, and its section made by the plane AQP will have with the corresponding section of the surface a contact of the second order. From the consideration of consecutive normals we can now find the expressions for the lengths of the principal radii of a surface at any point much more easily than by the former method. 204 233. To determine the radii of curvature at any point of a surface in terms of the co-ordinates of that point, from the consideration of consecutive normals. The equations to the normal at a point wyz of a curve surface are we —vt+p(s%-xs)=0, y -y+q(s'—-2)=0, and at a contiguous point vw +h, y +k, x+/, the equations to the normal are wv —v-h+p,(2'-x-l=0, y -y-k+9q,(% —z-/) =09. The values of w’, y’, 2 which simultaneously satisfy these four equations are the co-ordinates of the point of inter- section of the normals; subtracting the two former from the latter, we have —h+(pi-p)(® -2)-pl=0, -k+(u-9@ -*)-ul=0, both of which the co-ordinate x’ must satisfy. Now let the points be consecutive, and let the second point be situated on a line of curvature the projection of whose tangent on the plane of wy makes an 20 with the axis of w, then k = h tan 0 ultimately ; hence, expanding p,, q,, and J in the above equations, substituting for k, and then making h = 0, we find the equations (r + s tan 0) (x’ — 3) (s + ¢tan @) (x — z) = tan0 + pq + ¢’ tan 0, which must be satisfied by the same values of x’ and tan@ ; hence, eliminating tan @, we must have 1+ p’ + pq tan 0, Le Pe ee ae ee ne “@aas—pqg 9 14+ ¢- (= a)t? or (rt — 8°) (x! — x)? - $1 + p*)t- 2pqs+( + qr} (e — 2) +1+p?4+q°=0,.....(1),; the equation which determines z’— x, the projection of a principal radius of curvature on the axis of z, so that p=-(® -s)V1l +p ?+q. Hence if we find the two values of x’ — x from the above equation, and substitute them in the value of p, we shall 205 obtain the values of the two principal radii of the surface at the proposed point; also knowing z’ — z, the other co-ordinates w and y of the centers of curvature are known from the equations to the normal we —xv+p(s —x)=0, y —y+ q(x —2) =0. Cor. The roots of equation (1) have the same or different signs according as r¢—s° is positive or negative; therefore the curvatures are in the same or different directions (as we have before seen) according as rf — s’ is positive or negative. Also in order that the two radii of curvature may be equal but of different signs, we must have (1+ p*?)t-2pqs+ 1+q)r=0. 234. In order that the two radii of curvature may be equal but of the same signs, we must have (solving equation (1) and putting the part under the radical.equal to zero) ,(1+ p*)t-2pqs+(1+q")r}?-4(1 4+ p?+q’) (rt — s*)=0...... (1) or {(l+p)t-(1+q)r+2(1 + 9@°)r -2pqs}? —43(1 +p) (1+ 9°) -— p’g’t (rt —8*) = 0; and expanding the first term and reducing, we find S(i+p*)t-(1+q@)r}?4+43 +p’)s—pqr $(1+q°)s—pqt} =0...(2). Let (1+ p*)s —par=pqu, (1+ 9°)s— pat= pau; o Sl +p*)v— (1+ q)uh?+4uv = 0; Qu )* b l+p)v-(l+¢ Vice eae tes ‘ 4(1+4q°) , 4u =jJ(l+p)v-(C+q)ul? + 4uv— — —~ u? + ——__; ee a eae a {tp )o- (144 _ —) a + Canty 4u°=0...(3), an equation which can only be satisfied by putting u= OQ, v= 0, or (1 + p')s— pgr = 0, (1+q')s — pqt =0, 206 the conditions that the two radii of curvature may be equal and of the same sign. From this process it appears generally that the roots of the equations (Arts. 231, 233) which give the principal radii, and the directions of the principal sections, at any point of a surface are always real; for these roots involve the square roots of the polynomials (1) and (2), which are here shewn to be equivalent to one another, and to (3) which is a form that never can become negative. 235. The points of a surface at which the two radii of curvature are the same both in magnitude and sign, are called, as has been stated, umbilici. The co-ordinates of such points must satisfy the double equation l+p? pq 14+¢ 5 ry s t at which we may arrive immediately by observing that, since at such points all normal sections have the same curvature, the expression for the radius of curvature of any normal section, V1+pit+g fi +p +2pqm + (1 + g’)m"? y+ 2sm + tm R= must be independent of the quantity m which particularizes the normal sections; and for this it is necessary and sufficient that the coefficients of like powers of m in the numerator and denominator should be equal ; Pep) pgs A Also at these points the equation which determines the directions of the principal sections at any point, since it may be put under the form 1 Yo —- (1 K m + {utero m—-uU PY becomes identical, and therefore the directions of the principal sections indeterminate; in fact, all the normal sections having = 0, 207 the same curvature, each of them may be considered as a principal section. And the shortest distance between the normal at the um- bilicus and the normal at a consecutive point taken in any direction, will vanish as far as it depends upon small terms of the first order; and for certain directions it will vanish as far as it depends upon small terms of the second or higher orders likewise. For the first member of the equation (Art. 231) (h+pl)(1-9)-4&+a90(p-p)=9 (1) is the numerator of the shortest distance between the normals at wyx and any adjacent point ~+h, y+k, x47; and ex- panding p,, q,, / in powers of # and k as far as the second, we get P=p+dp+tdp, n=qt+dq+ hdq, l=dz+hd's where dp=rh+sk, @p=uh’ + 2vhk + wk’, and similarly for the others, so that the first member of (1) becomes sh+(p+dp+ 3 d’p) (dx + 4d’z)} (dq+4d’q) —jk+(q+dq+4d'q) (dz + £d’z)t (dp + 4d’p). But if wyz be an umbilicus, all the terms of less than three dimensions in h and k are identically zero; therefore writing down only the terms of three dimensions we get (h + pdx) d’q + pdzdq —(k + qdz)d'p — qdizdp, which put equal to zero gives a cubic for determining tand=k—h; so that in general there will be either one or three directions in which we may pass from an umbilicus to an adjacent point, so that the normals shall have a coincidence of a higher order than for all other adjacent points. If the coefficients of the cubic should be identically zero, we must have recourse to the terms of a higher order in the expansion of (1), and we shall generally determine a finite number of directions relative to which the normals coincide more closely with the normal at the umbilicus, than do the normals at other adjacent points ; 208 and those directions more especially deserve the name of lines of curvature through the umbilicus. 236. To find whether a given surface admits of umbilici, we must substitute the values of p, g, 7, 8, ¢, derived from its equation in the double equation = i and examine whether these together with the equation to the surface can be satisfied by real values of a, y, x. Consequently the number of umbilici of a given surface will in general be limited ; only if the double equation should be reducible to a single distinct equation, then that equation joined to the equation to the surface will determine a curve on the surface every point of which will be an umbilicus, and which is called a line of spherical curvature, because about each of its points the surface is uniformly curved like a sphere. 237. If through all the points of a line of curvature KPP (fig. 68) we draw normals to the surface, since every two consecutive oues intersect, they will form a developable surface whose edge of regression will be the locus of the centers of the first curvature relative to KPP’. Proceeding in a similar manner for each line of the same curvature LQQ’, &e., we shall obtain a series of developable surfaces of which the edges of regression will form by their assemblage a surface which is the locus of all the centers of the first curvature, and to which all the normals will be tangents. Also this surface will have a second sheet which will be the locus of the centers of the second curvature, resulting from the edges of regression of the surfaces generated by the normals along the lines of the second curvature RPQ, MP’Q’, &c.; and which will be touched by the same normals as the other. To obtain the equation to this surface, we must eliminate w, y, = between the equation which gives x’ — x, the projection of a principal radius upon the axis of x (Art. 233), and the equations to the normal. When the two sheets intersect, their intersection will be the locus of the centers relative to the line of spherical curvature of the surface under consideration. APPENDIX. Tur following Problems, distributed into Sections cor- responding to those into which the Treatise is divided, will furnish occasion for applying the results in the Text as they are successively obtained. ProspiteEMs on SeEctTion I. 1. To find the equation to a plane considered as generated by a straight line always passing through a fixed point and a fixed straight line. Let , y, x be co-ordinates of P any point in the generating line CQ (fig. 7) passing through the fixed point C in the axis of zx, and the fixed straight line AB in the plane of wy. Let x’, y’ be co-ordinates of Q, and a, b, c the lines OA, OB, OC; then oe yf ev ON -c—x# *y- ON. ¢-—z =+>1;, bout.—=—— = De es ee Fh A, a OR Cc y OQ G ~+0=1--, the equation required. 2. The equation to the plane generated by a straight line that always meets a line whose equations are v7 = mz +a, y=nx-+b, and that remains parallel to a line through the origin whose equations are w = m’z, y =n’, is e-mx—-a y—n'zs—b m — m' n—n' 3. To find geometrically the distance of a point from a plane. E4%, 210 Let a’, y’, x’ be the co-ordinates of the given point P (fig.9), s=Av+ By+c the equation to the given plane ABC. Through P draw a plane GAH perpendicular to the trace AR; then this plane is perpendicular both to the given plane ABC, and to AOB, and contains the ordinate PN. Let GH be the intersection of the two planes meeting PN in Q; draw PR perpendicular to GH. Then PR is perpendicular to the plane ABC; and PR = PQsin PQR = PQcos GHK. But PQ=PN- QN =2'- (Aa'+ By' +0), since Q is a point in the plane ABC, for which w=a’, y=y ; 1 and cosGHK = €Art. 29); J/i+ 4+ B e— Aa'— By -c_ 2/14 A+ Be the radical being taken with that sign which makes the whole expression positive. .. PR, the required distance, = 4. To find the equations to a straight line which cuts perpendicularly each of two straight lines not in the same plane, whose equations are given. Let / , L=MN2FZ+Aa VC=mese+a : \, : | be the equations to the y=nst+b y=ns+b given lines; then if x= da + By+e be the equation to a plane parallel to them both, dm+Bn=1, Am'+ Bn'=1, (Art. 23) , , n—-n m— mm = Saree O B=————_, 7° mi —- mn Mm — M7 Let x = A’a + By +c’ be the equation to a plane which contains the former of the given line and also the required line, and which therefore (Art. 41) cuts the plane parallel to both the given lines at right angles; therefore (Art. 23, and 35) AA'+ BB +1=0, Am+Bn-1=0, A’'a+Bb+c=0; hence c= — d'a— B’b, 211 and A'(dn - Bm)=-(B+n), B'(Bm-An) = -(4 4m); “. =A’ (w@—a) +B (y - bd), or = (dn - Bm) = - (e@- a) (B+) + (y—-b) (44m), or, substituting the values of 4 and B, x Ym (m'— m) +n (n'— n)} = (w — a) §m'—-m+n(m'n - mn')} + (y —b) j{n’-n+m(mn'—m'n)}, is the equation to the plane containing the former of the given lines and the required line. Similarly, the equation to the plane containing the latter of the given lines and the required line, by interchanging m,n, a, b, and m’, n’, a’, b,, is = }m'(m —m') +n (n-n')h =(w-a’) {m-m'+n'(mn'- m'n)} + (y-0)}n-n'+m'(m'n—-mn')t. These two equations determine the position of the line re- quired, since they are the equations to two planes each of which contains it; the equations to its projections may be found by Art. 20. 5. To find the length of the shortest distance between two straight lines whose equations are given. If a plane be drawn through each of the lines parallel to the other, the perpendicular distance of these planes which will of course be parallel to one another, will be the shortest distance of the lines (Art. 41). 8 I| Let meta) wvemsz+a’ ) : py be the equations to the y= no +b y=nszt+b lines, x = da + By +e the equation to the plane which is parallel to the second and contains the first, - Am'+ Bn'=1, Am+Bn=1, da+Bb+c=0, which give for A, B, c, the values n'—n m’ — m A =————— , B= - ——,, c=- Aa-— Bb. , a > , , b) mn—-m nr mn — mn? 14—2 212 Similarly, the equation to the plane which is parallel to the first and contains the second, is = dva+ By+c’, where A, B have the above values, and c’ = — Aa’ — Bb’; and the lengths of the perpendiculars dropped upon these planes from the origin are respectively (Art. 37. Cor.) — Cc —C V1 + ANE an lee — (c'-c) A (a’— a) + B(b'-b) nat at aera /1 + A+ B therefore, substituting for 4 and B their values, the shortest distance between the two lines (n’— n) (a’— a) — (m’'—™m) (b’— 5) V (m'— m) + (n'=n)? + (mn'— m'ny? ; Cor. When c in the equation to the first plane, and c the corresponding quantity for the second plane, have different signs, the origin is situated between the two planes ; and therefore the distance of the planes will be found by taking the sum of the perpendiculars. When the shortest distance vanishes, (n’— n) (a’— a) = (m’- m) (b'— b), which is the condition in order that the lines may intersect, found in Art. 22. whose difference = 6. Having given the lengths, the least distance, and the inclination of two opposite edges of a tetrahedron, to find its volume. Let 4 (fig. 11) be the vertex, and BOC the base of any tetrahedron; complete the parallelogram Cy, and join Ay; let Nw cut each of the opposite edges 4B, OC, at right angles; then OC is parallel, and Nw is perpendicular to the plane ABy. Hence, since they are on the same base, and their vertices are in a line parallel to the plane of the base, tetrah. dy BO = tetrah. dyBu = 4 AB. By.sin ABy. 4 Ne; but tetrahedron Ay BO = tretrahedron AOBC, 213 since they have a common vertex, and equal bases in the same plane ; ‘, tetrahedron AOBC = 1 4B. OC.Nvw.sin 8, where @ denotes the angle between AB and OC. 7. A sphere may be described touching each of the six edges of a tetrahedron, provided the sum of every two opposite edges be the same. 8. A cube may be cut by a plane so that the section shall be a square whose area is to that of the face of the cube as 9 to 8. 9. Ifa plane be drawn so that the sum of the perpen- diculars let fall upon it from m given points a’y’s’, vy’, &c. shall be always equal to a given line A, then it will always touch a sphere the co-ordinates of whose center, and whose 1 radius, are respectively F th of the quantities =(#’), = (y’), >= (z’), and R. 10. The shortest distance between a diagonal of a cube and an edge which it does not meet is a+»/2, @ being an edge. 11. If @ be the inclination of two planes /a+my+nz=a, lv + m’y + nx =, the distance of their line of intersection wns 1 > ik ea = Aes He from the origin = arp wo 2a(3 cos 0 + 2°. ProspLEMs ON Section II. The following Problems furnish examples of finding the equations to surfaces from the given geometrical mode of describing them. As the simplicity of the result will in every case greatly depend upon properly selecting the posi- tions of the co-ordinate axes, some instances are here intro- duced with that special object. 214 1. If a straight line have always three given points in three fixed planes at right angles to one another, to find the locus of any other point in it. Let the planes be taken for the co-ordinate planes, and let the line meet them in the points 4, B, C, (fig. 23); and let P be the describing point the co-ordinates of which are On=a, NQ=y, PQ=z2. Alsolet PA=a, PB=b, PC=c, ZACa=i, £Cbx=0, Ca being the projection of CA on wy. Then z=csini, w=acosicos#, y=bcosisin®@; 2 ne y Pe i aie —+—+— = 005 @+sin*2z=15 C a* b* hence the locus of P is an ellipsoid, 2. To find the locus of the intersection of two planes drawn through two given lines, so as always to be perpendi- cular to one another. Let 44’, BB’, (fig. 24) be the two given lines, AB their shortest distance which make the axis of x; also take the plane bisecting AB at right angles for that of «y, and the line bisecting the angle between the projections Oa, Ob of the lines, for the axis of « Then if O04 = OB=ce, and tanaOw=m, the equations to the lines 4d’, BB’, are Y=oMe, ®=C}; y= —Me, £=—C. Let zs = Av + By + € be the equation to the plane passing through 44’; .. d+ Bm=0; w-en A (v2) Bi PAS BAe ‘hit Se m me—y me—-—y Similarly, if * = A’w + B’y -c be the equation to the plane passing through BB’, d’- B'm=0; A eae: ppp TOS. me+y Lina ty) 215 But, because the planes are at right angles to one another, AA + BB+1=0, ©: ples aaa SES ee Tt eI), (mv) -—y (mx)? —y or (m* —1)2°+4+ ma’ — y? = (m? — 1) ce’, which, since only one of the coefficients is negative whether m> or <1, represents a hyperboloid of one sheet. Cor. It may be easily shewn that each of the lines AA’, BB’, lies in the surface, and that a plane perpendicular to either of them will cut it in a circle. If the lines are parallel, m=0, and vy’ + 3’ =c*, which represents a cylinder as it ought. 3. To find the locus of a point which is equidistant from two lines given in space. Retaining the same axes and notation as in Prob. 2, it will be found that the surface is a hyperbolic paraboloid having for equation may + (14+ m’)cz=0. 4. Two points move in straight lines with uniform velocities, to find the equation to the surface generated by the straight line which joins them. Retaining the same axes and notation as in Prob, 2, and besides denoting by a, a’, the initial values of w for the points moving in 44’, BB’, respectively, and by 7 : 1 the ratio of their velocities, it will be found that the equa- tion to the surface is (1 +1)c(cy—mazx)—(n—1)c(mea—zy)=m(a-na)(c? —32"). 5. ‘To find the equation to the surface generated by a straight line which constantly passes through three fixed straight lines. Let a parallelopiped be constructed, having its three edges (of which no two either intersect or are parallel to 216 one another) in the three given lines; take its center for origin and lines parallel to its edges for the axes; then the equations to the given lines will be Let the equations to the generating line in any posi- tion be a—h —k v=mzt+h, y=nzr+k, a“ atl ; m then since it meets each of the given lines, we have three equations of condition, viz. a—h b+k —-a@=mc+h, b=-ner+hk, = ; m n and it remains to eliminate m, n, h, k, between these and the equations to the generating line. The result is ayz+bzx+cxy+abe=0. Cor. If the signs of the quantities a, b, c be changed this equation is not altered; therefore the same surface would be generated by a straight line constantly passing through three different straight lines whose equations are v= ‘t hs Mat 3 3 * s=-€ Zi ,uc y= 6 The surface, as we know, is a hyperboloid of one sheet; the latter therefore is the only surface that can be generated in the manner described above; the origin is its center, and the axes of the co-ordinates are situated in the surface of the conical asymptote. 6. To find the surface generated by a line which always intersects two given lines, and is parallel to a fixed plane. Let the fixed plane be taken for that of yx, and let 4B (fig. 24), the line joining the traces of the given lines on that 217 plane, be taken for the axis of x, and its middle point for the origin. Also, let the plane of wy be parallel to the two given lines, and the axis of w bisect the angle between their pro- jections made parallel to Oz. Then referred to this system of oblique axes the equations to the given lines are =Me#, Z=C3 Y= —-MX, F=—=—C. > Let v=a, y=nz+b, be the equations to the generating line since it is parallel to yz; then since it intersects each of the given lines ma=ne+b, —-ma=—nc+b; “. b=0, and ma=ne, or eliminating a and n, mzx=cy is the required equation, representing, as we know, a hyper- bolic paraboloid. 7. To find the surface generated by a straight line con- stantly meeting three given lines that are parallel to the same plane. Take one of the given lines for the axis of wv, and a line intersecting it and parallel to another of the given lines for the axis of y, so that the plane of wy is that to which the three given straight lines are parallel; and for the axis of x take a line joining the origin with the point in which the third given line meets the plane containing the axis of y and the line to which it is parallel. Then the equations to the three given lines are y = 0 v= 0 Y= Ma ane set =k | and the equation to the surface will be found to be kyz+m(h—-k)ex=hky, representing a hyperbolic paraboloid since the surface has not a center, and is generated by a straight line that does not continue parallel to itself. No two of the fixed lines must be in the same plane; for if so, two planes will be generated, viz. that containing the two lines and meeting the third at an infinite distance, and that passing through the point of intersection of the two lines and the third line. 218 8. The locus of a point, whose distance from a fixed point is always equal to times its distance from a fixed line, ; y ne NC is a spheroid with semi-axes ——_,, —=—., c being the 1-n V/1 — n° perpendicular distance of the fixed point from the fixed line. Let A (fig. 6) be the given point, HA-=c the perpen- dicular from it on the given line 47’; produce 4H to O so that 40 = ——., and consequently HO = — ;- Draw Ox —n parallel to 47’ for axis of x, and take OA for axis of wv, and let P with co-ordinates wv, y, x be a point in the locus so that Ei ee), aE ne 2 2 ( - 2) +y +n’, MA =an’ ( _- 2) +n'y’; lL —-n : be Set ee or CU itp) Sia ste Stag (I sartt) rag oe —n which represents a spheroid with the semi-axes above stated. 9. To find the locus of a point whose distance from a given point always equals m times its distance (measured parallel to a fixed plane) from a given line. Take a plane through the given point parallel to the fixed plane for the plane of wy, and a plane through the given line perpendicular to the fixed plane for that of vg, and sup- pose the axis of y to contain the given point, then a+ (y—b)? +2? = n'y? +n? (mz+a- 2)? is the required equation, v= mz +a being the equation to the given line in the plane of vz. 10. The locus of a point whose distance from a fixed plane (that of wy) always equals its distance from a line inclined at an angle a to that plane, has for its equation y? — vy sin2a + (a — 2°) sin? a = 0. 219 11. To find the equation to the surface of a cable-ring, having given a and ¢ the radii of its inner and outer boundaries. Also to shew that the section made by a plane touching the inner boundary is a Lemniscata when ¢ = 3a. Let ACB a diameter of a circle whose center is C meet Oz, a line in the plane of the circle, at right angles in O; to find the equation to the surface generated by the revolution of the circle about Oz. Take ON, NP co-ordinates of P a point in the circle, then PN’ = AN.NB=(OA-ON).(ON — OB), or 3° = (w—a)(c - @) is the equation to the generating curve in the plane of za; therefore (Art. 52) the equation to the surface generated is = (/ a + y —a)(e- J v? + y). To determine the section made by a plane touching the inner boundary let v = a, then v= (fat y—a)(c- Vaty’), or (2° + 9’)? = (c + a) $(c — a) y’ — 242°}, which becomes the equation to the common Lemniscata if Cras SG: 12. If the axes of a hyperboloid become evanescent, always preserving a constant ratio to one another, the surface is changed into its asymptotic cone, Let the equation to the hyperboloid be 2 2 2 a, b, ec being certain given values of the semi-axes; and sup- pose the semi-axes to diminish continually, but always to be proportional to a, b, c; then they may be represented by na, nb, me, where m is a common multiplier continually growing 220 less; therefore the equation to the surface in its successive changes will be a y° x? x oe ie ie OTe na nb ae CD 3 a Cc therefore, making 2 = 0, or the semi-axes na, nb, ne, evan- escent, the equation to the surface becomes the same as that to its conical asymptote, viz. ei it 13. If two fixed lines pass through a point, the locus of a third line passing through the same point and making angles a, a’ with the former, such that tanda.tan4q’ is invariable, is a cone of the fourth order. 14. To find the locus of a circle whose center is the origin, and which always passes through the axis of z, and through an ellipse whose equation is a’y’ + b’a* = a®b’. 15. If three lines mutually at right angles and passing through a fixed point C intersect the surface of a sphere, then the center of gravity both of the triangle formed by joining any three points of intersection, and of the pyramid having C for its vertex and that triangle for base, will lie in spherical surfaces described about C as center with radii vas ae 2 ay 22 _ oO? =3V 3r 2d and 44/8r 2d respectively, » being the radius of the sphere and d the distance of its center from C. 16. If the paraboloid Ja + Uy? =Jlz be cut by planes through its axis, the directrices of the sections will lie in a surface whose equation is 4% (Ja + I'y”) + Ul’ (a + y’) =0. 17. If one of the co-ordinates of an ellipsoid be produced so that the part produced may equal the sum of the other two, its extremity will trace out an ellipsoid of the same volume as the original one. = 221 18. The locus of a circle that always touches the axis of x at the origin, and also passes through a fixed straight line ay + bw = ab in the plane of wy, has for its equation x (ay tbe) +(e + y’) (ay + bw - ab) = 0. Prospitems on Section III. We shall here give some applications of the principal results obtained in this section, relative to the projections of plane surfaces, the transformation of co-ordinates, and the intersections of surfaces by planes. 1, The pyramid whose vertex is the origin, and base any plane surface, is equal to the three pyramids whose common vertex is any point in the plane of the surface, and bases its three projections on the co-ordinate planes. Let d, a perpendicular from the origin upon the plane in which the surface lies, make angles a, 3, y with the axes of X,Y, %; also let A denote the area of the surface, and a, y, z the co-ordinates of any point in its plane, wcosat+ycosf + cosy =d. Hence, multiplying by 4.4, Acosa.tx+AcosB.4y+ Acosy.4%=A ve or, dy. + Ay. + Ay. Go | ee 1} ae which (since the volume of any pyramid is equal to the area of its base multiplied by one third of its height) ex- presses the property announced. The quantities d,, 4,, 4, will be positive or negative according as a, 3, y are acute or obtuse. 2. To find the sum of the projections of any number of areas upon a given plane. 222 Let D’, D", &c. denote the areas; a’, By’; a’, By’; &e. the angles which the perpendiculars upon their planes from the origin make with the rectangular axes of a, y, % produced in the positive directions. Also, let a, B, y be the angles which a perpendicular to the given plane makes with the same axes. ‘Then the areas must be multiplied respectively by the expressions f , , cos a cos a’ + cos (3 cos [3 + cos vy cos +y cos a cos «+ cos ( cos 3” + cosy cosy”, &e., which are the values of the cosines of the angles between the plane of projection, and the planes in which the areas lie. The sum of these products will be S' = cos a (D' cos a’ + D" cos a” + &e.) + cos 3 (D' cos 3’ + D” cos 3" + &c.) + cosy (D' cosy’ + D" cosy" + &e.) ; or, if we denote the sums of the projections on the co-ordinate planes of yz, za, vy by S,, S',, S, respectively, S' = S, cosa + S,, cos B + S, cos vy; which enables us, from knowing the projections of any areas upon three planes at right angles to one another, to find the sum of their projections upon any plane whose position is given relative to the same planes. In forming the values of S,, S,, S, regard must of course be had to the signs of cosa, cosa’, &c. 3. To find the position of the plane on which the sum of the projections of any number of areas is a maximum. Let S,, S,, S, denote the sums of the projections of the areas on the co-ordinate planes of yx z#, wy respec- tively; S,, S',, S, the sums of the projections on three other planes y'2’, s'v’, a'y’ at right angles to one another having the same origin, 223 . Sy = S, cos wae + S, cos v’y + S, cos v’x, t ’ , , S, = S,cosyxv+S,cosyy + S, cosy x, S,, = S, cos z'v + S', cosx’y + S, cos 2%. Hence, squaring and adding and taking account of the equa- tions of condition to which the cosines are subject, y2 2 2 2 2 2 Sy + 8, + S, = S, +S, + 8;. Hence, if the planes of y's’, 2’a’, ay’, be so situated that the sums of the projections on two of them vanish, the sum of the projections on the third will be the greatest possible ; let this be the plane of y’z’, then S,=0, S,=0, and the greatest sum S'= Sy = V/s? + 5) + S%. But if we had begun by supposing the projections on the planes of y's’, 2a’, ay’ given, we should have had the equation S, = Si cos va’ + S,,cos xy’ + S, cos wz’. In this case therefore, S$, = S, cos va’; similarl >) @ oH >) 3 S, = S,cosya’, S,= Sy cosza'; which three equations determine the position of the plane (or rather of its normal, viz. the axis of 2’) on which the sum of the projections is a maximum, since S‘y = JS? as SY + 82. Cor. Let S” denote the sum of the projections on a plane whose normal makes angles a’, 6’, y’ with the axes of w, y, x, and an Z=8 with the normal to the plane of greatest projection, and let the latter normal make angles a, 2, y with the axes; .. cos @ = cos a cos a’ + cos 3 cos 3’ + cosy cosy’ = S', cosa’ + S,, cos 3’ + S', cos ry’ ~~ S S SS” .*. S" = S'cos 0. 224 Hence, the sum of the projections vanishes on all planes per- pendicular to that on which the sum is a maximum. 4. To find the area of the surface of any portion of a right cone on a circular base. Let ACM (fig. 33) be a section through the axis of the cone perpendicular to the cutting plane CPB; AC=c, AB=c, LCAB=B; also let Cpb be the projection of CPB on a plane perpen- dicular to the axis, then Cb = (c +c’) sind B = major axis of projection ; and 4/ce' sin 1 =4 minor axis both of CPB and Cpb, since it Is a mean ioe between the perpendiculars dropped from B and C upon the axis of the cone; es (c+c) Vee . the area of projection Cpb = z sin® : : But Cpb is the projection of the conical BUMAEE CABP, every portion of which is inclined at an Z = 90°- +B to the plane CM, B(ce+ (c-+¢) See / ce ‘, area of surface CABP = r sine TC Ba 5. Three straight lines mutually at right angles and meeting in a point, constantly pass through a plane curve of the second order; to find the locus of their point of intersection. Let Aa’?+ BY =C = 0 h, k, & the co-ordinates of the point of intersection of the three lines; and let the curve be referred to that point as origin, and to the three straight lines as axes; let the co- sines of the angles formed by the new axes of a’, y’, 2’, with the axis of w, be denoted by m, m, 7; and similarly for the be equations to the plane curve, 225 axes of y and zs; therefore by the formule of Art. 90, the transformed equations to the curve are A(ma'+ ny + rx’ +h)? + B(m'a' + n'y + 7'2'4+ kP=C, mae + n'y +r es +1=0. In order that the axis of a may meet the curve, these two equations must agree in giving the same value for 2’, when y’ and x’ =0; “ y “. A(ma’ +h)?+ B(m'e' + ky =C, ma’ +1= 0, or a’ = —-—; m hence, substituting this value of a in the former equation, and proceeding in a similar manner with respect to the axes of y’ and 2’, we have A(m”h— ml)? + B(m’k —- m'l)? = Cm’”, A (n"h — nl)? + B (nk - n'l)? = Cn", A (rh — rl)? + B(r"k - 7'l? = Cr”. Therefore, adding these equations together and taking account of the equations of condition to which the quantities m,n, &c. are subject, A(h?+l?)+B(K+P)=C, the equation to the locus; which is therefore a surface of the second order concentric with the curve. Cor. If the given curve be an ellipse, this equation becomes, replacing h, k, 1 by «#, y, 2, Ba” + a’y’ + (a? + b*)2? = a’? b’," that to an ellipsoid; if a hyperbola, the equation to the sur- face is ba? — a’y? — (a — b*) 2? = a’d’, that to a hyperboloid of one or two sheets, according as the transverse axis of the hyperbola the conjugate axis. In the case of the ellipse, if we remove the origin to 15 226 the extremity of the major axis, by writing w — a for a, the equations become we | 02a : ‘ : ——-— + = = 0, Ba +a’y’ + (a? +0°)2 = 2ab'a; a a b- now let b°>=2ap —p’, p being the distance of the focus of the ellipse from its vertex, and make a infinite ; ~YyY=4pa, y+" =4pH; which shew that when the curve is a parabola, the locus required is the paraboloid which it would generate by re- volving about its axis. 6. Three straight lines mutually at right angles and meeting in a point are applied to a surface of the second order; to find the locus of their point of intersection. Let the equation to the surface be da’ + By? + Cx? =D; then referring it to the three straight lines as axes and to their point of intersection as origin, the transformed equation is A(ma’+ ny +rz' +h) + B(m'a! + n'y +7°x' +k) +C(m'e' + n'y +7's' +l)’ =D. Now making successively y’, 2’, =0, a’, 2’, =0, a’, y', =0, in order to determine the points where the axes of a’, y’, 2’ meet the surface, and calling Ah?+ Bh? + Cl’—- D=G, we have v?(m?A + m?B + m'?C)+ 220 (mAh +m Bk +m’Cl) + G=0, y? (nA +n? B+ n'?C) + 2y'(nAh+n' Be +n" Cl) + G=0, 8°? At rP B+ r?C)4+2ek(7Ah +r Bk + 7° Cl) + G=0. But since the axes of a’, y’, x’ touch the surface, the two points in which any axis meets it coincide in one; therefore the roots of each of the above equations are equal, and the first members are perfect squares ; “(mA + mB + m'C)G = (mAh + m'Bk + mC), (A+ n?B+n?C)G =(nAh +n'Bk +n’ Cl)’, Ate Bir * C)Ge (Ahk +r Bk +4 Cly: 2947 Hence, adding together these equations, and taking account of the equations of condition to which the quantities m, n, &c. are subject, (44 B+ C)G= Ah? + Bi’ + CP, or, restoring the value of G, and reducing, A(B+C)h?+ B(A+ OC) 4+C(44 BYP =(A4+B+C)D, the equation to the locus, which is therefore a surface of the second order concentric with the proposed one. Cor. For the ellipsoid, replacing h, k, 1 by a, y, 2, this equation becomes (P+e)a+(P+ec)y + (2 +0) = 0B? + ae? + b’c’, x’ y” 3? or Se aetee et & s- if g, g’, g” denote the three altitudes of the triangle formed by joining the vertices of the ellipsoid (Cor. Art. 82). =)15 ae) If we remove the origin to the extremity of the semi- axis a, and then make a infinite, the equations become 2 Pad 7 4 en =0, Yrs a4(pt pat app’; 2p ° 2p if therefore the given surface be a paraboloid, the locus required is a paraboloid of revolution. If the given surface be a hyperboloid of one sheet, making c’ negative, the equation to the locus is (P-—c)e + (@—e)y + (C84) =e -e(a +8); the locus is therefore an ellipsoid when c, the imaginary semi- ‘nat ie ab , axis, is less than —,———— and of course less than either J/ a +b? of the real semiaxes; a point, viz. the center, when ab F c¢ = ————; imaginary, till c = b the less of the real semi- a? + b° : 15—2 228 axes; a hyperboloid of two sheets till ce =@, when it becomes a hyperbolic cylinder; and a hyperboloid of one sheet for all future values of c, i. e. whenever the imaginary axis is greater than either of the real axes. If the given surface be a hyperboloid of two sheets, making 6b? and c’ both negative, the equation to the locus is (PP +c%)a7 4+ (P-a) + (CP -—a)P’=CWC (C+) - Be’, which is imaginary, as long as a the real semiaxis is less be : than ir: af ‘and of course less than either 6 or ec; a ge Oe, Cc : be ees 4 point, when a = ————-; afterwards an ellipsoid, till a =e \/ b+ & the less of the imaginary semiaxes, when it becomes an elliptic cylinder ; a hyperboloid of one sheet, till a = 6 when it becomes a hyperbolic cylinder; and for all future values of a, a hyperboloid of two sheets, i.e. whenever the real axis exceeds each of the imaginary axes. e a e e e e ] Also, if in the original equation we put D=0, 4 =—, = I 1 ; : B= Be? C= ai we have the equation to a conical sur- face of the second order, and the equation to the locus be- comes (0? + c*) a’ + (a? + c*)y? + (a’ + b’)2* = 0, which is also the equation to a conical surface of the second order; one, or two of the quantities a*, b’, c® being negative. 7. To find the equation to the section of an oblique cone on a circular base. Let the diameter of the base on which a perpendicular from the vertex falls, be taken for the axis of « Then (Art. 49) making 6 = 0, the equation to the surface is, putting a-2r=a, cy” = (ex — az) (er+ a's - ex); 229 therefore, the equation to the section made by a plane through the axis of y inclined at an 26 to the plane of the base, is cy” = (ecos 8 — asin@) }2cra’ + (a’ sin@ — ecos@)a"?. Cc a! As long as tan@ lies between - and , or 8 between SAB a and SEB (fig. 34), 7. e. as long as the plane cuts both sheets of the cone, the section is a hyperbola. When tan @ = an or the plane is parallel to ES‘, the a section is a parabola. In other cases it is an ellipse, which becomes a circle when c’ = (c cos 8 — asin 8) (c cos 8 — a’ sin @), or, c’?+c’ tan’@ =c’—c(a+a) tan@ + aa tan’@; c(a +a’) . tan@=0, or tan@ = —— ie aa —c The first value of 0 gives the base of the cone ; the second gives tan 9 = tan (SAB + SEB), or if DAy be the plane of the circular section, DAE = SAB + SEB, «. DAS = BES; that is, the cutting plane makes the same angle with one side of the principal section of the cone, that the base does with the other. ‘This circular section is called the subcon- trary section. 8. The sum of the squares of the reciprocals of any three semidiameters of an ellipsoid mutually at right angles, is equal to the sum of the squares of the reciprocals of the semlaxes. Let a, B, y be the angles which any semidiameter r makes with the axes, then the co-ordinates of its extremity arev=rcosa, y=rcosP, =7 cosy; 1 cosa cos" — cos" a = ae eae ++ : r a b? Cc ? 230 and forming similar equations for 7’ and 7’, and adding, since the semidiameters are mutually at right angles, we find 1 vk 1 1 1 sity APN Ota Fe 9. Ifa sphere be placed in a eee of revolution, the section of the paraboloid made by any plane touching the sphere is a conic section having the point of contact for a focus. ] eT ph inte 10. The eccentricity of any section of a paraboloid of revolution is equal to the cosine of the inclination of the cutting plane to the axis, 11. To shew that among the constants m, n, r &c. employed in passing from one rectangular system of co- ordinates to another, the following relations hold ; man — nr, m =n"r—nr"', m’=nr —n'r; and similarly for n,n’, n”; andr, 7’, 7’. Eliminating y’ and 2’ from equations (3) Art. 90 we get a’ = (n'r" — n"7') a + (nr —nr")y + (nr -—n'r)z =rAma«+rm'y +rm’sz from (5). Aman’ —-n"r, rm =n"r —nr”’, Am" =n —n'7; N= (nr — n"7')? + (nr — nr’)? + (n7! — n'ry? = (n? 47? +n’) (r+ 7? 4 7) sin? 6 (Cor. Art. 30) = 1,0 being the angle between the axes of y’ and 3’; and “, sin @=1. HenceA= = 1, and the proposed results are esta- blished. 12. ‘To shew that (mm'm’)? + (nn'n"”)? + (rer)? = (mar)? + (m'n’s’)? + (Mm nr’)? we have (mm’m”)? = mm’ (nn + rr")(nn'+ rr’) from (7) Lf Se AO =m m'n'n”’.n? + mm” rr? mn’. mm’ ny + m'nr’.m’n’r, 231 and forming similar expressions for (mm‘n”)* and (rrr?) and adding them together, observing that m?>m'm” (n'n” + 9’) = — (mm'm")?, we get 2(mm'm")? + 2(nn'n")? + 2(rr’r’)? pales ” , ” , ry Ay, ray ou , ” Woe =m n'r.m' nr +m’ nro mn'r’ + mn'r m'n"r + mn’. mn’ +m'n'y.mn'r + mn’r.m'nr’ ; and as the second member does not alter when we simul- taneously interchange m’ with », m” with r, n” with 7’, the proposed result is established. 13. To pass from the rectangular system of co-ordinates «, y, x, to another rectangular system a’, y’, 2°; having given the 2@ at which the planes of wy, ay’, are inclined to one another, and the angles @, vy, which their line of intersection makes with the axes of w and w’ respectively. Let the new plane of a’y’ intersect the plane of vy in the line Ov, (fig. 29), and be situated above it; and let Ox’ perpendicular to the plane a’y’ be the axis of x’, and Ox’, Oy’, the axes of w and y’; and let a sphere be described about O as a center with radius 1, cutting the axes in the points a, y, z, &c.; and suppose these points to be joined by arcs of great circles. Then if va, = ¢, and Za'a,y=0, or xz’ = 0, the plane of a’y’ and the axis Oz’ will be completely determined; also if wa, =, then the positions of Ow’, Oy’, in the plane w,Oy’ will be fixed. Also we have , , , , , , C= cosxxty cosyx+2 cosza, &C.; and our object is to express the nine cosines in terms of 0, p, , which may be done by means of the fundamental theorem in Spherical Trigonometry for finding one side of a triangle in terms of the other two and the included angle. First, from the triangle a’a,2, in which Zwv’a,x = 180° ~ 0, COS v'&@ = COs @ cos yy — sin @ sin W cos 0 ; 232 and changing W into 90° + w, cos ya” = — cos d sin Wy — sin d cos yy cos 8 ; also from the triangle z’v,a, in which 2’a, = 90°, za, = 90°— 0, cos *# = sin @ sin 0. Secondly, from the triangle w’x,y, in which a,y = 90° — @, cos wy = cos yy sin @ + sin Wy cos d cos 8 ; and changing y into 90° + Wp, cos y'y = — sin sin @ + cos ycos @ cos 0 ; also from the triangle x’a,y, in which z’a, = 90°, x’ay = 90° + 0, cos ¥y = — cos @ sin 0. Lastly, from the triangle w’v,z, in which za, = 90°, 2’v,z =90° - 0, cos a's = sin Wy sin 6; and changing vy into 90° + W, cos y's = cos vy sin @; also cos xs = cos @. PROBLEMS ON SeEcrTion I[Y. The following Examples will illustrate the method of drawing tangent planes and normals to curve surfaces, and finding their volumes and areas. 1. To find the equation to the tangent plane to an ellipsoid at a proposed point. 2 ® e @ y* Pod The equation to the surface is — + — + — =1, a Ds eae ve sdz y sds we — a tae sey ee O ae 8 oe coe a’ ede °° Bt ce’ dy Webicd dz dz ’ } Hence, substituting for — , —, their values in dx” dy dz ave, os 4 Tiere ae O — ¥)s 233 the equation to the plane which touches the ellipsoid at the point vys, Is fe + 5% (a0) 45% yy) =o, 2 , 2 so 8 we Cheer apr Raith: Tae Tia eset Lie v= 0, c c fi b? b? 7 YY 88 or — + ——+—=1. = b? Cc Cor. To find where the tangent plane cuts the axes, ; a making y’ = 0, s’=0, we get w =—, the distance from the @ center at which it cuts the axis of #; similarly the distances from the center at which it cuts the axes of y and x, are 5? Cc — and — y ra 2. To find the equations to the normal to an ellipsoid at a proposed point. dz ca das cy Since — = —- —-, —= ——%, the equations are dx a’ xs dy b? 3” ; Cw a 2 =[2(@'—2), f y= G2 Ge - 2). Cor. By making »’ = 0, we find the co-ordinates of the point where the normal meets the plane of wy, , c\ , Gi x“ =x(1-4), Yy -y(1-<)s also, the length of the normal, intercepted between the sur- face and the plane of wy, is ct a ca? ‘ x 2 eI Re eR Ber ea Kae & 3. To find the locus of the extremity of the perpen- dicular dropped from the center upon the tangent plane to an ellipsoid. The equations to a perpendicular from the center on the 23 4 Cw e AREY) 5 J tangent plane are wv = —-—2 = —~ 3", which, combined 8 P os ee J Bs ’ ‘ with the equation to the tangent plane, give by eliminating # and y, (a? + y”? +2”) = cs"; ce? 3 aa b?y’ i ye. f eo aengnat 9 a= : — in which expressions a’, y’, 2° are the co-ordinates of the point of intersection of the tangent plane and perpendicular ; and w, y, * the co-ordinates of the point in the surface to : au y gt which the tangent plane is applied so that = ae ia 4 mis 1; therefore the required equation is (aa’)? + (by’)? + (c2x')? = (w? + y? + 2)? Hence if R be the central distance of a point in the ellipsoid the normal N at which makes angles a, 3, y, with the axes, and P be the perpendicular on the tangent plane at that point, we get from equations (1), considering that P? = v? + y? + 2”, Pcosa=2, a’ cosa = Px, &e. fh) J im TLL me die 5, 7 =e i eS P as bt t a 2 Q 2 Gey P? =P? |—+ = +—)=a@ cosa + Bb’ cos’B +c’ cos’ y, ie en I pil hl oe R?P? = P’ (a? + y? +2") = a' cos’a + b' cos’ B + c* cos’. 4. If three tangent planes to an ellipsoid are mutually at right angles, their point of intersection will trace out a sphere concentric with the ellipsoid. Let P the perpendicular from the center on a tangent plane make angles a, 3, yy with the axes of a, y, x; then the equation to that plane is a’ cosa + y’cosB + 2’ cosy = P, which must be identical with the equation to the tangent plane expressed by the co-ordinates of the point of contact, viz. ve yy 82 Pnerre ean e o 2 a” b 235 xv P yP xP wana. «38 *. cosa = cos 3 =——, cosy = —, but —~+—4-—=1 a a. > B i? 2) Py Cc ? a b° Cc ) a P\* y P\? 3P\? P . Pia ( oo (=) 4s (==) =a’cos’a + b* cos’ 3 + ¢ cos? +y a Cc as before, which gives the length of the perpendicular in terms of its inclinations to the axes. Now let there be two other perpendiculars P’, P’, which make with the axes the angles a’, , , , ° B's a”, B’, ys respectively, “. P®? = a’ cos’a’ + b° cos’ 3’ + & cos’y’, P’”? = a? cosa’ + b’ cos? 3” + ¢* cos’ry’” ; and suppose the three perpendiculars to be mutually at right angles; then since a, a’, a’ are the angles which a line (viz. the axis of v) makes with three rectangular axes, viz. the three perpendiculars, if we take account of the condition to which they, as well as B, 8’; B’, y, y'5 y's are subject (Art. 5), and add, we find (Pete Pcs Pie Gay +e but the first member is equal to the square of the distance from the origin of the point of intersection of the three planes; this point therefore describes a spherical surface con- centric with the ellipsoid whose radius = J a + b? +c’. In the case of hyperboloids, one at least of the quantities a’, b’, ce’ is negative, and hence their sum may be negative or nothing ; in the former case there is no point in space through which three rectangular planes touching the hyperboloid can be drawn, and in the latter, the center is the only point which has that property. Cor. If we remove the origin to the extremity of the semiaxis a, and then make a infinite as in Art. 72, the equation to the ellipsoid will become 236 and the equation to the sphere will become oe —2an+y +2" =2ap — p> +2ap — p”, or, making a infinite, - # = p +p’; therefore the locus, in the case of paraboloids, is a plane perpendicular to the axis of the surface, and at a distance = — (p +p’) from the vertex. 5. Three planes mutually at right angles constantly touch the perimeter of a plane curve of the second order ; to find the locus of their point of intersection. Let the equation to the curve, and to one of the rect- angular planes, be respectively Li beet of; : : ; atiani, acosaty cos3 + % cosy =d; then making x’ = 0, the equation to the trace of the plane on wy is x cosa + y' cos3 = d, which must be identical with the equation to the tangent of the curve U , CR yy vd yd Sy yer Boel that cosa = GP? cos} = ra 2 way Ned ate = =") an (4) = a° cos’a + 6’ cos?3; and similarly , © id d® =a’ cos*a + 6 cos*', d’? = a’ cos’a” + b’ cos’. Hence, adding, d?+ d? +d”? = a? + b*, which shews that the locus is a sphere, concentric with the curve, whose ra- dius = \/a? + b°. Cor. If the curve be a hyperbola, the radius = J a — therefore we must have the transverse greater than the con- jugate axis, otherwise the problem is impossible; if a = 6, asin the rectangular hyperbola, the locus is a point, viz. the center. If the curve be a parabola, it may be shewn, as in Prob. 4, that the locus is a plane perpendicular to the plane of the parabola passing through its directrix. 6. If two concentric surfaces of the second order have 237 : the same foci for their principal sections, they will cut one another every where at right angles. x’ y 3? £ y 3 Behr ag chs bing sails oe y Ga = 1, be the equations to the surfaces; then the equations to the tangent planes are Po nx ie! y cena? © YY Se ‘ b) Sha. F Tiare 3 a? vege a’? b? ce? ‘ and in order that these may be at right angles, we must have Ga?” BBP ee? (hence one of the surfaces must be a hyperboloid, for some of the quantities a’, b*, &c. must be negative;) and this equation gives the relation among the co-ordinates of the points in which the surfaces may intersect at right angles. But by subtracting the two equations, we find 1 1 3 1 1 4 1 1 , aba as) ge OND Abbi F 2 Netacok eames for the relation among the co-ordinates of the actual points of intersection, which must be identical with the former if the surfaces intersect every where at right angles. Hence, equating the ratios of corresponding coefficients, a—a’?=? —-b? = — ec”, 6 , , iD , © a or @ —- P=a’*-b*, @-eC=aa*?—c*, Yb -—e’=b? —c?, which three equations express that the principal sections of the surfaces have the same foci, or that the surfaces are homofocal. Cor. In like manner, if the surfaces have not a center, and we represent their equations by 2 Y Ir la ars b*agiia i Za 3° (Sat aa ihe by eliminating w between these equations, and comparing the 238 result with the equation expressing that the tangent planes are at right angles, we find that the surfaces will intersect every where at right angles, provided BF 0 ae ee — 2p” 12 CC — aes 2a Pig at 2a 2a) that is, the foci of the principal sections of the surfaces must be coincident. 7. If three surfaces of the second order have the same foci for their principal sections, and if each of three planes mutually at right angles touch one of them, the point of inter- section of the planes will trace out a sphere. 8. If the tangent planes to two homofocal surfaces of the second order be parallel, the difference of the squares of the distances of those planes from the center of the surfaces is invariable. 9. The locus of the projection of the origin of co- ordinates upon the tangent planes to wyx = a*, has for its equation 27a°wyz = (a + y’ + 2°)’. 10. If the equation to a surface be \/v+\/y+\/ 2=\/a, the sum of the portions of the co-ordinate axes, intercepted between the origin and tangent plane at any point, is constant, 11. Ifwyszbe the point in a curve surface the tangent plane at which forms with the co-ordinate planes a pyramid of the least volume, the equation to that plane will be 12. The number of normals to a surface of the n‘" order that may pass through a given point cannot in general exceed V—-n +n. Take the given point for the origin, and suppose the equation to the surface to be w= 3" + Vis"! 4 Vis"? + ...4+ 0,124 V, =0, 239 where V, denotes an integral function of w and y of r dimensions; then du du R+@Q ie RS a5 ee © =0, or +Qp=0; similarly, S + Qq = 0; consequently the points of the surface through which the normal drawn from the origin may pass, will be determined by the values of w, y and x that simultaneously satisfy the three equations u= 0, Qe -Rz=0, Qy-—Szx=0; (1) and as each of these equations is of m dimensions, they will determine * points. But if x =0, w and Q are reduced to V, and V,_,; and equations (1) are replaced by %=0, y= Very 0s which determine m (m — 1) points in the plane of vy that may be foreign to the question, and such (not that the normal at them passes through the origin but) that at them the tangent plane is parallel to the axis of x; consequently the number of points determined by equations (1) that satisfy the proposed condition cannot in general exceed n’?— ”? +n. 13. If p, 7 be the perpendicular on the tangent plane and the radius vector at any point of a surface, then p* =r will equal the length of the perpendicular on the tangent plane at the corresponding point of the surface which is the locus of the extremity of p. Let wv, y, be the co-ordinates of a point in the first surface, a, 3, yy those of the corresponding point in the second surface; then we have the relation av+Bytys=p=ae+h?+y’, (1) since a, 3, y are the co-ordinates of the extremity of p. But this being the equation to the tangent plane at (wyz), the co- ordinates w, y, may receive indefinitely small variations 240 without the magnitude or position of p being altered; there- fore we may differentiate considering a, 3, -y as constant, . adw+ Bdyt+ydzx=0. (2). Now if V = f(a, B, y) = 0 be the equation to the surface traced out by the extremity of p, and P the perpendicular on the tangent pas to that surface, poate Bagt va, +N (a) + Ga) + Ge): and from the oats to fe surface ond +o 5 dB + dy =0 (3) But differentiating (1) pa it and taking account of (2) we get (w - 2a)da+ (y-—2B) dB + (z - 2y) dy =0; hence comparing the two latter equations we have d Therefore substituting for ae &c. in the value of P, and ‘A reducing by means of equation (1) we get _a(w7—2a)+BYy-2B)+y(8@-2y) P+ PhP +y Sa J (@—2a)* + (y— 23)? + (s-2y)? "a +y°42? 7 Application of the Formule for the Volumes and Areas of Surfaces to Examples. 1. To find the volume and area of the surface of a sphere, (fig. 41). Here av — — SL ae talers dudy od / a v Y's 241 + C, dV é . —-=taVYa-y-wv#+4(ae-y’) sin a” — y° and integrating between the limits 7 =0, r= MQ= /a — ¥°, *, integrating between the limits y = 0, y= CM, volume AMBQ = (a’y —4y’), 3 and, making y = a, vol. ABCD = = = = ae ae Andra *. vol. of whole sphere = 8 a= sos 2 as as\4 dz\? Ranger a, (<* ca oe dady / i e %3 i ds oe aot ib +C; dy / a? Le y? and taking the integral between the limits #7 = 0, v=\/ a?-y’, dS 1wa_ dy 2” .. integrating between the limits y = 0, y= CM, area of surface PBAQ = 7, ; Ta and making y = a, area of surf. DAB = ai 2 7 a ‘, area of surface of whole sphere = 8 ers 47a’. 16 242 2. The volume of the solid which is common to two equal spheres that have their centers in one another's surfaces, 5ara 12 1S 8. To find the area of the spherical surface intercepted by a right cone whose vertex is in the center of the sphere, and whose vertical angle is 2a. ds > @ As in Ex. 1, — = asin7=1———— +0, dy fay and integrating from # =0 to w= / a? sina — y’, ds Vie asin’ a — y” —— = asin~ ag y a —y? and again integrating from y=0 to y=asina (Integ. Cal. Art. 89), 9 4 a S= a (1 — cos a). 4. To find the area of the spherical surface intercepted between two meridians inclined at an angle @, and a plane perpendicular to one of them at a distance c from the center, and parallel to their line of intersection. | dS : = Pte See Bt Cc dx J a? — x : wv tan @ =a sin“} , from y=0 to y=a@£ tan@; \/ a — 2 x tan @ v = - @ tan @ f ——______ Va wv # (a? — x) \/a? — x sec? 0 * §'=awsin—} xv tan@ een es f SIN Ge eee OO, Va — a / a? — x? =aasin=! 243 and integrating from #=c to w=a, we get the surface required, Pie we OSI 7 : c tan @ 2S = 2a" sin~* ————— _— 2acsin™ en J/ ae aA ile? 5. To find the volume intercepted between the surfaces whose equations are Party, e=ar+y’ dxdy dV ——__— yt /e+ ye Ten dy Vara + hat log (VEVE EY) and taking the integral from y=0 to y =\/a? — a’, 9 dV = 3 a a — a Lan/@ — a + yeaa av therefore integrating between the limits vy = 0, w =a, we get (Integ. Calc. Ex. 5, p. 97) Te ra ra le ail See la ae ey rye Aa a *. volume required = 8V = 6. The axes of two equal cylinders intersect at right angles, to find the volume and superficial area of the portion which is common to both. Let OC (fig. 46) be the axis of one cylinder, O.4 that of the other; then the equation to AB or to the cylinder whose base it forms is s = 4/ a? — y’, av dV =<\/a’>—y’, and [ma Bais nk sal 8 ia dady — 16—2 24:4 integrating between the limits 7=0, w=ON= J a -y’; hence, integrating again between the limits y =0, y=a ae 16a3 vol. ADMB = oe and whole vol. intercepted = dz -y dz ad’ S a A 1 ——— ee | ee “gee 0, one = —_}~ ——_ § Bane? dy \/ a? a y : daw dady (yg y - ax + C=a from #=0 to va/fa-y ee dp by integrating which (since =f (p) by the equation to the generating curve), we obtain the equation to the projection. Ex. Let the surface be an oblate spheroid ; 2 pA , a —e to integrate this, make —, P 8 Rol 1h 2 , and the result is =U 260 If the surface be a right cone, the projection of the curve is an equiangular spiral; if a sphere radius a, the equation between the radius vector and perpendicular on the tangent to the projection, is 1 cosec’a cot’a — —__ a 10. To determine the shortest line that can be drawn between two given points on a surface of revolution. Let OC (fig. 48) be the axis of the surface which take for the axis of z, and a plane AOB perpendicular to it for that of wy; DP a portion of the shortest line=s, wv, y, 3 the co-ordinates of P, ON=p, 2AON=9, d.s\2 (apy* : pos el EP, a) +f ()}*, if s =f(p) be the equation to the generating curve, and dz ‘ dp =f (p); = Fay VA + ap? + {f' (wv)? = LV, ‘ d putting p=a, p=y,

hence an arc of a great circle is the shortest line that can connect two points on the surface of a sphere; this result may be obtained from the formula. Ex. 3. To determine the shortest line on the surface of an oblate spheroid. Let CEB (fig. 49) be the plane of the equator, PE, PN 262 two meridians, 4B a portion of the shortest line cutting PE perpendicularly at 4. Let the point M be determined by the angle @, so that CT'=p =asin0, ’'M=bcos@; and let A be similarly determined by the angle a, so that its distance from the axis = a sina; then by substitution we get ay OND cos’ 9 + 6? sin’ @ = sin@ , sin’ @ — sin’ a Now, since a is the least value that 0 can assume, let cos 9 = cosa cos Wy, i d in Pees ; ee. 4. beet Vals b? + (a? — b°) cos’ a cos” Ws ‘ cos a sin Vr ON: cos a sin Wy () Dre _Ja cos’ a + b* sin’ a — (a” — b*) cos’ a sin’ Wy = aft —c’sin* Wy, therefore s = arc of an ellipse, amplitude yy, whose semiaxes are a’ =/a? cos?a + b? sin? a, b, and eccentricity =c. At the point B, 6 = 90°, therefore y = 90°, and AB =a quadrant of the ellipse; also asina P ZL ABE = 90 - a. sin PMA = = sina at the point B, For the projection on the plane of the equator, dd : ds dp _ sina ds since p° — =asina do dp’ ms dw ~ asin? @ dy’ aD a a V/1—c' sin? adi, asina 1+msin* , where 2 = cot*a, 2 2 1 a r 1 i ere | AE + m sin’ hy) ~ asina ENE TE — ¢c’ sin’ PW ‘ p= {( +o) IT, (7, W- SF}, employing the notation of elliptic functions (Integ. Cale. p. 208.) Cor. If the eccentricity of the generating ellipse be very small, let a oe b? =e, and cos’acos* Wy =m; dp sina \/b? + (a? — b’) cos*acos’*W sina Ry/ Leh me diy a 1 — cos’ a cos’ vy l—m a rey sin a 1-m l1—m)(3+m) . 7 1) - Ee 4) EMO ACA) wee 1—m™m Q 8 sIn a sina o6 ¢& COSs’a = 2 ae a (« - a ee cos’ yy — &e.] : 1 — cos“ acos Vy 2 4A 8 tan sing 38e é . sing ; . p=tan( : 4 —y (6-5 | +o(ys "| sinacosta+ &e. sina 2 4, 16 2 The indefinite continuation of the curve will manifestly produce an infinite number of spires similar and equal, and contained between two parallels each at a distance = b cosa from the equator. 11. If a uniform and perfectly flexible thread of given length have its ends fastened in a surface of revolution whose axis 1s vertical, and rest upon that surface supposed perfectly smooth, to find the curve into which it will be formed. Taking the axis of the surface for that of z, and a plane perpendicular to it for that of wy, and proceeding as in Prob. 10, we have, # and & denoting constant quantities, to (noe) = J, (h +2) Ae + p* a4 + ta =a minimum ; 264 do dp\** (dz\* o* % h —_ = k 1 2 —— (=) Qa ee +e (7) +z) which, since we have x = f(p) by the equation to the gene- rating curve, is the differential equation to the projection of the required curve on a plane perpendicular to the axis of the surface. 12. The lines of greatest inclination on the surface \/ : . : : Seif 7) are given by the intersection of the surface with @ the cylinder x + y* = a’, where a is a parameter. 13. The tangent line at the point wyz to the curve 2 2 2 2 v4 @ Yy 20 x y & ne ae ee Ore +—-1 a? 2 a a ¢ x-v yy Y-y 2 2-8 1S > et Fe Ser ° a 6° a-—@ a c PROBLEMS ON Section VI. 1. Ir a surface of the second order having a center be cut by a plane which always passes through a fixed point, to find the locus of the centers of all the plane sections. Let Aa’? + By’ + Cz’ = D be the equation to the surface, z—c=A(w@7—a)+B (y-b) (1) the equation to the cutting plane, a, b, c being the co-ordinates of the point through which it always passes; therefore, elimi- nating y, the equation to the projection of the section on xw@ is Jez -c—dA'(w-a) + BO? =D. (2) z . B 24 Cx? A x + et 265 Let h, hk, J be the co-ordinates of the center of the section, then h and 7 are the co-ordinates of the center of the projec- tion; if therefore in equation (2) we substitute # +h for a, and x +/ for x, the projection will be referred to its center as origin, and therefore the coefficients of wv and x’ must each vanish; B B”? pete i(k 0) AP =1(B bee cy AR = 0, B , , Cl + Fe id — 0) + Bb- A (h—a)} =0. Multiply the latter of these equations by 4’ and add it to the former, : : Ah ee Ah+CAl=0, CM pint reat imilar] B re sImMarly, = ie Hence substituting these values of 4’ and B’ in (1), and expressing that it is satisfied by h, k, 1, we get Ah(h-—a)+ Bk (k- 6b) +Cl(l—-c) =0, the required equation, which represents a surface of the second order similar to the proposed one, the co-ordinates of its center being 4a, 3b, de. It is manifest that the two surfaces intersect in a plane of which the equation is daw + Bhy+Cez =D. 2. To find the locus of the middle points of chords of a surface of the second order that has a center, all passing through a given fixed point. Take the given point for the origin, and two conjugate diametral planes which pass through it for the planes of z@ and wy, and a plane parallel to the third conjugate plane for that of yx; then the equation to the surface will be of the form ax’ + by? +c2x°+2a v+d=0, 266 Let v= mz, y = nx, be the equations to any chord; then combining these equations with the former, we have (am? + bn? + c) 22 + 2a"ms+d=0, in which equation the values of = are the co-ordinates of the extremities of the chord; therefore the co-ordinates of its middle point are ” am , , , 3 vV= MB, Y=H=ns. , CS Se Sy ee am? + bn? + ¢ Hence, eliminating m and n, the required equation to the locus is an oy by” a ce” + ie 0; which represents a surface of the second order, similar to the proposed one, and similarly situated and passing through its center and through the origin. 8. The curve of contact of a surface of the second order, and of a cone the vertex of which is given, is a plane curve. Let aa’ + by +2? 4+2a'a +2by + 2e2+d=0, be the equation to the surface which may represent all surfaces of the second order. Then the equation to the tangent plane at a point vys is (ax+a)X+ (by+0') V+ (ex4+e) Z+ar+by+cx24d=0. Now because the cone and surface touch one another, at the points of contact their tangent planes coincide; and since the tangent plane of a conical surface always passes through its vertex, if h, k, 1 be the co-ordinates of the vertex they must satisfy the above equation ; “ (avt+ayjh+(byt+0)k+(exte)l+av+by+e2+d=0, which gives the relation among the co-ordinates wv, y, x of the points of contact, and shews that they all lie in the same plane. The curve of contact is therefore a plane curve resulting from 267 the intersection of the plane whose equation we have just found with the given surface. Cor. If the surface have a center, the line joining that point and the vertex of the enveloping cone, passes through the center of the curve of contact. Let C (fig. 52) be the center of the surface, 4 the vertex of the enveloping cone, Q2#S the plane of the curve of contact ; and let any plane drawn through 4 and C cut the surface in the line of the second order PQG, and the cone in the straight lines 4Q, AS’; then C is the center of the section PQG, and AQ, AS are tangents to it. Also, by a property of lines of the second order, QS, a line joining the points of contact, is an ordinate to the diameter PG which passes through the intersection of the tangents; therefore QS’ is parallel to the tangent at P, and is bisected in V, and CV.CA=CP’. Since therefore every chord of the curve of contact drawn through the point where a line joining the vertex of the cone and the center of the surface meets the plane of contact, is bisected in it, that point is the center of the curve of contact ; also the plane of contact is parallel to the plane touching the surface at the point where the same line meets the surface, and its distance from C is determined by the equation CV.CA = CP’. Similarly it may be shewn, that if a cone envelope an elliptic paraboloid, the line drawn through the vertex parallel to the axis of the surface passes through the center of the curve of contact; and the portion of it between that point and the vertex is bisected by the surface. 4. If the plane of contact of the enveloping cone of a surface of the second order always pass through a fixed point or a fixed line, the locus of the vertex of the cone will be respectively a plane or a straight line. Let 2’, y’, x be the co-ordinates of the fixed point through which the plane of contact always passes; then they must satisfy its equation, “. (av +a)hs (by +0)k 4+ (c2’4+0)l4+ a a'+b'y'+0'x’+d=0, 268 which gives the relation among the co-ordinates h, k, 1 of the vertex of the cone, and shews that the locus of the vertex is a plane; and that that plane is the plane of contact with the given surface of a cone whose vertex is in the given fixed point, for the equation may be put under the form (ah+a)a'+ (bk +0 )y'+(cl+c)x+adh+bk+cl+d=0. Again, let v=mz+ fy y=nz+g, be the equations to the line through which the plane of contact always passes. Then, in order that the plane may contain the line, we must have (Art. 23), (ah +a')m+ (bk +b')n + (cl + c’) =0, (ah+a)f+(bk+0)g+ah+0k+cl+d=0, which are the equations to the straight line in which the vertex moves, its co-ordinates being h, k, /. 5. If the plane of contact of the enveloping cone of a surface of the second order always touch a surface of the second order, the locus of the vertex will be another surface of the second order. Let the plane of contact always touch a surface of which the equation is me+ny +72? 4+2m'e+2n'y+2rx24+5=0......(1)3 then the equation to a plane touching this surface at a point VY 1s (ma+m')a'+(ny+n')y + (retr)s 4+m’'et+n'yt+r's+s=0, which must be identical with the equation to the plane of contact, namely, (ah +a')a'+ (bk +b)y +(cl tc) +adh+Vk+cl+d=0; met+tm ah+a ny+n bk+d t id eth oleae. eee en | Reet oy ley a. cree. metny+re+s ah+bk+cl+d me ees —_________ , TZ+7 el +c! 269 and it remains to eliminate v, y, x between these equations and equation (1). The result will be found to be m , n’ , ” r , , ‘ {i (ahta)+ (b+ U) 42 (el+e)—Whatkrcl+d| m2 mn? “2 1 1 1 rar. nea aide on h 4\2 yeni 2 a /\2 (= eee 5) > (a +a) +7 (bk +b) +i (else)t, which represents a surface of the second order. Cor. If the surfaces are concentric and av+by+ex,=1, ma’+ny? +72’ =1, their equations, the above result is reduced to 2 2 2 Cie TO ae —h?+—kh?+-—P=1. m n r 6. If two surfaces of the second order have each a prin- cipal section in the same plane, the projection of their curve of intersection on that plane is a curve of the second order. Suppose the plane of wy to contain a principal section of each surface, then no odd powers of z can enter into their equations, which will therefore be of the form Av’ + By? + Cx? +2Cay + 2A’a+ 2B"y + D=0, aa’ + by? + cz + 2Qcaey +20" a4 +2b' y+d=0, and if we multiply these equations respectively by ¢ and C and subtract, we shall find the equation to the projection of the curve of intersection of the surfaces on the plane of vy to be of the second degree in w and y. Hence if the axes of two surfaces of revolution intersect, since the plane containing the axes will be a principal plane of each surface, the projection of the curve of intersection on that plane will be a curve of the second order. 7. To find the diametral surface of Avy + es + yz) = vy, 270 which passes through the middle points of a system of chords equally inclined to the three co-ordinate axes. Here the equation to any chord, if aB-y be its middle point and m4/3 = 1, will be w-a=y-B=2s-y=mr, - a}(at+mr)(B+mr)+(atmr)(y+mr)+(B+mr)\(y+mr)} =(a+mr) (B+ mr) (y +mr), or m7 +(at+B+y—3a)m' r+ faBtay+By—-2a(atB+y) mr. +{aBy -a(aB+ay+By)$=0, or Ar? + Br? +Cr+D=0; and if p, —p, q be the roots, —Ag=B, —-Ap=C, Apq=D; «. AD=BC; . aBy —a(aBt+ay + By) =(a+B+y —-3a)$aB+ayt+ By -2a(at+ B+y)} the required equation. 8. The sum of the squares of the projections of an ellipsoid on any three planes at right angles to one another, is equal to the sum of the squares of its projections on its principal planes. Let the equation to the surface referred to any three rectangular axes through its center be av’ + by +c22 + 2ay%4+20ax+2cuy=d, (1). Since at those points in the surface for which cz +ay+ba=0, (2) dz dz dav dy are parallel to the axis of x; hence, eliminating = between (1) and (2), the equation to the projection of the ellipsoid on the plane of ay is become zero, the tangent planes at those points a (ac — b°) + y? (be — a”) + 2ay (ec —a'b’) = de; 271 .. square of area of this projection a’ dc” nw’ d’c ~ (ac — b”) (be — a”) - (ec’- ab’)? = re (S,5 S25 83, being the roots of the cubic in Art. 146) therefore sum of squares of projections on the co-ordinate planes w (a’ B? af ary? + By’) where a, 3, y are the three semi-axes of the ellipsoid. wd’ (a+b+ec) a 8 82 83 * 9. If O be a given point in a surface of the second order and OA, OB, OC any three chords passing through O, mutually at right angles, the plane ABC will always pass through a fixed point in the normal at O. Let the equation to the surface be au’ + by? + cz? + cay +c'2s=0, the co-ordinates being rectangular, and the axis of z a normal at the origin. Let the surface be referred to three new rect- angular axes ; 2. a(ma’ +ny +r2')+ b(m'a' +n'y' +17'')?+ c(m' a’ +n"y' +7"s")? +0 (ma! +ny' +73) (ma +n'y' +9'x') +0" (ma +n’y' +72’) =0. Let h, k, 1 be the parts of the axes of a’, y’, x’ intercepted by the surface M9 oe c'm m’) a em” = 0, “. h (am? + bm? + em k (an? + bn? + cn”? + cnn’) + cn" = 0, Lar + br? + er? err’) +67 = 03 therefore the equation to a plane passing through the ex- tremities of h, k, 7 is Ri wiloistetnk and referred to the original axes, it is metmy+m’s netnytn’s retrytr’s rey | 1g Ie i Il pd i) 272 and putting «=0, y =0, to find when it meets the axis x, we have ” A Ad \=1 (a+b+c) +c’ =0 — + —+—]j =1, or x(a c= 0, (s+ E45 ’ C) + consequently the plane cuts the normal in a fixed point. 10. An infinite number of surfaces of the second order may be made to pass through the sides of a given quadri- lateral in space. Let ¢=0, w=0, v=0, w=0, be the equations to four planes OAB, BAC, CAO, OBC (fig. 11) forming a tetrahe- dron; then the equations to the four edges AB, BC, CO, OA forming a quadrilateral in space, will be t=0 u=0 w=0 v=0 (1) =01 > ep = Ole tell Pele If therefore the equation to a surface of the second order be tw =p.Wv, it is evident that every one of the four straight lines (1) will be situated in the surface throughout its entire length ; and as uw is arbitrary, an infinite number of surfaces may so pass through the same four lines. 11. , <, or = a’7". In orthographic projection c= ©, # =0; . vw +y + (ax + b) =2°, the equation to an ellipse. It is to be observed that the optical cone, since it meets the surface of the sphere in a plane curve, will meet that surface again in a plane curve (Art. 78); so that the stereo- graphic projection relative to any point, of any circle of a sphere upon the surface of the sphere, is a circle. In the same manner it may be shewn that the stereo- graphic projection of any plane section of a paraboloid of revolution upon a plane perpendicular to its axis, isa circle, the eye being placed in the vertex; or that the stereographic projection of any section of an oblate spheroid upon the plane of the equator, is a circle, the eye being placed in the pole. We may employ the same method also to compare any angle on the surface of a sphere with its different projections ; this, however, may be done more simply by geometrical con- siderations, as will be seen in the following problem. 8. The gnomonic, orthographic, and stereographic pro- jections of the rhumb-line, which is a curve of double curvature traced on the surface of a sphere so as to cut all meridians under the same angle, are three of Cotes’s spirals. First to find the gnomonic projection. Let y Ba, (fig. 55) be the plane of projection perpendicular to BC, T¢ its intersection with a plane touching the sphere at a point P in the plane of za, then T¢ is parallel to By; Pt a tangent to the rhumb-line at P, Q the projection of P de- termined by producing CP to meet the plane wBy. Then ZTQ¢t is the projection of T'Pt; A ES Me CB and tan 7'Q TQ TQ TP TQ tan 7'Pt CQ tana, if a =the constant angle at which the rhumb-line cuts the 279 meridian. Let BQ=p, p=perpendicular from B on Qt which touches the projection of the rhumb-line at Q, BC =a, ei p atana Li cosec? q/\* cot’a Jip V/p+a ep p” rs ate? the equation to one of Cotes's spirals. Secondly to find the orthographic projection. Let # Cy be the plane of projection (fig. 56), T?¢ its inter- section with a plane touching the sphere at a point P, situated in the plane of za, then 7'¢ is parallel to Cy; Pt a tangent to the rhumb-line at P, N the projection of P, thenz TN¢ is the projection of 7'P¢; Ls Gene 0) Ege, fate iy GP t = SS ee ee SC >| . f = 3 and tan 7'N¢ TN Ty’ TP pn’ T'Pt py ane or if CN = p, p= perpendicular from C on N# which touches the projection at JV, p atana Dee cosec’ ai cot? a ye RU ar rR ca the equation to the hyperbolic spiral, as already found p. 260. Thirdly to find the stereographic projection, Let CO = CP be perpendicular to the plane of projection «Cy (fig. 56) ; join PO meeting the plane of projection in Q, which is therefore the projection of P, and 2 7'Qt of T'Pt. Then Z TPQ =complement of CPO =complement of COP =PQT, .«. PT=QT;; and since 7J'¢ is perpendicular both to PT and QT, .. 2TQt=T7Pt=a; therefore since the angle between the tangent and radius vector is constant, the curve is an equiangular spiral, its equation being p = p sin a. Gor. It appears, that if an angle a on the surface of a sphere have one of the arcs which contain it, a meridian, and if ry denote its gnomonic, w its orthographic projection, and @ the are of the containing meridian intercepted between its vertex and a plane through the center parallel to the plane of projec- 280 tion, then tan y=sin@tana, tanw= tana; also tana sin = /tanw. tan ry, that is, the tangent of the stereographic projection is a mean proportional between the tangents of the orthographic and gnomonic projections of the same angle. 9. To find the equation to the conoidal surface gene- rated by a horizontal straight line which constantly passes through a Helix, and the axis of the vertical cylinder on which it is traced. The equations to the Helix are (Prob. 2, p. 254) w e+y=a’, z=nacos'—; a let the equations to the generating line be s=, y= aa, then eliminating a, y, 2, a +a°’2 =a’, & ] 1 Te ee ig V1 a therefore, restoring the values of a and £, the equation to the surface is 1 ie cose =natan—'aq; Z= na tan) ; c This is the surface presented by the inferior superficies of a staircase, attached to a vertical column round which it winds. 10. To find the equation to the surface generated by the revolution of any straight line about a fixed axis. Let the fixed axis be taken for the axis of zx, and let the equations to the directing straight line be vw=Azth, y= Brr+k, and those to the generating circle x = 2, a + y° =a sethen eliminating a, y, x, we get (A468 +h)’?+(BB+k) =a, or, restoring the values of a and #, | (Az +h)? +(Bs+kye=a'’s+y’, or vw +4? — (47 + B’) 2 -2(AR+ Bh) 2 =h' + k’, 281. which is the equation to the surface, and manifestly represents a hyperboloid of one sheet with its center in the axis of gz. If in this equation we make w = 0, or y = 0, in order to find the equation to a section through the axis, the resulting equation represents a hyperbola; therefore the surface is a hyperboloid of revolution. Also if, retaining one of the above equations to the directrix, we change the signs of the second member in the other, the equation to the surface is not altered; therefore the same surface may be generated by two different straight lines, and through any one of its points two straight lines may be drawn so as to entirely coincide with it. This agrees with Art. 73. If we take for axis of x the shortest distance of the directrix from the fixed axis, the equations to the directrix, since it is parallel to the plane of yz, will be r=h, y= Bx; therefore putting 4=0, k=0, the equation to the surface (reckoned from its center as origin) becomes e+ yy? — Bs? = h’. 11. An oblate spheroid revolves about any diameter, to find the equation to the surface which envelopes it in every position. Let w=mz, y= nz be the equations to the diameter, c (a + y’) + ax? =a’ the equation to the spheroid ; then the equation (y—nz)p—(# —mz)q+my —nx=0 becomes (my — nx) (a? — c?) =0; so that the equations to the directrix are C(#+7)+asx =a, nx=my, and the equations to the generating circle, s+me+ny=2, wt+y +2 =a; eliminating w, y, x between these four equations, we find a? (a ~ c°) (m? + n°) = (Bae Va =a)" 282 therefore, restoring the values of a@ and #, the required equation is a? (a + y® + 3? — 0?) (m? + 2’) =S[(st+me+ny)/ae-@ —cV/a —a —y — #°) 12. The locus of the normal to the surface y = # tannz along a generating line is a hyperbolic paraboloid. PROBLEMS ON SeEctTion VIII. Tue following are some examples of finding the equations to twisted and developable surfaces, and envelopes. 1. To find the equation to the twisted surface of which the directrices are two vertical circles having the opposite sides of a horizontal parallelogram for diameters, and a straight line passing through the center of the parallelogram perpendicular to the planes of the circles. Let the center of the parallelogram be taken for origin and its plane for that of wy, and the rectilinear directrix for the axis of y; then the equations to the three directrices will be c=0 2=0, y=-b (#-a)+xX?=7, y=4+b (#+a)?+2? =2". Also let the equations to the moveable line be w=a(y-B), x=y(y- 8), so that it already fulfils the condition of meeting the axis of y, and one of the parameters consequently is eliminated; then, expressing that it passes through each of the circles, we have CC eae ane (1); 12 (6 B) 42h" 4 oO = Be ey Cees 283 therefore, by subtraction, B (ba? + aa + by’) =0; and rejecting the value 8 = 0 which would make the moveable line always pass through the origin and so generate an oblique cone, we have a” + 9° +o = Qe» (2)s by virtue of which, either of the equations (1) gives | aa (b®? — B’) = b(7* — a’)...(3); and it remains to eliminate a, 9, yy between (2) and (3) and the equations to the generating line. Substituting the values of a and vy given by the latter in (2) we find b a? + 2° a w =yt+- and then a = —— ———.; PAL, Gia alas b wv + 3? therefore, substituting these values of a and @ in (3) and re- ducing, we find for the equation to the surface faany + b (a + 2°) }? = br? ax + b? 2? (7? — a’). 2. To find the equation to the developable surface gene- rated by a straight line which constantly touches a Helix. The equations to the Helix being x ns v=acos—, y=asin—, na na at a point for which x = a the equations to its tangent are a Late ne v — a cos—— = — — sin — (z — a), na n na hag Oi 1 a y — asin — = — cos — (#—- a); Nad nm na a i oeet a ar\/ a + 2_ ge — aw , @ CoS — + Y sin — =a, and tan — Eyed da Yn & Gs na na na y - a 284 also, adding the squares of the above equations, we find 1 2 2 2 \2 ¢ + = ys eS, : vty —a 5 (% -a) therefore the equation to the surface is ——_——,, a/xv+y—-a—2x n/ x +y°—-a@’=2%—-—natan"' (oy eae : y—a 3. To find the nature of the curve traced upon a cylin- drical surface, when the surface is made plane. Let aw, y, = be the co-ordinates of any point in a curve traced upon a cylindrical surface; then the equations to the generating line which passes through that point are zg —-zs=m(a'—-2), y—-y=n(a'—-2@); and the equations to the tangent line of the curve are ped pike its Maenacly Be Pitts pe CED yy =— (a -2); let 7 denote the angle between these two lines, y the in- clination of the generating line to the axis of w, and s the length of the curve; then (Art. 30) cost = cosy ( La +1) i= cosy (m— +n — tT. TO da Now when the cylinder is developed, the generating lines preserve their parallelism, and may therefore be taken for the ordinates ; let a line perpendicular to them be the axis of the abscissee, and a, (3, the co-ordinates of the point in question ; d d d then “P = cos , bes oP — cosy (m S24 nts 1); also 1+ 68+ (= 65) +8) ~ Ge +2) 285 : ds\* : each being the value of (=) ; and if between these and wv the two equations to the curve we eliminate a, there will remain a differential equation between a and 3 which will be that to the required plane curve. But if the latter equation be given, that is, 8 = f(a), we may eliminate « between the two above equations, and there will remain a differential equation between a, y, and zs, which, with the given equation to the surface, will be the equations to the required curve of double curvature. Cor. If the curve on a plane be a straight line, the angle 7 is constant ; Pets (m dz dy : "ds eee CO == —— Se Y an 2) en } COS 2 ve is the equation for determining its nature, when the plane is applied to a cylinder ; ae curve is called the Helix, and we have for its length, by integration, cos y (ms + ny + #) = cosi(s + C). It is manifestly the shortest line which can join two points on the surface of the cylinder. 4. To find the nature of the curve traced upon a conical surface, when the surface is made plane. Let a, b, c be the co-ordinates of the vertex of a conical surface, v, y, x co-ordinates of a point in a curve traced upon it, and 7 the distance of that point from the vertex. When the cone is developed, the generating lines become radii vectores, and if @ be the angle which x forms with some fixed : ds\’ radius vector, equating the two values of (==) » we have a ESE ACI ale ee MOEN Nee LAE ae (a) + (3) fo uo a) +13 (5) : gun niente) (yi 2b)" 6 (iC) from which, together with the two equations to the curve we may eliminate «, y, x, and there will remain a relation 286 between x and 0, the equation to the plane curve when the surface is developed; or, if » = (0), and the equation to the conical surface be given, we may eliminate r and 0, and so arrive at the remaining equation to the curve of double curvature resulting from the application of a plane on which a given curve is traced to a given conical surface. Cor. All curves which make a constant angle with the generating line of the cone are characterized by the equation dr Ber ak i, and become equiangular spirals when the surface s is developed. Also all curves which become straight lines when the surface is developed, and which therefore are the shortest lines which can connect two points on its surface, are d 7 A gps Sak characterized by the equation aes /r — a’, a being the de dex length of the perpendicular let fall upon the line from the vertex as origin, or 1 ee Bag SE a é 2 2 (Jilin yoda zs (5=) ey (54) aaV? See et or finally, by developing this equation, Ot) eed FOS ee 3 Otter 0 \. es dz\* (dy\? Ca ea) + (ye 9s) = & {1+(3) + (5) . When the surface is a right cone on a circular base, the problem admits of a very simple solution, as will be seen in the following instances. 5. A given curve being traced on the surface of a right cone whose base is a circle, to determine its equation when the surface is developed. Let P (fig. 58) be a point of the curve situated on the generating line CR, Q its projection on a plane perpendicular to the axis; then our object is to find a relation between CP =r, and the angle which CP makes with CA, when the cone is developed ; call this angle 0, and let CQ =7', QCx = 6’, m=cosec ACD; 287 now, when the cone is developed, @ is subtended by a circular are, length AR; ~.0.4C =0'.AD, or, 0 =m@; also CP.sin PCD = CQ, or, est if, therefore, f(7’, 0°) = 0 be the given equation to the locus of Q, F(=, m8) =0 is the equation to the curve when the surface of the cone is developed. Conversely, if f(r, 9) =0 be the equation to a curve traced in a circular sector, and the sector be formed into a right cone, the equation to the projection of the curve on a plane perpen- , dicular to the axis of the cone will be f( mr", =) = 0. m Ex. 1. ae 4; also the equation to the semicircle is r = a cos 0, therefore the equation to its projection is , Sl 1 1 , r=7acos;6. 6. To determine the shortest line on the surface of an ellipsoid. av 7 Let — + a + — =1 be the equation to the surface; then a 1H q e e p=- bia a q= = es and the equations to the shortest line as & Ly eae ving ad’ x ; F papel waka y By, 7 =<" where a” = a &c., the arc s being Bs s an independent variable; also we have 3’ sol +qy’ which gives wx yy | 88 dle ne —— hk a = 01), ands a v wo y y"" 2 3"! y pe ar y 2 ) ag EY ate pee SE ae Tee | ey a B Y x z 3° f @) rom yw ru ” REAPER M EA od ig Fc a p + Po Be *y” Hence multiplying this by equation (2) so as to eliminate ys" x and integrating, we get Be A Bee ois ae : Ce eeeaiad oP) NAG: iG Cc between which and equation (1) together with the equation to the surface if we eliminate one of the variables, we shall obtain the required differential equation of the first order to the shortest line, or geodesic line as it is called. ? 289 7. The direction of the shortest line at any point of an ellipsoid is determined by the condition that the product - of the semidiameter of the surface which is parallel to that direction, multiplied by the perpendicular from the center on the tangent plane at the same point, is invariable. For if a semidiameter D be drawn parallel to the tangent of the shortest line at vyz, wv’, y’, x’ will be the cosines of the angles which it forms with the axes of the ellipsoid, and we shall have Sy ad | eg a sean 73 +—_= alii chats Coolie aii i from Prob, 3, p. 234, P being the perpendicular from the center on the tangent plane at wyz; therefore the result of the preceding Problem becomes PxDe=C. 8. The following are examples of Envelopes, when the equation to the series of surfaces contains only one parameter. Ex. 1. To find the equation to the envelope of all right cones of a constant volume, whose axes are in the same straight line, and bases in the same plane. eee b The equation to any cone is =@ , where a = altitude, 6 = radius of base, and the origin is in the center of the base. Let its volume always equal that of a hemi- sphere, diameter 3c; wah’ 7 (8e)? PM aon 27 ¢% J a+ y? %3 : Saeco or ety eae | Le (i-Y=*"), differentiate with respect to the parameter 6, 3 3 et — — + ee Ja +y, or b == Val ays a a (1 =) = a , the equation required. = 2b 276 2) 9c ADV ae carr, Cr abr 19 290 are the equations to the characteristic, that is, determine the radius and position of the center of the circle, in which the cone, the radius of whose base = b, is intersected by the consecutive cone of the same volume. Ex. 2. Ifa series of planes, passing through a fixed point in the axis of x, have their traces on the plane of wy all of the same length ; to find the equation to the developable surface formed by their intersections. If c= distance of the fixed point from the origin, a = length of the trace on the plane of wy, and a= the angle which it makes with the axis of x, the equation to the plane is v bask acosa asina ¢ which being differentiated with respect to a gives wtan’a = y; therefore substituting in the equation to the plane, we get Ps 1 — + — (#3 + y3)§ = 1, c hua the required equation to the surface. The equations to the characteristic in this case are x y -sePa+-=1, == tan’®a. a c av Ex. 3, If the center of a sphere whose radius =a, describe a curve in the plane of «xy, to find the equation to the annular surface which envelopes the sphere in every position. Let a and £3 be the co-ordinates of the center of the sphere in any position, and y=f(a#) the equation to the curve which it describes, therefore (3 = f(a) ; . (w-a)’+ fy -fla)P+=a is the equation to the surface, and by differentiating with respect to a, w-at fy—f(a)}f’ (a) =03 these are the two equations to the characteristic, whose 291 position and magnitude depend upon a; and if we elimi- nate a between them, we find the equation to the envelope. The characteristic in this case is manifestly a circle of con- stant radius whose center is a point in the curve which forms the axis of the surface, and whose plane is perpen- dicular to the tangent line at that point, as is expressed by the above equations; hence its nature is entirely inde- pendent of the curve whose equation is y= f(x); it is also the curve of greatest inclination, for its tangent line at every point is perpendicular to the intersection, with the plane of wy, of the plane touching the surface at the same point, therefore its equation is (p. 257) are The remaining equation is that to the envelope since the characteristic is situated upon it, which we obtain by observing that the normal always meets the plane of zy in the curve which forms the axis of the surface, and the length is equal to the radius of the generating sphere ; .=eVYltps+g=a, (Art.110) or #1 4+p'+q) = a3 this might also have been obtained by eliminating the arbitrary function from the two equations to the characteristic ; hence the integral of the equation z°(1 + p® + q*) =a’ is re- presented by the system of equations to the characteristic. Ex. 4. If the vertex of a right cone describe a given curve in the plane of wy, and its axis be always perpen- dicular to that plane, to find the equation to the surface which touches and envelopes it in every position, Let a and £ be the co-ordinates of the vertex, y = f(«) the equation to the curve which it describes, therefore 3 = f(a) ; a = tangent of the angle which the side of the cone makes with the plane of wy; then the equation to the surface is Pv (@- a)? + fy -f@}= 55 differentiate twice successively with respect to a, . Sy-fla)t f'(a)+a—-a=0, fy-f(a) $f" (a)-1- ff (a) }?=05 19—2 292 from which equations we may deduce as above the equation to the envelope, and the equations to its characteristic, and edge of regression. The tangent plane to the envelope at any point is also the tangent plane to the generating cone, and therefore makes a constant angle yy with the plane of wy; but sec y= V/l+p'+qs therefore, since a=tan-y, the differential equation to the surface is p+g=a’; the integral of which is of course represented by the system of equations to the characteristic. Hence, if S’ denote the area of any portion of the surface, as! dudy that is, any portion of the surface bears a constant ratio to its projection on the plane of wy. The characteristic in this case is evidently a side of the generating cone, and therefore perpendicular to the intersection of the tangent plane with the plane of wy; therefore its two equations are any +p +q= secry, or S = A, secy; dy q 2. 2 2 da p’ Pq = 2s hence the curve of greatest inclination of all surfaces generated in this manner is a straight line, inclined at a constant angle to the plane of vy. 9. The following are examples of envelopes when the equation to the series of surfaces contains two parameters. Ex. 1. To find the equation to the surface touched by a series of planes so drawn, that the product of the per- pendiculars let fall upon any one from two fixed points is invariable. Let the line joining the two points be the axis of «, 2a its length, and the origin in its middle point; z=Ax+ By +e 293 the equation to one of the planes; then the product of the perpendiculars upon it will be found to be ce — A’ a” : eT ae b° suppose, or c’= 6°(1 + 4°+ B’) + A’a?. Hence, differentiating the equation to the plane successively with respect to A and B considering ¢ as a function of A and B by virtue of the above equation, we have A i Ce) or ates GHA Se Cc B B : OD alin 12 a or y = — b’—; and consequently x =— ; r) Cc Cc 9 xv” . y i" 2 (P4+D)4+VP B+ ; * a? +6 BB - Cc ey the equation to a prolate spheroid, whose axis coincides with the line joining the two given points. Ex. 2. Let the equation to a surface be ™m v y” on ay a Ys bn + a” Pie to find the equation to the surface which it touches in all the positions it can assume, subject to the condition a”+ b"+c"=k" a constant quantity. Differentiating the equation to the surface successively with respect to a and b, regarding c as a function of a and b by virtue of the equation of condition, we have a™ gm q"-1 a” gm a” ake yas iC apemirian as Re er e e y” 2m br-} y” g™ b” m+1 m+l atts a 0, OF Fh Dn a b c c Cor'C Pare me Se Sa Ot ot et pee gen ete Aa ae Toe TGR as xe Om A nt J ee a b Cc c c AY»; k k 294 mn mn mn Z\ m+n Cc -nilarl U\ m+n a” y\ m+n b” or y. = ie 5 similar af ie = ie 9 4 — ke 9 mn mn man mn =, BM TB ymin 4 ymin — fe™+", the equation required. If m=n=1, the equation becomes \/ 8 + / a + V/y = / kes and agrees with Prob. 10, p. 238; ifm =n=2, itiszs+a@+y=hk, which shews that if a series of surfaces of the second order have the sum of the squares of their axes constant, they are all touched by a plane. Similarly, if the equation of condition be abe = k’*, it may 3 aye be shewn that the equation to the envelope is vyx = (=) ; and if m=1, we find wyz = Ss for the equation to the surface touched by a series of planes which form with the co-ordinate planes a pyramid of constant volume. ‘The surface repre- sented by the equation wyx =a constant, has also the property that the tangent plane at any point 2, y, x, forms with the co-ordinate planes a pyramid of smaller volume than any other plane drawn through the point of contact, the equation to the tangent plane being } , moth Ld poe leg, Ue Pm os Kix. 3. To find the equation to the surface touched by a series of planes which cut off from a given right cone, an oblique cone, such that its volume is constant, or such that the transverse axis of its elliptic base is constant. If the volume of the oblique cone be constant, then by a property of conic sections the transverse axis of the elliptic base is also constant, suppose it’... = 2c, and let 6 = semi-vertical angle of the cone; then it will be found that the equation to the touched surface is z* = cot” (a + y’ +c’), the origin being at the vertex ; which belongs to a hyperboloid of revolution about the axis of the cone, and to which the surface of the 295 cone is an asymptote, the 3 axes of the generating hyperbola being c and ccot 6; also the point of contact is the center of the elliptic section. In like manner if a series of planes cut off from a parabo- loid of revolution a segment of constant volume, the surface touched by them is a similar and equal paraboloid about the same axis, Ex. 4. When from three straight lines meeting in a point a plane cuts off three segments such that the product of two of them divided by the third is constant, the plane constantly touches a hyperbolic paraboloid. Ex, 5. To find the equation to the surface which is always touched by a plane whose equation is me+ny+iz=v,,.,..(1), m and 2 being independent parameters, and J and wv functions of m and m determined by the equations Differentiating (1) with respect to m and n, and taking account of (2), we have 7) dv n dv vex (-7 ree y+e(-7) ==. But differentiating (3) with respect to m, and putting for dv é —— the value found above, we have, making dm : m n ia eg bg toe A (v° he a’)? (v" = b*)? (v" = c’)? > 1 1 & & - Z of (e4- *) 3 al (hee a Co & ; similarly, 2s) tae A ie = *) v; 296 therefore, multiplying by m’ and n’, and adding, m” n* P—1 ?—1 +5 = A (mr t+ ny + Pe 0 o” —-q? a 4 ee z J y a s—a- 4(F-0)o, apo 4(Z-0)e. Therefore multiplying the squares of these three equations by , m*, n”, and adding, A= A (a +y' + 2° 4+v°—2v°)v* by (1) and (2), or 1= A (7 — wv’) v*, making r? =a? + y' 4+ 2”. pes lv Sm ay wt ea ra Hence from (4) =—, VOCS y— or gz—lu v~ zu — lv and adding unity to each side of these equations we find Yr —c Ps ce sv—lr rP—v g—lv’ vw xv—lv?’ aie aie C* % v (xv — Ir’) therefore eliminating z — Jv, we have ———. = —____ ; 2 2 2 2 Yr —C r—v . nilarl a? @ v (wv — mr’) bey v(yv —nr"’) similiar => — = —_—___—_— ; y 72 — QQ? yp? — a? y? — Bb? y2 — therefore multiplying by z, x, y, respectively, and adding, we have the required equation to the surface, Cc gy? a” uv’ b? y” v 2 2 + ——. + =" -,, = = ("0 - rv) = 0.~ (ee | ga ar = 2 ae | ) 10. To find the singular points of the surface a? x’ b? y Cc 3° Pic? Cea Tae or u=7"(a?a?+b°y’ +0") — a2a?(b’ +07) —b’y?(a? +0”) —c°s" (a? +b’) 40°’? =0, a, b, and ¢ being in descending order of magnitude. 297 We must have (Cor. Art. 108) ze = 0, ie = 0, = meri or w {a2(r?— b’ - c?) + a? a’ + by’ + c?2°} = 0, y {P(r — a? — oc) + a? a’ + Dy’ + c?s"t = 0, sf{e(r-a@-B)+ aa +b y+ &x*t =0. We can neither suppose the three variables to vanish together, nor two of them to vanish together, for by neither of these suppositions is the equation «=0 satisfied. Also if we make w or x vanish separately, the values of the other co- ordinates become Sat deh but if we make y = 0, we find a? — b? Bae vw=+c - ee Ea - a’ — which are real values and satisfy w=0; and at the point corresponding to these values, we get d? a—-B du Bese Wits GG RAY Sima saountrse 200) a’ sate? BP—c du ——— | C. SSS SS d 3° a’—c? dady ’ au a? + ¢ = a eae EE ea SI hens ah ie te VG . = Ponca)» S consequently the equation to the tangent cone at the singular point, taking that point as origin, becomes 2 uv? ae, a a +c os Boe ace 7 "PP * /(@- b?)(b?-c) ac Also for the distance of the singular point from the center we have, substituting its co-ordinates in (1), 2-h R-@& + : Poee gt FS =0; which gives 7? = b’, If fig. 60 represent the intersecting ellipse and circle that form the principal section of the surface in the plane of az, then the intersections of those curves, one of which is R, are 298 the singular points, and the two sheets of the surface meet in them. 11. To shew that tangent planes can be found which shall touch the surface in the preceding Problem in circles whose GEA) Ae radll ob The section by «z is an ellipse and a circle intersecting one another represented by the equations (a +2 — b’) (a?x 4 Ce? ac?) = 0 (fig. 60). Let PQT be a common tangent to this ellipse and circle, then CQ the perpendicular upon it = 6, let zQ7'C = 0, then 2 CT =bcosec@, T7'Q= bcotd, TP = — cot 6, and cot@ 6 oe Cc e ee = yf ——,,- We shall begin by determining the intersection a-— with the surface of a plane perpendicular to s@ passing through the common tangent PQT'; for which purpose we must, in the equation to the surface, change 2 into — x cos 9 + bcosec @, and x into wsin @ (Art. 101) which gives, dividing by 6° and observing that a” cos’ 0 + c’ sin’ 0 = BD’, 2a? (a + y? + cosec’?@ — 2bacot) (a +4" +a’ cosec’™ — yp weot 0) 2 2 a’c —2° (2 + ) yeye 2a” “t oe ( +c’) a cot 8 — ab’ cosec® 9 — a*c’cot?@ = 0. From this we get by successive reductions, taking notice of the value of cot 0, g (a +)? + (a +’) {G + b°) cosec” 9 — 2x cot 0 (2 oe =) + a°b’ cosec’O — 4.2.a7b cot @cosec?@ + 4a°x* cot’ @ — (x* + y*)(a" +c’) 2 + x°(a°—b") (1 ae, 207 2 . 7a) one (U' +c") cotO—a’b’cosec®8 ~a*c’cot*O=0, 299 or (wv +9’)? + 2(a° + y’)acot 6 {0 cot 0 — & (< ee -)| a ek egy. ey ay: + (a cot 0)* — 2(acot 6)° (5 ae -) v® + aa’ cot’@ ¢ ae = a a or | a’+y? , a AL v+y +tacot@iacotd—wx ae = 0, a which shews that for the assumed position of the cutting plane, the two curves of intersection become united in a circle (and consequently the plane touches the surface at every point in the circle) whose equation is a Ob y? = acot@ (; + -) 2 — a’ cot?@ — 2” 2 “= ( cot 8 — v) (@ — bcot 0) =(PT —- x)(« - QT), so that PQ is the diameter of the plane circle of contact; also 22 by the ellipse CP’ = a’ +c? — — 2A 1 . P@ =a - B+ 8 - ae =a (a - BY - e). ProspLeMs on: Section IX. 1. Asa first example of this theory, take the curve of double curvature, resulting from the intersection of a sphere and cylinder, considered in Prob. 1, p. 253. Its equations are y°=2av—a"*, x°=4a°—2aa, and the equation to the normal plane, as we have seen, is To obtain the equations to the line in which this is inter- 300 sected by the consecutive normal plane, we must join to it the derived equation with respect to #, namely 1 a-«a« dy a dz u(-f- GB) EE or, by substitution, 2 +4 — =0,......(2)5 fois and to find the equation to the surface generated by the per- petual intersection of the normal planes, we must eliminate v, y, and x, between these two equations, and the equations to the curve. Yuen Pee (ve Now from (2) (-%) == or — =2(%) : x x 3 , 2a a . v a—- @ and (1) may be written — ey hiss eee x %, a x é nq Nal uagip eae a & av Xv Xv 3 or — =(2-2) +%(1-=) + (%) = 0. x a a % a} x é 4 4 e e v e e Hence substituting for — its value, and reducing, we find for a the surface generated by the intersection of the normal planes 2 eT ET Te, 2 ee 2(e, - yp/ 23 -y)= 8/85 — y 3. a Al 1 - 2 2g sO +y° +a ay | a +2) Ad dA ") eel A ih de a a y+ @')+ y's ~2"y)'= (F5) (loa +52) 5 (2a 4+ 2x)? .. radius of curvature = —7—="—.. \/ 100+ 3H 2. On a given surface, to trace a curve of such a nature, that its involute shall be a plane curve. The curve of constant inclination, that is, one whose tan- gent line is inclined at a constant angle to the plane of ay, will have this property. For if s denote the length of the are, inter- 301 cepted between the plane of wy and a point whose ordinate is x, and + = the constant angle which the tangent line makes f ; dz with the axis of x, then rF = COS y ; Ss *. S=scosy, ands =~2 secy = the length of the tangent line between the point of con- tact and plane of wy; hence if a thread be applied to the curve of equal inclination, and then be unwound from it beginning from the plane of wy and be kept stretched in the direction of a tangent, the extremity of the thread will be always in the plane of wy, and will describe a curve which is manifestly the involute of the projection of the curve of equal inclination on the same plane; for the projection of the tangent line will touch the projection of the curve, and will be of the same length. The equations to the curve of equal inclination were found at page 257. Ex. Let the surface be a right cone and the origin in~ its vertex, so that p = x tana, 2 Fe eA oes Ue, cotta = cot! y 5» or Pp =pV/1- (tana cot y)’, the equation to an equiangular spiral. Hence it follows that this curve cuts the generating line of the cone under a con- stant angle, that is, it is the conical helix. Hence if a thread be applied to the surface of a cone, according to a helix traced on it, and then be unwound beginning from the vertex, its extremity will always be found in a plane perpendicular to the axis through the vertex, and will trace out an equiangular spiral on that plane, which is also the involute of the pro- jection of the conical helix on the same plane, and therefore similar to it. 3. If a uniform and flexible string be suspended from two points in the surface of a vertical cylinder whose base is a circle, it will form itself into a curve of double curvature, such that the involute lies on the surface of a sphere. 302 Let A (fig. 61) be the lowest point of the catenary, 4B the radius of curvature at that point; take the horizontal plane drawn through B for the plane of wy, and the axis of the cylinder for that of z, and let CM, MN, NP be the co- ordinates of any point P in the curve. Draw NT" per- pendicular to the tangent PT’ and join C7’; then PT is perpendicular to the plane C7'N, for a line drawn through NV parallel to PT’, and therefore coinciding with the tangent plane of the cylinder would be at right angles both to 7'N, CN, and therefore to the plane passing through them; and CNT is a right angle, therefore CT? = CN’ + NT® = CB’ + BA’ = CA’, because the perpendicular upon the tangent from the foot of the ordinate, is constant by the nature of the catenary. Hence the locus of 7’ is on a sphere radius CA, and since CT'P is a right angle, therefore PT' touches the sphere. But PT'=arc AP; if therefore a thread be unwound from AP beginning from A, and be always kept stretched in the direction of a tangent, its extremity will trace out a curve situated on a sphere and the thread itself will be a tangent to the sphere. 4. In a curve of double curvature, if S'Y be a perpen- dicular on the tangent at any point P from a fixed point S, and SY" be a perpendicular on YY’ the tangent to the locus of Y, then, 2 being the perpendicular from S$ on the plane through YY’ and the middle point of SP, SY! = SP’ (SY? -— h’). In fig. 55 bis take Q, R, points of the curve on opposite sides of P and very near to it, and let qg, 7, be the points of intersection of the perpendiculars from §' on the chords PQ, PR produced; these points lie on a sphere diameter SP. Through O the middle point of SP, and through the line qr draw a plane cutting this sphere in the circle gYr. Now let Q; R, move up to and coincide with P; then Pq, Pr both coincide with PY the tangent to the curve at P, and qr coincides with YY’ the tangent to the locus of Y; and SY’ 303 being the perpendicular from the vertex of a cone upon a tan- gent to the circular base, it is easily shewn that SY‘ = SP*. (SY? —i’). Hence the normal plane to the locus of the foot of the per- pendicular on the tangent to any curve from the pole, always bisects the corresponding radius vector. ProBLEMS ON SECTION X. WE shall next apply the results obtained in Section X. to several particular cases. 1. To find the equation to a plane which shall have a contact of the first order with a surface at a proposed point. Since the equation to a plane (which here takes the place of V=0, Art. 212) contains only three arbitrary constants, ‘a plane can in general have only a contact of the first order with a surface completely given. We have therefore s=Av+By+c, P=A, Q=B, from which there results between the constants A, B, c, and the co-ordinates x, y, s of the proposed point in the surface, the relations s =Axv+ By+c, A=p, B=q; subtracting therefore the value of x from that of x’, and putting for A and B their values, the equation to the plane having a contact of the first order with a given surface at a point vyz, the same as that to the tangent plane, is s—#= p(w —ax)+q(y'—y). 2. The equation to a sphere is (2 - a)’ + (y' - BY’ + (@ - 7) =, and since it contains only four arbitrary constants, a sphere cannot generally have a complete contact of the second order with a surface. Suppose it required only to have a contact of the first order ; 304 then ia SE, inne? dx s—y dy em—+ Hence there results between the constants a, 3, y, 6, and the co-ordinates wv, y, %, of the proposed point of the surface the following relations, (7 —a)*+(y — B)?+ (2 -y)P= oe, — G—-a y-p — P> —_— —___ = e-¥ s—¥ or w-a+tp(e-—y)=0, y-B+q(e- +) =9, which equations shew that the centers of all the spheres lie in the normal to the point of contact. q> r) Also (z- iF l+p+q = 0, ors— ee mt z Vit pre po qo and az=@&#-4+ SS 9 Bp=y+— ———.. ASE rig \/ Le rae Hence the number of spheres, which may have a simple contact with a surface at the same point, is unlimited; the radius of the sphere being assigned, the co-ordinates of its center are given by the above equations. 3. To find the radius of curvature of any normal section, at a given point of an oblate spheroid. (Art. 229.) Let /= 2 PGA, PG being a normal (fig. 67); and let the section PQ make an angle a with the meridian, and its radius of curvature = R; ; a a(l-eé ‘+, radius of curvature of meridian PO = p = eed : (1 — e’ sin?/)3 c a radius of curv. perp. to meridian PO’ = p’ LIP 1 — e’ sin® pp. a 1-é iz p sin’a + p cos’a oY Ay esin?] 1 ~ e? + e’cos’acos*] 4. To find the umbilici of an ellipsoid. These are points (Art. 226) at which, if the ellipsoid were 305 a generated by a circle of variable radius moving parallel to itself, the plane of the generating circle would become the tangent plane. Let M (fig. 67) be one of these points which is necessarily in the principal section containing the greatest and least axes, and let AZ7' be the tangent at MW; then c a—b ec av tanta / Oo F oe GPa = (Art. 104) of 7 = A c 9 6 if CN =a, since the equation to BPM is x = —\/a?— a’; a hence the co-ordinates of M, and of the three other umbilici similarly situated in the other quadrants of the principal section, are a — b amy e=+ta ‘ 2 0° z<=tec — C™ a a” — Also the radius of curvature at an umbilicus since it is the radius of curvature of the principal section cor- responding to these co-ordinates. 5. To determine the lines of curvature of an ellipsoid. 9 ~~ A Wik Let the equation be —+5 4 —-—=1; +3 dx y v ne yy poly dp az OY chi PB 'sF dp ne 2 dp a ate Ora y)3"8 dy ape OG, sep ty MY 2» ERIE Na ee therefore, by substitution in the equation of Art. 231, a(B-y) avy eS) 20 306 + {B(a-~7)w*—a(B-y)¥"-aB(a-B)} —" - Bla-y) ay =o, dy\* dy A sonrs 2 — Ay*® — B) — -: =O: eae ey) gre : De + ? tate) Sig Gk ea B(a-) Cary which will be both positive if we suppose the plane of pro- jection, that is, the plane of wy to contain the greatest and the mean axis, so that a, 8, y are in descending order of magnitude. calling > ; ; { d LU > To integrate this equation, let d =» UE being a v Yy function of v; x? 2 * vy (4 3 - 1) +5" (@*- dy'- B) =o, or Aa’ue—y’?+u(a’? —-Ay’® — B)=0, Bu o P(l = au (1 _ 1 OF i= oe y(1+ Au) =a?u(1+ dAu)— Bu, or y=2°u my du B du : tent d Hence differentiating 2y = 2UUu + x ain Guna 7? r ek Fe i 0, since 2 dy g OC e — = ; (1+ Auyf ’ Scar oa . oa : : of 2° . u =C, which gives y c (a a) the complete integral, C being the arbitrary constant. Also taking the other factor # = j , we have + Au y? =u (0° - #\/B) = 7 (@- VB) (VB -2), or (yA) + (@- VB) =0 307 the singular solution, which resolves itself into the two mir ee BP y =0, Pa aly Sora asn ale a — Cc and indicates the umbilici of the surface (Prob. IV.), the normals at which, as we know, are intersected by the normals drawn from consecutive points in every direction on the surface. 6. To construct the projections of the lines of curvature of an ellipsoid upon the plane containing the greatest and the mean axis. First, to determine the constant in the complete integral, let a, y’ be the given co-ordinates of the point through which the lines of curvature pass, A ey B Ayiiti Bi whol gh te Meaeee 10) oe Cais tases Sa Ay?+B— ax 1 Aa LI ee Li a oe a pag Ave + BE ah aday Ay” + B+x? ; and Lig hea sre 3G? (Ay? +B +a) — 4Ba”, Hence, we see that C has two real values one +, and the other —, and that 1 + AC is always +; hence the complete integral represents an ellipse when C is —, and a hyperbola when C is +; and therefore the projections of the lines of curvature passing through any point on the surface of an ellipsoid upon the principal plane containing the greatest and mean axes, are an ellipse and hyperbola. In the first case, if m and m be the semi-axes of the ellipse, is the equation by which m and m are connected; in the second case, if m and m be the semi-axes of the hyperbola, n* : B Win Ae Wee AC? Bt BS 20—2 308 Hence the semi-axes for each ellipse in the projection are the co-ordinates of a point in the same given hyperbola, and for each hyperbola the semi-axes are the co-ordinates of a point in the same given ellipse; these are called the auxiliary ellipse and hyperbola; they are concentric with the ellipsoid, and their semi-axes major and minor (which are the same for both curves, and coincide in direction with those of the ellipsoid) have the following values SJB =o oa ana M/E nent a Cc” a Hence we have the following construction for the pro- jection, upon the plane containing the greatest and mean axes, of the lines of curvature of an ellipsoid. Let ACB (fig. 68) be this plane (that of wy), and with the axes found above construct the auxiliary ellipse and hyperbola GO, HO. From any point J in the hyperbola, and 7 in the ellipse, let fall perpendiculars on the axes CA, CB; and with semi-axes CN, CM construct the ellipse JZN, this is the projection of a line of the first curvature; and in the same manner the projections of any number of lines of the first curvature may be constructed. Again with semi-axes mi, ni construct the hyperbola »X, this is the projection of a line of the second curvature; and similarly any number of projections of lines of the second curvature may be con- structed. As the points J and i approach O and G, the ellipse and hyperbola continually approach the lines CO, CB, and are at last confounded with them; therefore CO, CB are the pro- jections of lines of curvature, and consequently the curves in which the surface is cut by the principal planes of wz, yx are lines of curvature; also if in the equation to the auxiliary hyperbola we make m = a, we find n = b, therefore the ellipse AB is included in the construction, and is itself a line of curvature. Hence the intersections of the surface with its three principal planes are all lines of curvature. Since CO ~ Bb a= wih 72 , and is therefore “ -—y=0; = (=!) Ay (=) pee dx J but at an umbilicus when y= 0, a= £1/B this equation is decomposed into if wv dx is the only admissible value, which when integrated and cor- rected so as to pass through an umbilicus, gives y=0, and indicates the vertical ellipse which is projected into COA. d dy\? d an, 4 (=) +1=0; therefore = 0 8. The lines of curvature of an ellipsoid are the curves in which it is intersected by homofocal surfaces of the second order. Let the equations to a given ellipsoid and to another homofocal surface be 2 2 2 z 2 eh ap tis PMA Dee mg Ta ta Oe Gaolegad pam de) then the plane of wy being supposed to contain the greatest and least axes of the ellipsoid, the constants a”, b%, c”® are subject to the conditions 2 2yeee 2 Ae 2 PE Pak. ae er 2 Qian 12 a a baa." — b*, a —~c= a4? — 07, 1c bv c* —bee) 313 which amount only to two distinct equations, so that one of those constants is arbitrary. ‘The co-ordinates of all the points of intersection of (1) and (2) by Prob. VI., p. 237, satisfy the equation ae y” 3? eat ppt poe? which shews that some of the constants a”, 6”, c’® are negative; and from equations (3) it must be either 6” alone, or b” and c¢” together that are negative. Hence the curve of intersection of (1) and (4) which is the same thing as that of (1) and (2) has for its projection the equation @ ; art peaiom 2 e-—-B)=1, (5 on J (8-8) =1, ©) which represents an ellipse since 6” is always negative. Let X and Y denote the semi-axes of this ellipse, then 2 2 ; Bae eck =a? — b, a” b? so that the semi-axes of (5) are subject to the same condition as are the semi-axes of the projections of the lines of curvature of the ellipsoid by Prob. VII.; consequently the curves in which 2 2 2 o ~ ~ e ° e e v e ° 6 . the ellipsoid is intersected by — — 7 + —; = 1, in which a” is oe Yas arbitrary but 6” and c” subject to the conditions a” + 6” = a? — b’, a® —¢c® =a’ —c’, are its first lines of curvature. Similarly the 2 2 2 . ° . dh J] S$ second lines of curvature lie in the surface — — co oe 1; where a” is arbitrary, but 6” and c” subject to the conditions a’ 4b? =a?—0, a? +c* =a’ —c’. And equation (4) shews that those lines of curvature are likewise situated on a conical surface of the second order concentric with the ellipsoid. 314 9. To determine the radii of curvature of an ellipsoid. . of ies : Let the equation be — + B 4+ —=1,then taking the values of a w the differential coefficients from Ex. 5, we have ’ 2 2 2 ee) t= — x (~) (=) _# G4 pyt=- laa (2) (E) @- a}, ry) (x)' aaa oe appa acd A 2? a BY lc 9 ae ra «)° apt spate #+(2) (f) @- 99}, But in taking the sum of these three equations, the co- 2 efficient of (~) within the brackets is g a are S+5- (Sef) (C48) (.-£-8)-o off) (+ pi 2pqs+ + eee = Pa fetBry- Way} =o also rt — s? = (45) {(B-y)(a- 4) - wy} = : oe a“ y 1 md taped S (54 B +3) yee D being the distance of wy from the center, P the perpen- dicular from the center on the tangent plane, and 4?=a+6+ y3 therefore by substitution in the equation of Art. 233 and re- duction we have (4 @- x)? + tf - D*) (z’ — #) + + Be = 0. , But Rea~(/-) /ispaeg = -2=9%, z 315 A’ — D’ ary ag 3) ageieaie 2 = kt P R+ pe = 0 (1) Let R’, R” denote the values of # in this equation, then A2 — D? P ? abe\? - tore 2 , Reg R.R (=) h+ which is the simplest form to which the result can be reduced ; the first equation shews that at points where the tangent planes are equidistant from the center, the product of the radii is constant; and the second that at points which are themselves equidistant from the center, the sum of the radii varies inversely as the perpendicular on the tangent plane. Hence the volume of the ellipsoid may be expressed by the two radii of curvature at any point, and the perpendicular on the tangent plane, for it 4cabe _4 7pm ./PR SSS =— PS RR’. Also the radius of curvature at an umbilicus (since the two radii become equal to one another) b 2 2 Gb? pein (= +5) = —, as before. Pew iy ac The equation (1) for determining the two values of R may be presented under a different shape, leading to several im- portant results; and may be investigated by the following direct method that is equally applicable to all surfaces whose equations are of a particular form. 10. To find the two radii of curvature at any point of a surface whose equation is of the form f (#7) +) +x) = 0. In this case the quantities w’, v’, w’ vanish in the formula of Art. 217, and we have ‘ TA APE Ss 1 oie Pu+tmv +n'w = ous + V* + W? = P suppose, 316 which must be a maximum or minimum by the variations of l, m, m under the conditions (Art. 24) Pamin?=1, 1U+mV+nWe=0, (2) by virtue of which m and m may be considered as functions of 7; hence differentiating the above three equations with respect to 7, multiplying the two latter by indeterminate coefficients } and «, adding them to the former and then d dn making the coefficients of a and Pr vanish, we get for finding J, m, m the equations lu+«U +aAl =0 mv +«V+Am=0> which give P+rA=0; nmw+kW+rAn =0 =P o— PP 0 — P ‘ob pa, es U Ve W consequently from (2) we get the equation furnishing p, p’ the maximum and minimum value of & : 1 Seo (since P = aU + V* + W*), U° V? Ww? i ht gaa Oar eres a Te x’ y° 3° In the case of the ellipsoid — + = + —=1, this becomes ham ao eey x y° Prd BGeiRp) a PCB) ay CReaR gy a p being the perpendicular on the tangent plane at wyz. We shall next deduce some results both with regard to the radii of curvature and the lines of curvature of an ellipsoid, from applying the general equation to lines of curvature given in Cor. Art. 231, to that surface. . @& 317 11. To find the equations to the straight lines which touch the lines of curvature at any point of an ellipsoid. The equation of Cor. Art. 231 when applied to the ellipsoid 2 2 2 —+>+—=1 gives (considering each co-ordinate as a func- ¥4 a ‘ fer Gee tion of a new variable ¢ and making sates: &c.) y¥x(y—- Beta (a-y)y+ay(B-a)x=0, yy ee’ casera Now the equations to the tangent line at vy are (Art. 124) OY — ye Lee ap eee Se pas a’ y’ 2! , to which must be joined SETS as a therefore eliminating w’, y’, ’, the tangent lines to the lines of curvature at a point wy are determined by the intersections of the cone and plane (Y-y)(Z-32)(y—- B)vt+(X-2)(Z-2z)(a-y)y +(X-«)(Y-y) (6 -a)%=0, (¥-«)"4(¥-y) 54 (Z-s)==0, a B ye the cone having its vertex in the proposed point, and the plane being the tangent plane to the surface at that point. 12. The tangents to the lines of curvature at any point of an ellipsoid are parallel to the axes of the diametral section made by a plane parallel to the tangent plane at that point. OR diy? > 3? Detect B + — =1 be the equation to the surface, ryz a 5. tide sie. oC ele ZZ j the point in it, and —- +— ee the equation to the a cutting plane; then r= X* + Y?+Z? is to be maximum, 318 X, Y, Z being taken so as to satisfy each of these equations ; therefore proceeding as in Ex. 10, (a+A)X+Kv=0, (B+A)V+ny=0, (y+A)Z+nx=0, (1) and eliminating \ and « we get YZ(y—-P)v7+XZ(a-y)y+ XY (B-a)x=0; ; woaekw | oY: ZZ t ; : which with — + nes +— =0 determine two straight lines a 6 which are the axes of the diametral section; and these lines are evidently parallel to the tangents to the lines of curvature determined in the preceding Problem. Also for the magnitude of the axes we get from equations (1) 7+rA=0, < (a=) = B=) =ly-0)3 ERE coe it, Uolss y y a ee “a(a-r) B(B-1r) yy-n ” which determines ”. 13. The radius of curvature of any normal section of an ellipsoid = D’ +p, D being the semidiameter of the surface which is parallel to the tangent line of the normal section, and p the perpendicular from the center on the tangent plane. For comparing the above equation for finding the axes of the diametral section, with that for finding the radii of curvature in Problem X. we have #p =r, which shews that the radii of 2 2 curvature at any point of an ellipsoid are equal to aa and —, p p being the perpendicular on the tangent plane at that point, and 4A and B the semiaxes of the diametral section parallel to that plane. Also for the radius of curvature R of any normal section at the same point inclined at an angle @ to a principal section, we shall have (Art. 222) 1 (cos’@ sin’ @ p rae e* =) a 319 14. For different points taken along the same line of curvature of an ellipsoid, the product of the diameter of the surface that is parallel to the direction of the line of curvature at any point, and the perpendicular from the center on the tangent plane at the same point, is invariable. As in Problem XI., the equations to the lines of curva- ture are a (sy'— ys’) a + B (wa! — xa’) y+ y (ya’— wy’) x'=0, (1) ae yi ee 0. Ce Tas ame cy Hence eliminating by cross multiplication, and taking » to denote an arbitrary multiplier, we get a’ = B (#2 - Bees ie y (yx - vy) 5 = Boye (“Ss het =)- Bryw (= +e +), 2 2 2 or putting = + ie a = =u, and \=pn.aBy we get for a’, o res and similarly for y’ and 2’, x a old Baye dhs ore/’ par x v=—uU——U y¥=—-t -—-—4& s=—uw—-—UuU, 2 lu Pte a ad a eg aca (2) Hence multiplying these equations respectively by wv”, y”, x and adding, and supposing the arc to be independent variable EST so that a’ a” + y’y” + ’3" = 0, and observing that “v 320 or PD = C as in Prob. VII. p. 289. Hence all properties de- duced from this equation for the geodesic line of the ellipsoid, will hold also for the lines of curvature. Also this process leads to the integral of (1) and conse- quently to the determination of the lines of curvature. For equations (2) give , , : , VU ; u SU a 9 2Y mB en! 3 Of a uU+ ma uw+uld Ut ey hence multiplying by i ; Bi and "e , and adding and dividing a Y by w’ we get 2 2 ‘ x v ——_—_— + s Ee a’(w+ua) SP (wtup) xy (u+py) which by substituting for w its value may be reduced to the form x - 3 SEL Ses Ve EP a- ., -- es SS ee Paes ol Es + ee ee ataByn B+raByp ytaByu which shews that the intersections of confocal surfaces of the second order are lines of curvature, agreeably to Prob. VIII. =1, 15. The curvature of the shortest line through any point of an ellipsoid varies as the cube of the distance of the tangent plane at that point from the center of the surface. Since the osculating plane of the shortest line contains the normal to the surface, the radius of curvature of the shortest line will be the same as that of the normal section of the surface which passes through its tangent, and therefore will Woe Ge : equal Soe ae rs PD=C by Prob. VII. p. 289. In the case of a surface of revolution, the perpendicular on the tangent plane coincides with the perpendicular on the tangent line to the meridian, and consequently the radius of curvature of the a?b? meridian = —_, and the curvature of the shortest line is proportional to the curvature of the meridian through the same point. 321 16. If a’, a’, be the parameters of the lines of curvature that cross one another at any point of an ellipsoid (that is, the semi-major-axes of the confocal hyperboloids on which they are respectively situated) and @ and 4 7 — @, the angles which the geodesic line through the same point makes with the lines of curvature, then the value of a” sin®@ +a” cos’ @ is the same for every point of the geodesic line. It is shewn Prob. VIII. that the lines of curvature which cross one another at any point of an ellipsoid are the curves in which the ellipsoid is intersected by homofocal hyperboloids of one, and of two sheets, respectively. If therefore we take for the equation to the ellipsoid i, 2 oe at ow de ao: we may consider the two lines of curvature at any point to be determined by the hyperboloids whose equations are we y 2 “ y? “Sy eee g fone A a” Pea? f—a” = -a toa Le gre BFL 2 cig 7—* and a’, a’, are called the parameters of those lines of cur- vature. If we subtract each of these latter equations from the equation to the ellipsoid, we get ue y” Poa 2, /2 (2 par (pe a, + 73 2) (,,/2 ae (2), a”a (a -—b)(h -—a*) (a&-Cc)(a*-©&) a y = Es ata 6%) (62. 052) (abot) (Gia) © Hats) But between the co-ordinates wv, y, x of a point in the ellipsoid (1),.and 7° the square of a semi-axis of the diametral section drawn parallel to the tangent plane at that point, we have by Prob. XII. the relation P x’ y rf Gof) @-A@F-F) | @-A@oésr) which compared with (2) and (3) gives for 7° the values a’ —a’®, and a’ —a’™”. Now let P be the perpendicular from _ the center on the tangent plane at ays, and 4, B, D the semi- _ diameters of the diametral section, respectively parallel to the lines of curvature and the geodesic line passing through zyz ; then 21 a 322 AB? ‘ par vir 2. | rapes se = 4’ sin’ 0 + B’ cos’@ = (a — a”) sin’? 0 + (a — a”) cos’O (ABPY? or a’? — a” sin’ @ — a’” cos’*0 = (PD) = a constant, since ABP is proportional to the volume of the ellipsoid, and PD is constant by Prob. VII. p. 289. The result a” sin? @ + a” cos’@ = constant, which is the integral of the differential equation of the second order to the geodesic line on an ellipsoid, was first given by Jacobi; and the result PD = constant, as well as several other Problems here given, are due to Joachimsthal (see Crelle’s Journal, Vol. 26). 17. The tangents to the lines of curvature at any point of an ellipsoid bisect the angles between the tangent lines to the two circular sections through the same point. For if the diametral section be drawn relative to that point, the tangent lines to the circular sections are parallel to the equal diameters of the section, and the tangents to the lines of curvature to the axes of the section; and in an ellipse the equal diameters are equally inclined to either axis. 18. To find the radii of curvature at any point of a paraboloid. Pr 2 Let the equation to the surface be — + = + 2%= 0; then a it may be shewn, as in Example 9, that the maximum and minimum radii of curvature at a point wysx, are va Sula genbisnee " ape 2 a+b—2% y | a GB Jab (S48 41). Hence if the radii be bestia on the axis of x, the sum of the projections = a + b — 22. 19. In surfaces of the second order, the two curvatures are in the same direction at every point of the same surface, or they are in opposite directions at every point. 323 The curvatures are in the same or opposite directions at any point of a surface, according as r¢ — s’ for that point is positive or negative. But, substituting for 7, s, ¢ their values from 2 1 ee ey we find rt—s? = X, ; ; Camny 3.8 iry x apy a quantity which can never change its sign by the variation of 2, y, x, and will always have the same sign as ary; therefore for the ellipsoid and hyperboloid of two sheets, it will be positive, or the two curvatures at every point of the surface will be in the same direction ; and for the hyperboloid of one sheet it will be negative, or the two curvatures at every point will be in opposite directions, Similarly, for the elliptic and hyperbolic paraboloids the curvatures are respectively in the same, and opposite directions, at every point. 20. If # be the radius of absolute curvature at any point of a curve defined by the intersection of two surfaces u,=0, U,=0; and r,7,, be the radii of curvature of the sections of w, = 0, w,.=0 made by the tangent planes to U,=0, U,=0 respectively, at that point; it is required to express R in terms of r,7,, and @ the angle between the tangent planes. Let OP, O,P (fig. 54 bis) be the normals at P to the two surfaces, PV, PV, the radii of curvature of the sections of each surface made by the tangent plane to the other; then if VO. V,O, be perpendicular respectively to PV, PV,, by Meunier’s Theorem O, O, will be the centers of curvature of the normal sections; and if P# be perpendicular to OO,, R is the center of curvature of their intersection; .«. PR=R PV=*r PV,=r, OPO, =0, OPR = a suppose, O,PR = 6-a, PVcosa rcosa cosa. sin@ SLE RO 8 008, rear a ott EW R — ) cos(@—a) sin@ cos@cosa_ sin@sina cos@sin@_ sin@sina > = = qi —4+ ——_ = —_—___—_— ite v; R R r POA 1 1 2cos@ 1 SS = oo ke 7 Pr rv; r, 21. The locus of the focus of an ellipse rolling along a straight line is a curve such that if it revolve about that line, the sum of the curvatures of any two normal sections at right angles to each other will be the same at every point of the surface generated. Suppose the ellipse to touch the line ON along which it rolls in P; join SP and draw SN perpendicular to ON, then SP is a normal to the locus of S, and if ON=2, NS =y, ds and the arc of the curve described by S =s, we have SP=y—; faa tel Nie Ca aR or aiteknaan WP SN” ds yy eae cL aM Pai de ae Oe ds°’ ds y’ r y if r be the radius of curvature at S' to the locus of S; ts 2a (25+) =2 08g, ee ae Ses SP. %2-a which shews, since SP and r are the two radii of curvature at SS’ of the surface of revolution, that the sum of the curvatures of any two normal sections at right angles to one another is the same at every one of its points. 22. If # be the radius of curvature at a point whose distance from ‘the axis =p, of the shortest line traced on a surface of revolution; and if the radius of curvature of the meridian at that point = r, and the part of the normal inter- ‘ : 1 1 1 ] c cepted between it and the axis =m, then — = — + e _ | = ) Kier Tee e being the value of p where the shortest line cuts the meridian at right angles. 23. If a line of curvature be a plane curve, the tangent planes applied to the surface at its various points will all make the same angle with its plane. Jas! Necle teulp. 362 Strand . Hymeas Analytical Geometry Jos! Neele teudp. 352 Strand . a a Plate 3 Ja*Nede se. Burleigh Str. Strand Hymers Analy tical Geomeby Plate 3 Ja? Neele se Burleigh Str, Strand > rar bs S ne Sd E 3 5 % = S! 8 S Flate 4. ale sc. Burleiah Sta: Sand JatNve Plate &. Jos Neele teulp. 862 Strand. Liymers Analytical Geometry Plate 4. 56 bis Jost Neele teulp. 862 Strand. & 3 & J] f ¥ Hymers Analytical Geomary r y See Jost Neele fiulp, 852 Strand. = Sp wee Cambridge. LIST OF BOOKS PUBLISHED BY MACMILLAN AND CO. Axspy on Roman Civil Procedure Aischyli Eumenides. Drake. 8vo. cloth . Aristotle on the Vital Principle. Collier. Cn. 8vo. cl. . Breamont’s Story of Catherine. Fep. cloth Birks’ Difficulties of Belief. Crown 8vo. cloth . Bolton’s Evidences of Christianity Boole’s Philosophy of Logic. Preparing Differential Equations. Preparing ‘ Mathematical Analysis of Logic, sewed . Brave Words for Brave Soldiers and Sailors, per 100 Butler’s (Archer) Sermons. Ist Series, 8vo. cloth —— 2d Series, 8vo. cloth —— Lectures on Philosophy. 2 vols. 8vo. cl. . Letters on Romanism, 8vo. cloth CAMBRIDGE Theological Papers. Moor. 8vo. cloth. Cambridge Senate-House’ Problems and Riders, with Solutions :— 1848 to 1851.—Problems. Ferrers and Jackson 1848 to 1851.—Riders. Jameson S&S 2-6 ts OS Bri oa eS Oo © i SP IS ont pp 2 B&B @& So kG (OS Oe Zz LIST OF BOOKS PUBLISHED BY Cambridge Senate-House Propione and Riders, with Solutions :— 1854.—Problems and Riders. Walton and Mackenzie 1857.—Problems and Riders. Campion and Walton Cambridge Fitzwilliam Museum Hand-book, en. 8vo. sd. Cambridge Mathematical Journal. Vol. L., 8vo. cloth. Cambridge and Dublin Mathematical Journal. 9 vols. Campbell on the Atonement. 8vo. cloth Cicero on Old Age, Translated. 12mo. sd. Cicero on Friendship, Translated. 12mo. sd. Colenso’s Journal in Natal. Fep. cloth — Village Sermons. Fep. eluth . Ordination Sermons. 18mo. sd. Companion to Communion. 18mo. cloth — morocco ——: limp cloth Cooper’s Geometrical Conic Sections. Cn. 8vo. cl. . Cotton’s Sermons at Marlborough. Fep. cloth Crosse’s Analysis of Paley’s Evidences. 18mo. . Davies on St. Paul and Modern Thought. 8vo. Demosthenes de Corona. Drake. Cn. 8vo. cl. —— Translated by Norris. Cn. 8vo. cloth Drake’s Notes on Jonah and Hosea. 8vo. cloth Drew’s Geometrical Conic Sections. Cn. 8vo. cl. Evans’ Sonnets on Wellington, 8vo. sd. FARRAR on Classical Studies. 8vo. sd. 2 & 0 10 Oies Ove 0 18 7 -4 0 10 0 2 0.2 0 5 ee OPst Orage Was Use t er) Oe? 0. 5 Os aig 0 4 ee Ort Os Ge Ge Gs Cy 80 Fes FCs OI OO Gs, Ca Ce Con ery SS} Nees aes =n SS, ee MACMILLAN AND CO. Farrar on Truth making Free, sd. . : Flowers of the Forest. Cn. 8vo. sd. Frost’s First Three Sections of Newton. Cn. 8vo. GopFray on the Lunar Theory. 8vo. cloth Grant’s Plane Astronomy. 8vo. HALE on Transportation. Crown 8vo. sd. Hallifax’s Analysis of Civil Law. Geldart. 8vo. Hamilton on Truth and Error. Crown 8vo. cloth Hare’s (Archdeacon) Charges. 3 vols. 8vo. cloth . —— Charges for 1843-45-46.: 8vo. cloth —— Miscellaneous Pamphlets. 8vo. cloth Victory of Faith. 8vo. cloth . — Mission of the Comforter. 8vo. cloth — Vindication of Luther. 8vo. cloth . Contest with Rome. 8vo. cloth - Parish Sermons. 2d Series. 8vo. cloth Portions of the Psalms. 18mo. cloth Hardwick’s Middle Age Church History. Crown 8vo. History of the Reformation. Crown 8vo. —— Twenty Sermons. Crown 8vo. cloth en Christ and Other Masters :— Part I. Jntroduction. Part Il. The Religeons of India. Part IIT. The Religions of China. 8vo. cloth. each Hemmine’s Differential and Integral Calculus. 8vo.. Hervey on the Genealogies of our Lord. 8vo. cloth . Sermons on Inspiration. 8vo. cloth on University Lecturers, 8vo. sd. a Kae Se an HS SF CO SS LOS eee OS et i OS ao aS aS 10 Or io) 10 aa Co >) Oo OS-O A 2) Bm - SO 6) OO OF eer cS ‘ane (2p eth Ne ee 4 LIST OF BOOKS PUBLISHED BY Howard on the Book of Genesis. Crown 8vo. cloth . Exodus and Leviticus. Crown 8vo. — Numbers and Deuteronomy. Crown 8vo. Howes’ Chureh History of First Six Centuries . Humphreys’ Exercitationes Iambice. Fep. cloth IneteBy’s Outlines of Logic. Fep. eloth . JuWELL’S Apology. Translated by Russell. Fep. cd. Journal of Classical and Sacred Philology, Vols. I to Lites . , . , each Three Numbers. Published yearly, 4s. each. Justini Martyris Apologia Prima. Trollope. 8vo. . Juvenal, with Notes, by Mayor. Crown 8vo. cloth . KENNEDY on International Law. Crown 8vo. cloth . Kingsley’s Two Years Ago. 3 Vols. crown 8vo. cloth —— Westward Ho! Crown 8vo. cloth —— The Heroes. Crown 8vo. cloth, gilt leaves —— Glaucus. Fep. cloth, gilt leaves —— Alexandria. Crown 8vo. cloth —— Phaethon. Crown 8vo. . Law on the Fable of the Bees. Maurice. Fep. Lectures to Ladies on Practical Subjects. Crown 8vo. Lushington’s Poems. Fep. 8vo. cloth Mackenzie on the Influence of the Clergy Maclear on Incentives to Virtue 2 se 0 8 6 010 6G 0 5 6 0 3 6 Oy nt ou 012 6 0 7 010 6 URNA es 9 Le Live Or) Zeneg 0°76 > os 0 5 0 O- 2-90 0 4 6 Oa 3 0 O38, Goma Quy Leah MACMILLAN AND CO. Mansfield’s Paraguay, Brazil, and the Plate Mansfield on the Constitution of Salts. Preparing. M‘Coy’s Contributions to Paleontology. 8vo. cloth . Masson’s Essays on the English Poets. 8vo. cloth MauRIcE ON THE OLD TESTAMENT :-— The Patriarchs ana Lawgivers. Crown 8vo. cl. The Prophets and Kings. Crown 8vo. cloth —— on THE New TESTAMENT :— Gospels of St. Matthew, Mark, and Luke, and the Epistles of St. Paul, Peter, James and Jude. 8vo, cloth ‘ Gospel of St. John. Crown 8vo. cloth Epistles of St. John. Crown 8vo. cloth —— Theological Essays. Crown 8vo. cloth —— The Prayer-Book. Fep. cloth —— The Church a Family. Fep. cloth —— The Lord’s Prayer. Fep. 8vo. cloth —— The Sabbath Day. Fep. 8vo. cloth —— Christmas Day, and other Sermons. 8vo. cloth —— Doctrine of Sacrifice. Crown 8vo. cloth . —— Religions of the World. Fep. cloth — Kcclesiastical History. 8vo. cloth. . —— Learning and Working. Crown 8vo. cloth —— A Photograph Portrait . Mayor’s Lives of Ferrar. Fep. cloth Autobiography of Robinson. Fep. cloth . Juvenal. Crown 8vo. cloth Minucius Felix, translated by Lord Hailes. Fep. cl. oo Oe On On ONO £O8° Sono gO § oS SO 10 10 Soe Coe Cle Co pec BeO> BC: GECo gC>: FS -_-~ we 6 LIST OF BOOKS PUBLISHED BY Napikr on Bacon and Raleigh. Crown 8vo. cloth Nind’s Sonnets of Cambridge Life. —— Odes of Klopstock. Fep. cloth Norris’ Ten School Room Addresses. Norway and Sweden, Long Vacation Parkinson's Elementary Mechanics. Crown 8vo. cloth German Lyrist. Crown 8vo. cloth . Letters from Italy and Vienna. Fep. cloth 18mo. sewed . Ramble in Crown 8vo. cl. —— Elementary Optics. Crown 8vo. cloth Parminter’s English Grammar. Fecp Payn’s Poems. Fep. cloth . cloth Payne’s Decaeus and The Bond Child. 18mo, cloth . Peace in War. Crown 8vo. sewed Pearson’s Finite Differences, 8vo. Perowne’s Elementary Arabic Grammar. §8vo. cloth. Perry’s University Sermons. Crown 8vo. cloth Phear’s Elementary Mechanics. 8vo. cloth —— Elementary Hydrostatics. Crown 8vo. cloth Plain Rules on Registration. Crown 8vo. Per 100 . Plato’s Republic. Translated by Davies and Vaughan. Principles of Ethics. Crown 8vo. sewed . Procter on the Prayer-Book. Crown 8vo. cloth Puckle’s Conic Sections. Crown 8vo. cloth Ramsay’s Catechiser’s Manual. 18m o. cloth Reichel’s Sermons on the Lord’s Prayer. Crown 8vo, Robinson on Missions and the State. SaLLust, with Notes, by Merivale. Fep. 8vo. cloth. Crown 8vo. cloth ram at AN = EY om Re ne RE de a or pe ee ee aes aes Eee 0 0 0 0 0 A ognmnnrnrocdcoen ss key er ep) Mesh eee) ieee tee fee =) MACMILLAN AND CO. Selwyn’s Work of Christ in the World. Cn. 8vo. sd. Verbal Analysis of the Bible Simpson’s Epitome of Church History. Fep. cloth Smith’s (Alexander) City Poems. Fep. cloth. . Smith’s (Barnard) Arithmetic and Algebra. Cn. 8vo, Arithmetic for Schools. Crown 8vo. . Key to Arithmetic. Crown 8vo. Snowball’s Trigonometry. Crown 8vo. cloth. Introduction to Trigonometry. 8vo. sewed Cambridge Course of Natural Philosophy Swainson’s Hand-Book to Butler’s Analogy. Cn. 8vo. Tait and STEELE’s Dynamics. Crown 8vo. cloth. Taylor's Restoration of Belief. Crown 8vo. cloth THEOLOGICAL MANUALS :-— Hardwick’s Church History of the Middle Ages The Reformation ; A : J Howes’ Church History of the First Six Centuries . Procter on the Prayer-Book Westcott on the New Testament Canon Thring’s Construing Book. Fep. 8vo. cloth —— Elements of Grammar. 18mo. cloth —— Child’s Grammar. 18mo. cloth Thrupp’s Psalms and Hymns. 18mo. cloth limp cloth Antient Jerusalem. 8vo. cloth Thucydides, Book VI. Frost. 8vo. cloth Todhunter’s Differential Calculus. Crown 8vo. cloth . —— Integral Calculus. Crown 8vo, cloth So cofo. 6 S&S SFC OBA Se —) = os oe BS ee O 2 OLS ES laa az H Ci 4 = St Sr bee EO 10 12 = bS — bo 15 “I 10 10 for) Sse sy Oe ee OS i Ga C2: 9 8 BOOKS PUBLISHED BY MACMILLAN AND CO. Todhunter’s Analytical Statics. Crown 8vo. cloth —— Conic Sections. Crown 8vo. cloth . —— Algebra. Crown 8vo. Tom Brown’s School Days. Crown 8vo, cloth . Trench’s Synonyms of the New Testament. Fep. cl. —— Hulsean Lectures. Fep. cloth —— University Sermons. Fep. 8vo. cloth Vaughan’s Parish Sermons. Crown 8vo. cloth . Waters of Comfort. Fep. cloth Westcott on the New Testament Canon. Crown 8vo. —— Introduction to the Study of the Gospels. Westcott and Hort’s Greek Testament. Crown 8vo. cl. Wilson’s Five Gateways of Knowledge. Fep. cloth —— People’s Edition, ornamental stiff’ covers Wilson’s Dynamics. 8vo. boards Wright’s Hellenica. Fep. 8vo. cloth —— Help to Latin Grammar. Crown 8vo. cloth —— Seven Kings of Rome. Fep. cloth . —— Vocabulary and Exercises. Fep. cloth Rk. CLAY,’ PRINTER, BREAD BTREET HILL, LONDON, £ s. 0 10 0 10 0 10 0 5 0.5 0 2 0.5 0 4 012 0 2 0 1 0 9 0 3 Ogee 0 3 0 2 a QS Ba GB ep Sy ery eS ep SOS ; : . ( ie | a iM | ¢ 1 | , me : iv i Hovis y Rae ag 7” te WN eee . if m He ai 4 fe} Ni SEN A) mn ‘he ig 5 ty oe i" pe eee i , rar ul UAE eee ia ay i fog see ees f fia AL i VE i | . ’ 76 a eat an on ity ei ( Cui an een es i XY { ‘ ie e,/ behet ve ur .*) j ‘ wy : wegen | i oe f Pay ; ; ha , Y j pi ( my bt fu isi te ee teed ed . vie ( . ; ] i! 4 \ OMAR HACERA LHCD RR Senne §=—516.33H99T1848 Dra ete este sOA bast bier f Weer richeice deurnsedens yur leuskedsbeizdetsfybrieheats A TREATISE ON ANALYTIC rela Be Al My “ah ert shy ystiin Serra be Ca any ( : mhatiatts int “ir ADA tiles tah ite he pee mts Beir asnt an ; been chet se Fatah Fur hs 1 si ithe fh Ge iQ tied by kill? hime ds ( ier dda a ere a Ah rhs ieaes Carpe ced ‘7 DP ¥ M4 Se Pas " \ i t Ay j ( ’ vel Byun PAM vi 3 0112 Hy ' seit rk H i ‘ Ws Ps ane ue j hpi rue i fehl tebe tl i wide bet Artbed hehe A Ye i { ae et ea Ry Ry , Girt t Lf pe Le ayy! selieheie ts he | { A e WV {alts Wien | Weld i a HURL! is Pedegets i rs : A) ntyrye I vet vet et" L Te ee Wee htt Liha) ! iu Cee Pe aba depp \ yeh t Vey Mish uP i the |} ¥4 i ah bes 4 “Witch he re aN $ A vayiaiist iitiedie es bedeliit j PN hed Vasant a ect rete y Hy purl a Alb ie aa Nea alte A Joie tein bebe t pee “yet ie betvievelt reed Hs tte wold ed! fe : \ He jieily tir beeeged ) pave ea re a fe Gen Wig eds ihe vi ure ii! iva bate been ts (Fie i] beet ff ' Hee WEE MOIS Ie Be thelial Hi(PERE LED Mee pad ei a det Wane die het iebtetié } + petals te Pete dete eat Meus : Petcnht Welt ey hat 1 th “ rsies ap ye hide 18 Abele tava ‘ ih} : Dies aa et pyle it Payee ; eitocgeweeshinht ptaehet Peta pape leery ‘ b ybely pile : ir a BEAT AL Pe be He tee yet eee pM i Wh : ny untae fl Aga) ‘eh beh shit sin Puce 1 Wetint) Qepomedt- tev Milk We Ply heey Deg we el th iY He edith eyeye eee ben Watied ent! writs ete i i t Vata te We Ah { “line it (ih ih Neh ¥ “Ele Maly MOM At {) Heaps eis nah Ge thet hath | Tea ie thle Thetis Pi debe hehe: \| Ped tp heel i a lt Dont * PA a ot nae oO A CU vary Ny §) iite ped “et ie tlefi POMS MER de ftied eben eds eH et Ale edie She tie WAGER Be fle fie Gs atti 1A iat Rohertety Age we Avett rPfediay ths te i AP A” i ‘ Wee ites eh Seiten | jenn Pele tye de He pivede Wye Nr ba We Wed { eHEM i Al rY Uy Fabs (et Ve tha ths i [ VE tlette ry ete et y HY fehited ty ¢ " HM ia ATTA Thy Sha eae UME fivk fy DHA AS ee Di left Bete & Hetbegi-He\ Hie Heyy } ‘ Mi tesa dene he Mie Ginliepe ont et $ vninanany fi ( Web 64 aie Wey Media teMe beg ie {it a ae Me tvegir He a he hetisleedbe Getto g i aanies (st hehe “AP edi fb ied tiene ‘ ’ : ee Reel nw Depeliebed Pein U hia janice eee be i vele ie (tore! Hi ii hea fhe iinfarte (ee edger 4, sete pe tehh UA Teese ise Ke de ted cdiotis teil de he i) At i ana { ted WAN Me Be fi ite Ube Hittite he abcd Hef Dp ed ened Osher eeGee Wey ht Ht ines etitets Sth Meat fled epee hey ' i te lis \ a im fief tt eh ‘\ rit ye BiH} Wen M (A ¥ Aye a Gj Uy jotpeiey! M4 { vat uata| ibe hit ne an d u t Wedd dy eye preted! se fet tye fled #fls fi a aie fi hh Ph helatey Hal Py asia) iy AA " Ahem ah Aes ote Hehe ee i HO ik rete t f aa ee ia. hi Lak jietiaet Ness (EAE Ie Yb th Atte hehe wer tif itp fe [Cine edicte We ke W ey) A BeVe db Py hd GMD EYER HED ete 8A) Ue oh) et) Rv fe feeriedey { es hed Hi ) at Gira iehes Pe heb see Mog ay iby Me be tele eget cok ied Ha eon We, be vate Ce oan} rin aba it Wo mbral the abet Yad é ; } Nigh ‘ HN fre ple Ssh hee Pye ef HN Pelle het F Wd [Eble uy 4 To aawet é Hi ad Pea len ee BY Sebel Woh ble peseqe Ri td ehh ica ith Poien its f| yA Ae3 dani Hehofated, CRA Tes Oe W i } Jit hie am tia iy hen geal) penesie he has Vest aay Teen TL MR Vilernok | shy) l Pare eh ee ee ' J Heh fon fee h Bopha | wi gOS eae i wie ii jiates Hi¥) iM bt ta geiinde tout ae ui tone et } CSW Ay APR ALE Sd ah he Ge Babine ate th iy) (Oo tb. Beach eh (ted rt) 4 Nia! rae Heterdle Le tt uJ 1a t Hetiel ah f yay Nitra en | rfletlepede te hepatic i ht Cater A HE Le bebed hehe tebe poflbbie de” [eiRecwell pete the ieee the woh gpl ag (frre tegit Ue Pee AE Oiletio’ ba Dep ete J 6 ‘pee {Sit De hepewe sacl a a Wee tied bebe eet ( it ito isi eb wt f fet eh He ed tered etl iets aly ' +f i net ei (the i yi) Heh! ei? it vie ahh ib aie ted Cid es ead (a a 6 Pee uid Hh help eehte ahr he Neti he ijell Hopepbataet it dhe Rieke te be ied pricy She Getuned: thy tt ( Mifieht ie 1 a i PAL He jlr De t he pity eae f i} feditien ata ie eWele Hedin fesy Hehe hed if ave ¢ at HEME AMY edb i i GA LENS eA beep Levee a! oe LEASE elfemd may eM at af j as et wt v4 phe hag ibe Ooh Hopieedel Wed ott de tofle } ; i ¢ 4 Vege eu ] 4 dee ; on if, rai Pitan We { ; Jrebale bes Meni Aa oe ary 4 ie ava hee BORE WON Fgh TP Poa payed oy