OL/L ,e_ v_ THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES 'ick is DUE on the last date stamped below SOUTHERN BRANCH, iJNIVERSITY OF CALIFORNIA, LIBRARY, ILOS ANGELES. CALIF. 0. 1^ ^^ A><.^t ^ ELEMENTS OF QUATERNION^S. BY A. S. HAEDY, Ph.D., PROFESSOR OF MATHEJIATICS, DARTMOUTH COLLEGE. 45043 BOSTON: PUBLISHED BY GINN & COMPANY. 188 7. Entered according to Act of Congress, in the year 1881, by A. S. HARDY, in the office of the Librarian of Congress, at Washington. J. S. Gushing & Co., Printers, Boston. i Engineering & Mathematical Sciences • Library a A PREFACE. rpHE object of the following treatise is to exhibit the elementary principles and notation of the Quaternion Calculus, so as to meet the wants of beginners in the class-room. The Elements and Lectures of Sir William Rowan Hamilton, while they may be said to contain the ^ suggestion of all that will be done in the way of Quater- \ nion research and application, are not, for this reason, as also on account of their cliffuseness of style, suitable for the purposes of elementary instruction. Tait's work on Quaternions is also, in its originality and conciseness, beyond the time and needs of the beginner. In addition to the above, the foUowmg works have been consulted: 6^ Calcolo del Quaternione. Bellavitis; Modena, 1858. Exposition de la Metliode des Equipollences. Traduit de ritalien de Giusto Bellavitis, par C.-A. Laisant ; Paris, 1874. (Original memoir in the Memoirs of the Italian Society. 1854.) Theorie Elementaire des Quantites Complexes. J. Hoiiel; Paris, 1874. Essai sur une Mani^re de Representer les Quantites Imaginaires dans les Construction GSometriques. Par R. Argand ; Paris, 1806. Second edition, with j^reface IV PREFACE. l)y J. Iloiiel; Paris, 1874. Translated, with notes, from the French, by A. S. Hardy. Van Nostrand's Science Series, No. 52; 1881. Kurze Anleitung zum Rechnen mit den (^Hamilton' sclieri) Qunternionen. J. Odstrcil ; Halle, 1879. Applications Mecaniques du Calcul des Quaternions. Laisant; Paris, 1877. Introduction to Quaternions. KeUand and Tait; Lon- don, 1873. A free use has been made of the examples and exercises of the last work ; and, in Article 87, is given, by permis- sion, the substance of a paper from Volume I., page 379, American Journal of Mathematics, illustrating admirably the simplicity and brevity of the Quaternion method. If this presentation of the principles shall afford the undergraduate student a glimpse of this elegant and pow- erful instrument of analytical research, or lead him to follow their more extended api)lication in the works above cited, the aim of this treatise will have been accomplished. The author expresses his obligation to Mr. T. W. D. Worthen for valuable assistance in the prejDaration of this work, and to Mr. J. S. Gushing for whatever of typographical excellence it possesses. A. S. HARDY. Hanover, N.H., June 21, 1881. COJSTTEI^TS. CHAPTER I. Addition and Subtraction of Vectors ; or, Geometric Addition and Subtraction. Article. Page. 1. Definition of a vector. Effect of the minus sign before a vector . 1 2. Equal vectors 2 3. Unequal vectors. Vector addition 2 4. Vector addition, commutative 3 6. Vector addition, associative 3 6. Transposition of terms in a vector equation 4 7. Definition of a tensor 4 8. Definition of a scalar 5 9. Distributive law in the multiplication of vector by scalar quantities 6 10. If 2a + 2,3 = 0, then Sa = and 2,3 = 7 11. Examples 8 12. Complanar vectors. Condition of complanarity 15 13. Co-initial vectors. Condition of coUinearity 16 14. Examples 17 15. Expression for a medial vector 24 16. Expression for an angle-bisector 25 17. Examples 26 18. Mean point 28 19. Examples 28 20. Exercises 30 CPIAPTER II. Multiplication and Division of Vectors ; or, Geometric Multiplication and Division. 21. Elements of a quaternion 32 22. Equal quaternions 34 23. Positive rotation 35 n CONTENTS. Article. Page. 24. Analytical expression for a quaternion. Product and quo- tient of rectangular unit-vectors. Tensor and versor of a quaternion 36 25. Symbolic notation q = TqVq 39 26. Reciprocal of a quaternion 39 27. Quadrantal versors, i, j, k 40 28. Whole powers of unit vectors. Square of a unit vector is — 1, 41 29. Associative law in the multiplication of rectangular unit- vectors 42 30. Negative sign commutative with i, j, k 43 31. Commutative law not true for the products of i, j, k . . . . 43 32. Reciprocal of a unit vector 44 33. A unit vector commutative with its reciprocal. Reciprocal of any vector 45 34. Product and quotient of any two rectangular vectors .... 47 35. Square of any vector 47 36. Distributive law true of the products of i. j. k 48 37. Exercises 1 48 38. Symbolic notation, q = Sq + Yq 49 39. De Moivre's theorem 50 40. Products of two vectors. Symbolic notation 56 41. General principles and formulae 58 42. Powers of vectors and quaternions Gl 43. Relation between the vector and Cartesian determination of a point G4 44. Right, complanar. diplanar and collinear quaternions. Any two quaternions reducible to the forms -, 1 and -, 1 . . 65 /3 /? yd 45. Reciprocal of a vector, scalar and quaternion 66 46. Conjugate of a vector, scalar and quaternion 68 47. Opposite quaternions 70 48. Vl\.q = V- = — = KVq : . . 71 q Vq 49. Representation of versors by spherical arcs 71 60. Addition and subtraction of quaternions. K, S and V dis- tributive symbols. K commutative with S and V. T and U not distributive symbols 72 51. Multiplication of quaternions; not commutative. \jUq = U\]q. TTlq = X\Tq. K(/r= KrK'/. Product or quotient of com- planar quaternions 74 52. Distributive and associative laws in quaternion and vector multiplication 79 53. General formulae 82 CONTENTS. Til Article. Page. 54. Applications 82 55. rormulae relating to the products of two or more vectors . . 99 56. Exercises 106 57. Examples. Applications to Spherical Trigonometry .... 108 58. General Formulae 119 59. Applications to Plane Trigonometry 125 CHAPTER III. Applications to Loci. 60. General equations of a line and surfiice 131 61. Use of Cartesian forms in conjunction with quaternion rea- soning • . 132 62. Non-commutative law in quaternion diflferentiation. Differ- entiation of scalar fimctious 132 63. Quaternion differentiation 134 64. Illustratiou 13G 65. Distributive principle 138 66. Quadrinomial form 139 67. Examples 140 The Right Line. 68. Right line through the origin 145 69. Parallel lines 145 70. Right line through two given points 146 71. Perpendicular to a given line ; its length 147 72. Equations of a right line are linear and involve one indepen- dent variable scalar 150 The Plane. 73. Equation of a plane 151 74. Plane making equal angles with three given lines 152 75. Plane through three given points 153 76. Equations of a plane are linear and involve two independent variable scalars 154 77. Exercises and problems on the right line and plane 154 The Circle and Sphere. 78. Equations of the circle 164 79. Equations of the sphere 165 viii CONTENTS. Article. Pi(W. 80. Tangent line and plane 1(!G 81. Chords of contact 1G7 82. Exercises and problems on the circle and the sphere .... 167 83. Exercises in the transformation and interpretation of ele- meutaiy symbolic forms 176 The Conic Sections. Cartesian Forms. 84. The parabola 178 85. Tangent to the parabola , . . . 178 86. Examples on the parabola 180 87. Relations between three intersecting tangents to the para- bola 185 88. The ellipse 101 89. Examples on the ellipse 102 90. The hj'perbola 105 91. Examples on the h3'perbola .... 105 92. Linear equation in quaternions. Conjugate and self-conju- gate functions 100 93. General equations of the conic sections 201 94. The ellipse 204 95. Examples 206 96. The parabola 2U 97. Examples 216 98. The cycloid 222 99. Elementarj' applications to mechanics 223 100. Miscellaneous Examples 231 ELEMENTS OF QUATERNIONS. QUATERNIOI^S. CHAPTER I. Addition and Subtraction of Vectors, or Geometric Addition and Subtraction. 1. A Vector is the representative of transference throvgh a given distance in a given direction. Thus, if A, B are anj'- two points, vector ab implies a trans- lation from A to B. A vectd" ma}" be represented geometricall}' b}- a right line, whose length denotes the distance over which transference takes place, and whose dii-ection denotes the direction of the trans- ference. In thus designating a rector, the direction is indicated by the order of the letters. Thus, AB (Fig. 1) denotes transference ^^^- ^- from A to B, and ba from b to a. Retaining the algebraic signification of the signs + and — , if AB denotes motion from a to b, then — ab will denote motion from B to A, and AB=— BA, — AB = BA .... (1). Hence, the effect of a minus sign before a vector is to reverse its direction. The conception of a vector, therefore, implies that of its two elements, distance and direction; it was fii'st defined as a directed right line. It is now applied more generally to all quantities determined by magnitude and direction. Thus, force, the path QUATERNIONS. of a moving bod}', velocity, an electric cun-eut, etc., are vector quantities. Anal^'ticall}-, vectors arc represented by the letters of the Greek alphabet, a, p, y, etc. 2. It follows, from the definition of a vector, that all lines u'hich are equal and parallel may he represented by the same vec- tor symbol loith like or tinlike signs. If equal and drawn in the same direction, they will have the same sign. Hence an equality between two vectors implies equalit}' in dis- tance with the same direction. Thus, if AB (Fig. 2), cd, be, ef and iiG are equal and drawn in the same direction, thc^' ma}' be represented by the same vector symbol, and AB = CD = BE = EF = no = a . . . . (2) . 3. It follows also from the definition of a A'cctor that, if vec- tors are not parallel, the}- cannot be represented b}' the same vector symbol. Thus, if the point a (Fig. 3) move over the right line ab, from A to b, and then over the right line bc, from b to c, and AB = a, BC must be denoted by some other spubol, as /3. The result of these two succes- sive translations of the point a is the same as that of the single and direct translation AC=y, from a to c ; in either case a is found at the extremitj' of the diagonal of the parallelogram of which ab and bc are the sides. This combina- tion of successive translations is called addition, and is written in the ordinarv wav, , o /o\ a-f /i = y (3). This expression would be absurd if the s^-mbols denoted mag- nitudes only. It means that transference from a to b, followed GEOJIETEIC ADDITIOlSr AND SUBTEACTIOX. 3 bj transference from b to c, is equivalent to transference from A to c. The sign -|- does not therefore denote a numerical ad- dition, or the sign = an equality' between magnitudes. It is, however, called an equation, and read, as usual, "a plus fi is equal to y." This kind of addition is called geometric addition. 4. If the point a (Fig. 3), instead of moving over the sides AB, BC of the parallelogram abcd, had moved in succession over the other two sides, ad and dc, the result would still have been the same as that of the single translation over the diagonal ac. But since ab and bc are equal in length to dc and ad respect- ively, and are drawn in the same direction, we have (Art. 2) AB = DC and BC = ad, and if the first two translations are represented by ab and bc, the second two maj' be represented hj bc and ab, or a + /3 = /3 + a=y (4). Hence the ox>eration of vector addition is commutative^ or the sum of any number of given vectors is independent of their order. 5. If the point a (Fig. 4) move in succession over the three edges AB, bc, cg of a parallelopiped, Fijr. 4. we have and or AB -{- BC = AC, AC + CG = AG, (ab + BC) + CG = AG. In like manner BC + CG = BG, AB + BG = AG, Hence AB + (bc + cg) = AG. (aB + BC) 4- CG = AB + (bc + Cg) . . • (5) , and the operation of vector addition is. associative, or the sum of any number of given vectors is independent of the mode of grouping them. 4 QUATERNIONS. 6. Since, if AC = y (Fig. 3), then ca= — y, we have or, comparing with equation (3), a + /3 = y, a term may he transposed from one member to another in a vector equation by changing its sign. Also, in every triangle, any side may be considered as the sum or difference of the other two, depending upon their direc- tions as vectors. Thus (Fig. 3) y — /3=a, 7 - tt = /?. It is to be obser\'ed tliat no one direction is assumed as posi- tive, as in Cartesian Gcomotr}-. The only assumption is that opposite directions sliall liave opposite signs. ' The results must, of course, be interpreted in accordance with the primitive as- sumptions. Thus, had we assumed ba = a (Fig. 3) , y and ^ being as before, then p — « = y. a-l3=-y. 7. If two vectors having the same direction be added together, the sum will be a vector in the same direction. If the vectors be also equal in length, the length of the vector sum will be twice the length of either. If n vectors, of equal length and drawn in the same direction, be added together, the sum will be the product of one of these vectors by «, or a vector having the same direction and whose length is n times the common length. If then (Fig. 2) AF = xxn = XCV = Xa, where a, b and f are in the same straight line, cd = ab, and x is a positive whole number, x expresses the ratio of the lengths of AF and a. From the case in which x is an integer we pass, by the usual reasoning, to that in which it is fractional or in- commensurable. Vectors, then, in the same direction, have the same ratio as the corresponding lengths. GEOMETEIC ADDITION AND SUBTEACTION. 5 If AB = a be assumed as the uuit vector, then AF = ma, in which m is a positive numerical quantity and is called the Tensor. It is the ratio of the length of the vector ma to that of the unit vector a, or the numerical factor by which the unit vector is multiplied to produce the given vector. Any vector, as /?, may be written in general notation ^ = T/3U/3. In this notation, T^S (read "tensor of y8") is the numerical factor which stretches the unit vector so that it shall have the proper length ; hence its name, tensor. It is, strictly speaking, an abstract number without sign, but, to distinguish between it and the negative of algebra, it may be said to be always posi- tive. U/3 (read " versor of /?") is the unit vector having the direction of /3 ; the reason for the name versor will appear later. T and U are also general symbols of operation. Written be- fore an expression, they denote the operations of taking the tensor and versor, respectively. Thus, if the length of ji is n times that of the unit vector, T(/3) = n, where T denotes the operation of taking the stretching factor, i.e. the tensor. "While r(/?)=r/3 indicates the operation of taking the unit vector, that is, of reducing a vector jB to its unit of length without changing its direction. 8. If BC (Fig. 5) be any vector, and ba = ?/bc, then — BA = AB = — ?/BC ; yjg 5^ and, in general, if ba and bc be 5 c A any two real vectors, parallel and of unequal length, we may always conceive of a coefficient y which shaU satisfy the equation BA = ?/BC, 6 QUATERNIONS. where y is plus or minus, according as the vectors have the same or opposite directions, y may bo called the geometric quotient, and is a real number, plus or minus, expressing numerically the ratio of the vector lengths. This quotient of 2yaraUel vectors, which may be positive or negative, whole, fractional or incom- mensurable, but which is always real, is called a Scalar, because it may be always found by the actual comparison of the parallel vectors with a parallel right line as a scale. It is to be observed that tensors are pure numbers, or signless numbers, operating onl}- metrically on the lengths of the vectors of which the}' are coefficients : while scalars are sign-bearing numbers, or the reals of Algebra, and are combined with each other by the ordinar}' rules of Algebra ; they may be regarded as the product of tensors and the signs of direction. Thus, let a= aUa. ® Then Ta = a. If we increase the length of a b}' the factor &, 6 is a tensor, but the tensor of the resulting vector is ba. If we operate with — 6, — 6 is not a tensor, for a is not onl}' stretched but also reversed ; the tensor of the resulting vector is as before ha ; in other words, direction does not enter into the conception of a tensor. As the product of a sign and a tensor, —h in a scalar. The operation of taking the scalar terms of an expres- sion is indicated by the sj'mbol S. Thus, if c be an}' real alge- braic quantity, S ( — &a Va -\-c) = c, for — ba Ua is a vector, and the only scalar term in the expres- sion is c. 9. It is evident from Art. 7 that if a, &, c are scalar coeffi- cients, and tt any vector, we have {a + b + c) a= aa-\-ba + Ca . . . . (G). Furthermore, if (Fig. 6) OA = a, AB = /3, BC = y, Oa' = ma. GEOIHETRIC ADDITION AND SUBTKACTION. then, a'b' being drawn parallel to ab and b'c' to BC, a'b' = 7JI/3, b'c' = my. Now OC = a + ^ + y, Fig. 6. and oc' = moc = m (a -\- (3 + y) , But we have also oc' = oa' + a'b' + b'c' = ma + m/S + my. Hence 7n (a + /3 + y) = t^ioL + m/S + my . . . ( 7) , or the distributive laiv holds good for the multiplication of scalaY and vector quantities. 10. It is clear that while a — a = 0, a ± /3 cannot be zero, since no amount of transference in a direc- tion not parallel to a can affect a. Hence, if na + mft = 0, since a and y8 are entirel}' independent of each other, we must have na = and mfi = 0, or 91 = and m = 0. Or, if ma + n(3 = m'a-j- n' ^, then m = m' and n = n'. And, in general, if 2a + 2^ = 0, then [ . . . . (8). 2a=0 and 2/3 = QUATERNIONS. Three or more vectors ma}-, liowcvcr, neutralize eaeh other. Thus (Fig. 7) ^'^■''- a + ^ + y + 8=0, €-/3-a=0, ii^^-^O-^ Y and this whether abcd be plane or ^/\y^'' gauche. In any closed figure, there- _^ , fore, we have a + /3 + y + S+ =0, where a, (3, y, 8, , are the vector sides m order. 11. Examples. 1. The right lines joining the extremities of equal and jKtrallcl right lines are equal and parallel. Fig. 8. q ^^^ _^ Let OA and no (Fig. 8) be the given lines, and oa = a, ^^ J\^ /^y BO = yS, DA = y. Then, b}- condition, ud = a. Now, BA = BO + OA = ^ + a ; also, , , BA = BD + DA = a -|- y ; or, equating the values of ba, /3 + a = a + y. Hence (Art. 2), y = ^, and bo is parallel and equal to da. 2. The diagonals of a parallelogram bisect each other. In Fig. 8 we have bd = oa = OP + PA ; also BD = BP + PD ; .'. OP + PA = BP + PD. But, OP and pd being in the same right line, OP = ?«PD. Similarly " PA = »BP. GEOMETRIC ADDITION AND SUBTRACTION. Hence and mPD + nuF = PD + BP, 7ft = 1 , n= 1, OP = PD, BP = PA. 3. If tioo triangles, having an angle in each equal and the inchiding sides proportional, be joined at one angle so as to have their homologous sides parallel, the remaining sides will he in a straight line. Let (Fig. 9) AB = a, AE = j3. Then, by condition, do = xa, db = x^. Now Fif?. 9. CB = CD -|- DB = CC (/? — a) , But BE = ^ — a. Hence (Art. 2), b being a common point, cb and be are one and the same right Hne. 4. If tivo right lines join the alternate extremities of two parallels, the line joining their centers is half the difference of the parallels. We have (Fig. 10) AB = AD + DC + CB, and, also, AB = AE + EF + FB. Addino; Fig. 10. or, as lines. 2 AB = (ad + Ae) + (dC + Ef) + (CB + Fb) = EF — CD ; AB = ^ (eF — CD) , 10 QUATERinONS. 5. The meclials of a triangle meet in a point and trisect each other. ^'"■"" Let (Fig. 11) BO = a, CD = /3. Then OC = a, DA = (3. Now BA=2a+2/3=2(a + (3), and, since od = (a + /3). ba and od are parallel. Again BP + PA = BA = 2 OB = 2 (op + Pd) . But BP and pd, as also op and pa, lie in the same direction, and therefore BP = 2 PD and pa = 2 op. Hence the medials oa and db trisect each other. Draw cp and pe. Then and BP=2PD = fBD = |(2a + ^), CP=CB + BP=| (2a + y8)-2a = |- (/3 - a) , PE = PB + 15E = a + /? - f (2 a + /5) = i (/3 - a) , Hence pe and cp are in the same straight lino, or the medials meet in a point. 6. In any qnadrilateral, plane or gaitcJie, the bisectors of opposite sides bisect each other. "We will first find a value for op (Fig. 12) under the supposi- tion that p is the middle point of ge. We shall then find a value for OP, under the supposition that p is the middle point of fii. If these expressions prove to be identical, these middle points must coincide. In this, as in man}' other pi'oblems, the solution depends upon reaching the same point b}' difierent routes and comparing the results. Fisr. 1-2 GEOMETRIC ADDITION AND SUBTRACTION. 11 Let OA = a, OB = |S, OC = y. 1st. OC + CG=OE+EG. (a) But which, in (a), gives y + i(f3 — y)=ia + Ea. .-. EP = ^EG = i(y + ^ — a), OP = OE + EP = |a + i (y + y8 - a) = i(a + /3 + 7). (&) or 2d. FH — |-AB = FO + OA, rH-|-(/3-a) = -|y + a. .'. FP = |FH = :|:(a + ^ — y), OP = OF + FP = ly + i (a + ^ — y) = i(a + /3 + y), which is identical with (5). Hence, the middle points of fh and GE coincide. 7. If ABCD (Eig. 13) be any parallelogram, and op a^iy line parallel to dc, and the indicated lines he draiun, then will mn be p^ct^rallel to ad. Let AM = a, BM = fi. Then AO = ma, AD = na +_2)y3, OD = — ma + 7la +p>l^' "We have NM = NO + OM = NP + PM, in which NO =x{— ma + ?Za + pfi) , OM= (1 — m) a, NP = aj ( — ?«/? + ?la +pjB) , P3I = (1 —m)j3. 12 QUATERNIONS. Substituting in the above equation, we obtain, by Art. 10, 1 — 7n X = 111 Substituting this value in KM = NO -f CM, ( — ma + na +i>/5) + (1 — in) a Fig . 13. M T> ^ C A \/ 7\ B / L V^ N KM = 1 — m 1)1 1_ — m m {na +Pp) = AD. m Hence ad and nji are parallel. 8. If, through any jJOint in a 2^ciranelog7'am^ lines be drawn parallel to the sides, the diagonals of the two non-adjacent 2Kirallelograms so formed will intersect on the diagonal of the original parallelogram . Fig. 14. A I' and Let (Fig. 14) OA = a, on = /8. c Then oii = ma, oe = ??/?. AVc have RD=RO +OK -I- Er)= ??/34- ( 1 —m) a, ES =E0 + oK + us =ma+ i\ — n) /?. Also FO = FR + RO = CTRD + RO = o; [»/3 + ( 1 — ??? ) a] — ma, (a) FO = FE 4- EO = ?/ES 4- EO = ?/ [??la + (1 — 7i) /?] — n{3. (6) From (a) and (b) nx = y (l — n) — 71 and x{l—m) — m = ym. Eliminatinj? y X = 1 — ??l — M GEOMETRIC ADDITION AND SUBTEACTION, Substituting this value of x in (a) FO = [n/3 + (1 — in) a] — ma (/3 + a), 13 1 — m — 71 mn 1 — m or, FO and oc = (^ + a) are in the same straight line 9. 7/", in any triangle oab (Fig. 15), a line od he drawn to the middle point of as, and he produced to any point, as f, and the sides of the triangle be produced to meet af and bf in h and R, theii ivill HR be parallel to ab. Let OA = a, OB = /?. Then or = xa, OH = yj3, AB = (3 — a. Now o OD = OA + ^ AB = |- (a + /5) . Also, OF = z (a-f^), that is, some multiple of od. Then, 1st. BR=pBF, — f3 + Xa:=p ( — /? + Of) =p[-/? + ^(^ + /3)]; .-. x=2)z and —l=pz—p. Fis. 15. Eliminating z And, 2d. p = cc + 1. AH = gAF, -a + ?//?=g (-a + OF) = g[_a + 2;(a + /3)]; .-. y — qz and — 1 = g2 — g. Eliminating 2 From (a) and (6) g = 2/ + i. cc ?/ (a) (&) 14 QUATERNIONS. and, since j) = a' + 1 and q = i/ + 1, Fig. 15. a; = y and P = Q- •• KII = KO + OH = y/3 — Xa = X (/3 — a) = .r.vu, or, ini and ah are parallel. 10. If any line pr (Fig. IG) be draicn, cutting the tiro sides of any triangle abc, and he produced to meet the third side in q, then ^'S- 16. PC . BQ . KA = CR . AQ . liP. Let ijp = a, CR = (3. Then rc=pa, RA = rfi and ba = hc + ca = (1 +^>) a + (!+>•)/?• "We have AQ = .TBA = X [(1 + ;>) a + (1 + r) ft], as also AQ = AR + RQ = — rft + ?/I'R = — rft + ?/ (;)a + ft) . .-. X (1 +2)) = yp aiifi ^' (1 + '■) = - '■ + y- Eliminating y whence or a-= (1 + x)pr; AQ_BQ PC RA BA BA BP Cr' PC . BQ . RA = CR . AQ . BP. 11. If triangles are equiangular, the sides about the equal angles are proportional. Let (Fig. 17) 15C = a, c\ = ft. Then be = ?na, ed = nft, BD = ma + nft and nx — a + ft. Now BD = pBA, ma + nft = ]) {a + ft). "WTience m = p, n = j> and m .*. BE : BC : : ED : CA. GEOMETBIC ADDITION AND SUBTRACTION. 15 12. If, throvgh any point o (Fig. 17), xvithin a triangle abc, lines he draivn piarallel to the sides, then icill ^ + ^^ + ^ = 2. CA CB AB Let CA = /3, CB = a. Then ab = a — /?, ED = m/3, HI = j) (a — /5) and GF = na. We have CO = CG + GO = cii + HO. (a) Now, as Imes, ^^ <5A /-.NO = = n, .'. CG=: CA — GA= (1— 70 p. CB CA V / r EB ED CB CA~ ' DB _ DE AB AC ' .'. GO = CE = CB — EB = (1— m) a. .-. HO = AD = AB — DB — (1 — m) (a — (3). Substituting in (a) (1 _ ,,) /3 + (1 - on) a = pf3 + {l- m) (a - ^) , or (Art. 10) , , o 12. Complanar vectors are those which lie in, or parallel to, the same j^htne. If a, ^, y are any vectors in space, thej^ are complanar when equal vectors, drawn from a common origin, lie in the same plane. If a, y8, y are complanar, but not parallel, a triangle can al- ways be constructed, having its sides parallel to and some mul- tiple of a, /3, y, as aa, b/3, cy. If we go round the sides of the triangle in order, we Ijave aa + 5^ -f cy = 0. If a, /3, y are not complanar, conceive a plane parallel to two of them, as a and (3. In this plane two lines may be drawn parallel to and some multiple of a and f3, as aa and h/S ; and these two vectors may be represented by pS (Art. 3) . 16 QUATERNIONS. Now p8, being in the same plane with aa and b(3, cannot theivfore be equal to y, or to any multiple of it ; j?S and y can- not therefore (Art. 10) neutralize each other. Hence 2)8 + cy = aa + Z>/3 + cy cannot be zero. If, then, ice have the relation aa + b(S + cy = hetiveen non-parallel vectors, they are complanar; or, if a, (i, y be not complauar, and the above relation Ije true, then, also, a = 0, b = 0, c = 0. 13. Co-initial vectors are those ivhich denote transference from the same pioint. (a). If three co-initial vectors are complanar, and give the relations, . . , i o , a -> (a) aa + h(3 + cy = \ ,^. (6) a + b + c = J ^ ^' they xoill terminate in a straight line. For, let ox — a (Fig. 15) , ob = /3, od = y. Then da = a — y, BA = a — /?. From Equation (9), {b) (a + & -f c) a = 0, from which, subti'acting (a) of Equation (9), 6 (a - /?) + C (a - y) = 0, 6ba + CDA = ; and, since these two vectors neutralize each other, and have a common point, thej' are on the same straight hue. Hence, A, D and 15 arc in the same straight line. (6). Conversely, if a, f3, y are co-initial, complanar and ter- minate in the same straight line, and a, b, c have such values as to render aa -f 6^ -f- cy = 0, ^^'^'' ^'"^ a + 6 + c = 0. DA = a — y and BA = a — /?. GEOIVIETEIC ADDITION AND SUBTRACTION. But, by condition, 17 or in which 14. Examples. a — (3 = X (a — y) , {1 — x) a — /3 + Xy = 0, (1 — a-) — 1 + a; = 0. 1. The extremities of the adjacent sides of a parallelogram and the middle 'point of the diagonal between them lie in the same straight line. Fi- IS. Let OA Then a, OB = /3, OC = y. OD = OB + BD, 2y-/3-a = 0. But, also, 2 — 1 — 1 = hence, b, c and a are in the same straight line (Art. 13). 2. If two triangles, abc and smn (Fig. 19), are so situated that lines joining corresponding angles meet in a point, as o, then the pairs of corresponding sides x>roduced will meet in three points, p, Q, R, ivhich lie in the same straight line. Let OA = a, OB = /?, OC = y. Then os = ma, om = nji, ON = py, BA = a — /?, MS = ma — nfi, BR = a; (a — fS) and 1st. BM = BR — MR, or n{3 — I3 = x (a — (3) —y (ma - n(3) , .'. 7i — 1 = — cc + yn, X — my = 0. Eliminating y m {n — V) 18 QUATERNIONS. Also m (n — 1) OU = OB + BR - /? -h X {a - (i) - (i ^^^ _ ^^ (a -/?), whence „ (^^-1) (i-m {n -\) a OR = 7/6 — IL («) 2d. CN = CP — NP, or 2iy-y=v {jB-y)-w {n(3 - X)y). . p — 1 = — V + v:}), V — wn = 0. Eliminating w ,( ( ^, _ i ) V = Also n — jj 9? (p — 1) OP = OC + CP = y + r (/? - y) = y ^^ _ ^^ (/? - y), ^^^^^^ ^^_ j>(»-l)y-.(;>-l)/3 (^^ 3cl. In the same manner, we obtain -//i (/( — 1) a — p (?H — 1) y OQ = 2) — til (c) From (a) , (b) and (c) vrc observe that, clearing of fractions, and multiplying (o) by i> — 1, ('->) by ?/i — 1, (c) by 7i — 1, and adding the three resulting equations, member by member, the collected coefficients of a, (3, y, in the second member of the final equation, are separately' equal to zero. Hence the fu-st member OR (m — n) (2) — 1) + OP (n — p) (m — 1) + oq (p—m) (n — 1) = 0. But (m - ») (;)-!) + (n -p) (m - 1) + (P - m) (?i - 1) = 0. Hence, r, p and q are in the same straight line. ' GEOMETEIC ADDITION AND SUBTKACTION. 19 3. Given the relation aa -j- b(3 -\- Cy = 0. Then a, ft, y are eomplanar ; but, if co-initial (as they may be made to be, since a vector is not changed by motion parallel to itself, i.e. hy translation without rotation) , and a + '^" ' 6 + c is not zero, they do not terminate in a straight line. Hence, if o is the ori- gin, and A, B, c, their ter- minal points, A, B and c are not collinear. Let these points be joined, forming the triangle abc (Fig. 20), and OA, OB, oc prolonged to meet the sides in a', b^ c! To find the relation between the segments of the sides, let whence OA'=a'=ri'a, ob'=/3'= ?//?, oc'=y'=2!y, «' p /8' y' a=-, p = -, 7=-- Substituting these in succession in the given relation, -a'+6^-f-Cy = 0, aa + -B'+ Cy = 0, y whence, since a,' c, b are to be collinear, _ + 6 + c = 0, X 20 QUATERNIONS. and, for a like reason, h a-\ 1- c = 0, y a + i + - = 0. z Whence « „ ^ -,. p h + c a + c a + b and h-Y-c a + c a + h or, from the given relation, ,_ h(i + Cy „,_ Cy 4- Ct« ,_ «a + hji °-~ h + c ' ^~ c + a ' ■^"~ a + 6 ' "Whence 6 (a'-^)=C (y-a'), C (^'-y)=a(a-^'), a(y'-a)=H/3-7'), and ba' _ c cb' _ a Ac' _ 6 a'c "" V b'a ~~ c' c'b a' or, multiplying, ba' . cb' . Ac' . = a'c . b'a . c'b. 4. If o (Fig. 20) he any pointy and abc any triangle, the transversals through o and the vertices divide the sides into seg- ments having the relation ba' . cb' . ac' . = a'c . b'a . c'b. Let a'c = a, bc = aa, cb'= (3, ca = h(3. Then ba = aa + h/S. Also let BO = XBbJ OA = ^a'a, BC'= ?JIBA, CC'= ZCO. GEOMETRIC ADDITION AND STJBTEACTION. 21 Then BO = xbb' = X (bc + cb') = X (aa +/S) , OA = 2/a'a = y (a'c + ca) =?/(a + 6/5), Bc'= mBA = m (aa + b/3) , cc'= zco = z (cB -f- bo) = z [— aa + x (aa + /5)] . From the triangle boa we have bo + OA + AB = 0, X (aa -\- f3) + 7j (a + bfS) — bj3 — aa = 0. .'. xa-\-y — a=0, x -\-yb — b = 0. Eliminating y From the triang-le bcc' 6(1- a) 1 — ba bc + cc'+ c'b = 0, aa-i-z [— aa-{-x (aa + ;8)] — m (aa + b(3) = 0, whence, as usual, and substituting the above value of x, . 6(1— a) 1 — a l — ')n = z — z—^ — ; — -, m = z- or l-6a ' 1 — m 1—6 1— 6a m 1— a Substituting for m, 6 and a, C A_ AB' CA' Bc'"~ b'c a'b' which is the required relation. 5. If (Fig. 20) U7ies be draion through a' •&', c' and produced to meet the opposite sides of the triangle in f, q, R, then are p, q and r col- linear. Fig. 20. 22 QUATERNIONS. "With the notation of the last example, Bc' = mux = — — — -— (cia + bB) . a-\-b — 2 1st. From the triangle c'ba' c'a'= c'b + B\' a + b a + b -2 -(aa + &^) + («-!) a ~l(b-2)a-b(3l Also a'r = a;c'A'= a'c + cr = a'c — yf3, a -I + ^ 1; A ih-2)a-bf3:\ = a-yf3, a + b — 2 b y = b-2' '''"'^ BR = BC + CR = aa - -A_ R, (a) b-2'^ ^ ^ 2cl. From the triangle c'ab' c'b'= c'a + ab' = (l-7n)(aa + 6i3) + (l-5)/3 b-\ Also [«a-(«-2)/3]. a + b — 2 b'q = xc'b'= b'c + cq = b'c + 2/a, x-^-^=l-{_aa-{a-2)(i-] = -li + ya, a + b — 2 a .-. y = a-2' and 3d. BQ = BC + CQ = (a +y)a = — i —^a. {0) Ct "~~ ^ a'p = .rA'B'= x(a + f3), a'p = a'b + v.r = {l — a)a + y{aa + bfi) , n-1 a — b GEOMETRIC ADDITION AND SUBTRACTION. 23 and BP = ?/BA = (aa + bB) , a — b (c) Multipl^-ing the second members of (a), (h), (c), by (a—l) (& — 2), — (a — 2) (6 — 1), (a — b) respective!}', theii- sum is zero. Hence (a - 1) (5 - 2) BR - (a - 2) (& - 1) bq + (a - 6) bp = 0. But (a - 1) (6 - 2) - (a - 2) (6 -1) + (a - &) = 0. Hence r, q and p are collinear. 6. If FC (Fig. 20) and po be produced to meet aa' and bc, tJien T CDirt s are collinear ivith cl A similar proposition would obtain for q and r. With the following notation, we have BA = a, ba'= /?, bb'= aa + b^f BO = BA + Ab'+ b'o = Ba'+ a'o, a + 6/3 - (1 - a) a + x{aa + b,8) = /3 + y (a - /3). a ■•• y ^^Jyp + aa also a + b ' Fig. 20. BP = ba'+ a'p = BA + ap ; /3 + 2[aa+(/j — l)^]=a + Wa, a — \ + b 1-6 ' a a and 1-6' BC = BA'+ A'C = BA + AC, /3 + ^J/3 = a + 16 [(1 - a) a - 6/3], 24 QUATERNIONS, I- a BC = \-a Now to find BS, Bc' and bt, we have Fig. 20. 1st. BS = x'ba' = i;r + y'vo, . .., h 1 -26- 1 - a BS bft 1- -■2b- a 2d. BC' •. v'ba = v'— ■■ BC 4- a «co, 2 a + b- 1' V.C ,/ aa 2a-\-b-l 3d. BT = BA'+ a't = Ba'-|- z'a'o = BP -f- iv'pc, I n + b a — b bB - aa bt = -t! b — a Clearing of fractious and adding (1 - 2 6 - o) BS + (2 o -I- & - 1 ) Bc'+ (6 - a) bt = 0, (\-2b-a) + (2a-\-b-l) + (h -a) = 0. Hence s, c' and t are collinear. as also 15, A medial vector is one drawn from the origin of two co- initial vectors to the middle point of the line joining their extremities. GEOMETRIC ADDITION AXD SUBTRACTION. 'ZO Thus (Fig. 21), if p is the middle point of ab, op is a medial vector. To find an expression for it, let oa = a, ob = yS, then or, adding, OP = OA + AP = a -f- AP, OP = OB + BP =/3 — AP, 0P = a + /3 (10), The signs in this expression will, of coijrse, depend upon the original assumptions. Thus, if ao = a, OP = — a + AP = /3 — AP, op = ' 2 16. An Angle-Bisector is a line which bisects an angle. To find an expression for an angle-bi- sector as a vector, let oe = a (Fig. 21) and of = ^ be unit vectors along oa and OB. Complete the rhombus oedf. Since tlie diagonal of a rhombus bisects the angle, od is a multiple of op. Now od = a + (3, hence OP = x(a-\-(3) (11). In this expression op is of an}' length and x is indeterminate. If op is limited, as by the line ab, then AP = x{a + 13) — aa, AP = l/AB = ?/ (&^ — aa) , x{a + /3) — aa = y{b/3 — aa) , (a) or Eliminating x X — a = — ya and x = yb. a y = - a Substituting in (a) a + b a-\-b (12). 26 QUATERNIONS. 17. Examples. 1. If 2'>o,'rcdMograms, whose sides are 2'>ciTallel to tico given lines, be described upon each of the sides of a triangle as diago- nals, the other diagonals icill intersect in a j^oint. Fitr. 22. Let ABC (Fig. 22) be the given tri- angle. Let the diagonals u'f and a'd intersect in p, and suppose oe to meet a'd in some point as p! Let OA = a, ob'= /?, whence oa'= ?na, OB = n^. Now b'p — DP = a. But And d'p = ?/b'q = y • i (b'c + b'b) = ^y[ma + (n-l)/3]. DP= ZDll = Z .^ (dC + C.v') = i/[(m-l)a-^]. («) (Art, 15) Substituting in (a) , we obtain, as usual, Again But 2 (\-n) z— — i '- — \-\-')>in — n op'— dp'= a + (3. 0P'= .TOG = X .^ (OA + Ob) = ^.r (a + »/3). (h) Substituting in (h) this value of op' and dp'= i'DII, we obtain as before, 2 (1— u) v = 1 + mti Or, ron = zdii = dp'= dp. Hence, p and p' coincide, and the three diagonals meet in a point. 2. A triangle can alwaj/s be constructed lohose sides are equal and parallel to the medials of any triangle. GEOMETEIC ADDITION" AND SUBTRACTION. 27 In Fig. 23 we have aa'= ab + ba'= ab + I^BC. bb'= bc + ^CA. CC'= CA +iAB. .*. aa'+ bb'+cc'= f (ab + BC + ca) = 0. (Art. ■ 3. The angle-bisectors of a triangle meet in a point. Let a, yff, y be unit vectors along bc, j,. 23 AC, AB (Fig. 23). Then (Art. 16) AP = X (y + ^) , BP = ?/ (a - y) . 10), Now (a) BC = AC — AB, aa =hfi — Cy (&) where a, 6, c are the lengths of the sides. Substituting a from (6) in (a) 'h(i - cy We have also cp = AP — AC = a; (y + ^) — b(3, fb(S -cy \ CP = BP + CB = ?/ ( ~ y 1 -I- cy _ /j^. Eliminating y Substituting in (c) CP = c6 a + & + c cb a + b -\-c _ 5 a + ?> + c ^ 6 a + 6 4- c = i^(a + /S) (y + ^)-6/? [cy_(a + 6)/3] ( — aa — ttyS) (c) Hence (Art. 16) cp is an angle-bisector. 28 QUATERNIONS. 18. The Mean Point of any jwlygon is that to ichich the vector is the mean of the vectors to the angles. Hence, to find the mean point, add the vectors to the angles and divide l\y the number of the angles. Thus, if oj, a,. 03 .... be the vectors to the angles, the vector to the mean point is a, 4- ao+ 03 + .... +a„ (13), where n is the number of the angles. The mean i)oint of a polyedron is similarl}- defined. It co- incides in either case, as will ajipoar later, with the center of gravity of a system of equal particles situated at the vertices of the polygon or polyedron. 19. Examples. 1 . The mean point of a tetraedron is the mean point of the tetraedron formed by joining the mean points of the faces. Let (Fig. 24) oa = a, ob = /3, oc = y. The vcctore from o to the mean points of the faces are Ha + iS + y), H^^ + y). hiy + P). and that to the mean point of the tetraedron formed b}- joining them is ' a + ft + y a + (S a + y 7 + ft' 3 "^ 3 "^ 3 "^ 3 H« + /3 + y), which is the vector to the mean point of gabc. The same is true of the tetraedron formed by joining the mean pouits of the edges ab, bo and ca with o, since 'a+/3 13 + y g + y" 2 2 2 = H« + ^ + 7)- GEOMETRIC ADDITION AND SUBTRACTION. 29 The above is, of course, independent of the origin, and would be true were o not talien at one of the vertices. 2. The intersection of the bisectors of the sides of a quadri- lateral is the mean point. Let (Fig. 25) oa = a, or = /?, oc = y, 0D = 8, OR = p. Then (Art. 15) o p = 1 (of + oe) = i[i('x + S)+Hy + /5)] = i(a + /8+y + 3). If o is at A, then oa = a = 0, and P = H/? + 7 + S). 3. If the sides (in order) of a quadrilateral be divided propor- tionately, and a neiv quadrilatercd formed by joining the points of division, then will both quadrilaterals have the same mean point. Let a, /?, y, 8 be the vectors to the vertices of the given quadrilatex'al, from any initial point o. Then, for the vector to the mean point, we have i(a + /3 + y + 8). If m be the given ratio, and a', ^', y' 8' the vectors to the ver- tices of the second quadrilateral, then a'= a-j-m (/3 — a) = (l— m) a + m/3, /3'=(l-m)/? + my, y'= (1 — m)y -\- mS, 8' = a + (1-771) (8 - a) = 8 - m (8 - a) ; whence i (/8'+ y'+ S'-f a') = Ha 4- /5 + y 4- 8). 30 QUATERNIONS. 4. In any quadrilateral^ plane or gauche, the middle point of the bisector of the diagonals is the mean point. Let (Fig. 26) oa = a, ob = /9, oc = y, os = ^y. ^'^•2«- Then (Art. 15) OP = \ (OQ + os) 5. If the tico opposite sides of a quadrilateral be divided pro- portionately, and the points of division joined, the mean i)oints of the three quadrilaterals ivill lie in the same straight line. Let c' a' (Fig. 27) be the points of division, and m the given ratio- Then, if OA = a, HC = y, Oa'= Wla, c'c = my, AB = /3 and o is the in- itial point, the vectors to the mean points p, p' p" are OP =i(3a + 2^ + y), OP' =^[(r,i + 2)a + 2/3 + (2-m)y], op"=i[(m + 3)a + 2/3+(l-m)y]; , I — m . s ••• pp = — r—(7-«)^ 4 p p = 4 (y - ") • Therefore, pJ p',' p are in the same straight line. 20. Exercises. 1 . The diagonals of a parallelopiped bisect each other. 2. In Fig. 58, show that bg and cii are parallel. 3. If the adjacent sides of a quadrilateral be divided propotx tionately, the line joining the points of division is parallel to the diagonal joining their extremities. GEOMETRIC ADDITION AND SUBTRACTION. 31 4. The medial to the base of an isosceles triangle is an angle- bisector. 5. In any right-angled triangle abc (Fig. 58), the lines bk, CF, AL meet in a point. ' 6. Any angle-bisector of a triangle divides the opposite side into segments proportional to the other two sides. ^ 7. The line joining the middle point of the side of any paral- lelogram with one of its opposite angles, and the diagonal which it intersects, trisect each other. - 8. If the middle points of the sides of any quadrilateral be joined in succession, the resulting figiu'e wUl be a parallelogram with the same mean point. 9. The intersections of the bisectors of the exterior angles of any triangle with the opposite sides are in the same straight line. 10. If AB be the common base of two triangles whose vertices are c and d, and lines be drawn from any point e of the base parallel to ad and AC intersecting bd and bc iu f and g, then is FG parallel to dg. CHAPTER II. Multiplication and Division of Vectors, or Geometric Multipli- cation and Division. 21. Elements of a Quaternion. The quotient of tico vectors is called (i Quaternion. "We arc now to see -what is meant l)y the quotient of two vectors, and what are its elements. Let a and /?' (Fig. 28) be two vec- tors drawn from o and o' respectively and not lying in the same plane ; and let their quotient be designated in the usual wa}' b^' — ^. \ "Whatever their relative positions, we o'' T ^'' m^*^}' alwaj'S conceive that one of these vectors, as /3', ma^' be moved parallel to itself so that the point o' shall move over the line o'o to o. The vectors will then lie in the same plane. Since neither the length or du'ection of /3' has been changed during this parallel motion, we have (^ = /3', and the quotient of an}' two vectors, a, P', will be the same as that of two equal co-initial vectors, as a and B. "We are then to determine the ratio — , in which a and yS lie in the same plane and have a conomon origin o. "V^Hiatever the nature of this quotient, we are to regard it as some factor which operating on the divisor produces the dividend^ i.e. causes ji to coincide with a in direction and length, so that if this quotient be q, we shall have, by definition, gB = a when —=q (1^)' 32 ^^ /? GEOMETRIC MULTIPLICATION AND DIVISION. 33 If at the poiut o' we suppose a vector o'c = y to be drawn, not parallel to the plane aob, and that this vector be moved as before, so that o' falls at o, the plane which, after this motion, y will determine with a, will differ from the plane of a and jj, so that if the quotient q and q' will differ because their planes differ. Hence we con- clude that the quotients q and q' cannot be the same if a, ^ and y are not parallel to one plane, and therefore that the position of the plane of a and yS must enter into our conception of the quotient q. Again, if y be a vector o'c, parallel to the plane aob, but differing as a vector from (i', then when moved, as before, into the plane aob, it will make with a an angle other than boa. Hence the angle between a and y8 must also enter into our con- ception of q. This is not onl}- true as regards the magnitude of the angle, but also its di recti on. If, for example, y have such a direction that, when moved into the plane aob, it lies on the other side of a, so that aoc on the left of a is equal to aob, then the quotient g' of -, in operating on y to produce a must turn y . y in a du'ection opposite to that in which q=- turns /3 to produce a. Therefore q and q' will differ unless the angles between the vector dividend and divisor are in each the same, both as regards magnitude and direction of rotation. Of the two angles through which one vector may be turned so as to coincide with the other is meant the lesser, and it will therefore, generally, be < 180? Finally, if the lengths of ^ and y differ, then - = q will still differ from - = g! Therefore the ratio of the lengths of the vec- y tors must also enter into the conception of q. We have thus found the quotient g, regarded as an operator which changes (3 into a, to depend upon, the plane of the vectors, the angle between them and the ratio of their len2;ths. Since 34 QUATERNIONS. two angles are requisite to fix a plane, it is evident that q depends upon four elements, and performs two distinct opera- tions : 1st. A stretching (or shortening) of /3, so as to make it of the same length as a ; 2d. A turning of /3, so as to cause it to coincide with a in direction, the order of these two operations being a matter of indiffer- ence. Of the four elements, tlio turning operation depends upon three ; two angles to fix the plane of rotation, and one angle to fix the amount of rotation in that Fig. 28. plane. The stretcliing operation de- pends onl3' upon the remaining one, I.e., upon the ratio of the vector lengths. As depending upon four 'T *" " elements we observe one reason for \ calling q a (juaternion. The two ope- ,, " "^1 — 15' rations of which q is the symbol being entirely independent of each other, a quaternion is a complex qnantiti/, decomposable, as will be seen, into two factors, one of which stretches or shortens the vector divisor so that its length shall equal that of the vector dividend, and is a signless number called the Tensor of the quaternion ; the other turns the vector divisor so that it shall coincide with the vector dividend, and is therefore called the Versor of the quaternion. These factors are symbolicalh' repre- sented bj- Tq and Vq, read " tensor of q" and "versor of q" and q ma}' be written q = Tq . Tq. 22. An equality bettoeen tico quaternions may be defined di- rectly from the foregoing considerations. If the plane of a and (3 be moA-ed parallel to itself; or if the angle aob (Fig. "28), remaining constant in magnitude and esti- mated in the same direction, be rotated about an axis tlu'ough o perpendicular to the plane ; or the absolute lengths of a and y3 GEOMETKIC ISrULTIPLICATION AND DmSION. 35 vaiy so that their ratio remains constant, q will remain the same. Hence if , a T a' I - = g and — -= q, then will when 1st. Tlie vector lengths are in the same ratio, and 2d. The vectors are in the same or parallel planes, and 3d. The vectors make with each other the same angle both as to magnitude and direction. The plane of the v,ectors and the angle between them are called, respectively, the plane and angle of the quaternion, and the expression -, a geometric fraction or quotient. It is to be observed that q has been regarded as the opei'ator on /?, produc- ing a. This must be constantl}' borne in mind, for it will sub- sequentl}' appear that if we write q/3 = a to express the operation b}' which q converts /3 into a, q(3 and I3q will not in general be equal. 23. Since q, in operating upon /5 to produce a, must not only turn /3 through a definite angle but also in a definite direction, some convention defining positive and negative rotation with reference to an axis is necessary. B}' x)ositive rotation with reference to an axis is meant left- handed rotation when the direction of the axis is from the plane of rotation towards the eye of a person who stands on the axis facing the plane of rotation. [If the direction of the axis is regarded as from the eye towards the plane of rotation, positive rotation is righthanded. Thus, in facing the dial of a watch, the motion of the hands is positive rotation relatively to an axis from the q\q towards the dial. For an axis pointing from the dial to the e^-e, the motion of the hands is negative rotation. Or again, the rotation of the earth from west to east is negative relative to an axis from north to south, but positive relative to an axis from south to north.] On the aljove assumption, if a person stand on the axis, fac- ing the positive direction of rotation, the positive direction of 36 QUATERNIONS. Fig. 31 (bis). J the axis will always be from the place where he stands towards the left. K ?, A-, j (Fig. 31) be three axes at right angles to each other, with directions as indicated in the figure, then positive rotation is from i to ^*, from j to k, and from k to i, rela- tiveh' to the axes A', ?', j respcctiveh'. A prcciseh" opposite assumption would be equally proper. The above is in accordance with the usual method of estimating positive angles in Trigo- nometry and Mechanics. 24. Let OA and on (Fig. 29) be any two co-initial vectors whose lengths are a and 6, a and /? being unit vectors along oa and oa, so that OA = aa, OB = &/3. Let the angle boa between the vectors be represented by ^ ; also draw AT) perpendicular to ob, and let the unit vector along da be 8. /£ ~ The tensor of on is evidently a cos (^ and that of da a sin . If we assume that, as in Algebra, geometrical quotients which have a common divisor are added and subtracted by adding and subtracting the numerators over the common denominator, so that then, since we have OA OB OA = OD + DA, CD + DA OB a cos . /? a /'cos <^ h _ OD DA ~ OB OB + P + a sin <^ . S sin . S\ ^ GEOMETRIC MULTIPLICATION AND DIVISION. 37 We liaA'e already defined (Art. 8) the quotient of two parallel vectors as a scalar, and in the first term of the parenthesis, jB beino; a unit vector, " = 1 , and ox a f , , . , 8\ . . — = 7 cos<^ + sm is the versor of q, its plane, deter- mined b}- the axis e, and angle <{> being the plane and angle of the quaternion. "When a and (3 are of the same length, or Ta=T/?, Tq= — = 1, and the effect of g as a factor, or operator, is simply one of version. Like T, the s^-mbol TJ is one of operation, indicating the oper- ation of taking the versor, so that Ug = cos . This operation takes into account but one of the two distinct acts which we have seen the quotient q must perform, as an agent converting /8 into a, nameh*, the act of version ; it thus eliminates the quantitative element of length. In this respect it is similar to the reduction of a a'cc- tor to its unit of length, an oi)cra- tion which also eliminates this same element of length, and has been designated by the same sj-mbol U. "When a and /3 are at right angles / to each other, ) = g (Eq. 15) in the symloolic notation P /3 ^l (16), or q = Tq, Vq and say that the quotient of two vectors is the product of a tensor and a versor; and that 1st. The tensor of the quotient, ( — ), is the ratio of their tensors; ^ ^-^ 2d. The versor of the quotient, (cos ^ + e sin ^) , is the cosine of the contained angle 2')lus the 2Woduct of its sine and a unit vector, at right angles to their plane and such that the rotation which causes the divisor to coincide in direction ivith the dividend shall be positive. 26. If, for ^ = g, we write '- = q[ it is evident that g' differs (i a- from q both in the act of tension and ver- sion ; tlie tensor of q' being the reciprocal of the tensor of g, and the unit vector e, while still parallel to its former position, is reversed in direction (Art. 23) since the direction of rotation is reversed (Fig. 30) . Hence ^ = ^(cos<^-esin0) .... (17). a Ta '1 is called the recijyrocal of -. As already remarked, the a jB positive direction of € is a matter of choice. It is only neces- sary that if we have -f- e in U-, we must have — e in U _, or conversely. 40 QUATERNIONS. Fig. 31. J 27. Let ?, J, k (Fig. 31) represent unit vectors at right angles to eiich other. The eliect of any unit vector acting as a multi- plier upon another at right angles to it, has l)een defined (Art. 24) to be the turning of the latter in a positive direc- tion in a plane perijendicular to the ope- rator or multiplier through an angle of 90? Thus, / operating on ,/ produces k. J. This operation is called multijjlica- tion, and the result the product, and is expressed as usual V= ^ (18). The quotient of tn^o vectors being a factor which converts the divisor into the dividend, we have also k^ . J (10), either the product or quotient of two unit vectors at right angles to each other being a unit vector 2wr pen dicular to their i^lane. This multiplication is evidently not that of algebra ; it is a revolution, which for rectangular vectors extends thi'ough 90? Nor is k in Equation (18) a numerical product, nor i in Equa- tion (19) a numerical quotient. This kind of multiplication and division is called geometric. In accordance with the above definition we may write the fol- lowing equations (20). ij = k k_ . J jk = i I ki=J i=k i ji = -k -k . kj = — i -''-k GEOMETBIC MULTIPLICATION AND DIVISION. 41 ^•(-0 = -i ^ = ^ 1- . . . . (20). Jc(-j) = i Since the effect of i, Jc, J as operators is to turn a line from one direction into another wlaich differs from it by 90° they are called quadrantal versors. -J = i -A: y -./ -k = i — j = k — i i = k -J — i -k =J k =j — i 28. Since i Xj = k and i x k = — j = —1 X j, we have iXiXj=—lXJ, or i X I = — 1 . "We may denote the continued use of i as an operator by an exponent which indicates the number of times it is so used. This is consistent with the meaning of an exponent in algebraic notation. In both cases it denotes the number of times the operator is used, in one instance as a numerical factor, in the other as a versor. Thus .0 ui /'^ , jjm = j-i^^ _=-, etc. JJ r In confonnity to this notation the above equation becomes i'=-l (21), 42 QUATERNIONS, aud in a similar manner, (22). Ti'r. 31. J Hence the square of a unit vector is — I. The meaning of tlie word " square " is more general than that which it possesses in Algebra, as was that of tlie word " product" in Art. 27. The propriet}' of this ex- tension of meaning lies in the fact that for certain special cases, the processes above defined reduce to the usual alge- braic processes to which these tenns were originally' restricted. The condu- ^^ sion i- = — l is seen to follow directly from the definition, since if i operates twice in succession on either ± j or ± k, it turns the vector, in either case suc- cessively' through two right angles, so tliat after the operation it points in the opposite direction. A similar reversal would have resulted if the minus sign had been written before the vector. Thus — {±j) = T j- Hence i x i, or i-, as an operator, has the effect of the minus sign in revers- ing the direction of a line. 29. It is to be observed that so long as the cyclical order i, j, A', t, J, A;, j, .... is maintained, the product of any two of these three vectors gives the third ; thus and therefore as also ij=k, jk=i, ki = j; {ij)k = kk = h~=-l, (jk)i= a = r = — 1, {^OJ = JJ = f = - 1 ; i(jk)= a = r-= — 1, k{ij) =AA- = /r=-l, GEOMETRIC MULTIPLICATION AND DIVISION. 43 hence iU^c) = {iJ)k, which involves the Associative laiv. "We may therefore omit the parentheses and write yk^JJd = kij=-i (23), or, the continued product of three rectangidar tinit vectors is the same so long as the cyclical order is maintained. But kUi)=k{-l-) = -l'=l .... (24), or, a change in the cyclical order reverses the sign of the loroduct. 30. In Equation (24) we have assumed that A-(-A') = -K-. That this is the case appears from the fact that i operating on — / produces — A:, or i{-j) = -k, and that the same result would be obtained b}' operating with i on J, producing yfc, and then reversing k. That is, to turn the negative, or reverse, of a vector through a right angle, is the same as turning the vector through a right angle and then re- versing it. Tlie negative sign is, therefore, commutcdive tvith i, ;, k, or ^■(-i) = -0' = -^' (25). 31. It follows du-ectl}' from the definition of multiplication, as applied to rectangular unit vectors, that the commutative prop- erty of algebraic factors does not hold good. For y = ^, but Ji = -k. 44 QUATERNIONS. Hence, to cJiange the order of the factors is to reverse the sign of the 2)7'oduct. The operator is always "oi-ittcn first; and, since the order cannot be changed without affecting the result, in reading such an expression as ij = 7c, this sequence of the factors must be indicated by saying " i into j equals 7c" and not " i multiplied by j equals 7i," the latter not being true. Hence also the conception of a quo- tient as a factor requires a similar dis- tinction, which in Algebra is unncces- Fi;:. 31. ji = k is not true. sary. In the latter, from - = a we h have, indifferentl}', ah = c and ha = c. But from - = i, while iJ = k is true, J In expressing therefore the relations be- tween i, j and k by multiplication instead of division, care must be taken to conform to the definition, the quotient being used as the multiplier or operator on the divisor. This non-com- mutative propcrt}' of rectangular unit vectors, which results directl}' from the primary definition of the operation of multipli- cation, will be seen hereafter to extend to vectors in general and to quaternions, whose multiplication is not commutative except in special cases. The quotient then being a factor which operates on the divisor to produce the dividend, we have ^;./=A-, that is, y^=k .... (26), the cancelling being performed by an upward right-handed stroke. But'^^ = 7i' is not time, for this would involve Jt= ij. 32. It follows also that the directions of rotation of a fraction. as -, and its reciprocal are opposite. J Thus -= • k (27), GEOMETRIC MULTIPLICATION AND DIVISION. 45 and therefore that the reciprocal of the quotient i is — ^■, or \--i (28); I that is, the recij)rocal-of a unit vector is the vector reversed. Tliis maj' be written \^i-'^-i ...... (29), the exponent denoting that, as a factor or versor, i is used once, wliile the minus sign before the exponent indicates a reversal in the direction of rotation. 33. If a be any unit A'ector, we obtain from the preceding Ai'ticle a — = a( — a) But hence 1 1 -a:^a — (30), or, a unit vector and its reciprocal are commutative and their product plus unity. If a is not a unit vector, a = TttUa, -=-^ = -— Ua (31), the tensor of the reciproccd of a vector being the reciprocal of its tensor. k It must be care full}' observed that a fraction, as -, cannot be 1 1 *' written indifferenth^ k- or -/c, for this would involve ki~^ = i~^k, i i which is not true. 46 QUATERNIONS. B}' definition A' (—?') = — /. or /./"' = k- = —/ = -. Hence, k 1 ' i ' i - = ]c- or ki \ From the meaning attached to the ordinary' I i notation of algebra, .A- A- . i i (a) would appear to be coiTect ; for, cancelling, we have A; = A\ k 1 "Whereas, since - must be written A;-, we should have i i or iki-^ [= - ikq = A-r' / [= A-] jt[=-A-]=A;, which is not true. Of course that equation (a) is false is directl3' evident from the fact that -7= —j, and (a) involves i ( — ./) = ( —j) i or ij =ji. The above, however, shows that, as cancelling must be performed by an upward right-handed stroke when the exi)ressiou is in the form of a quotient or fraction, so when ex- pressed in the form of multiplication, the cancelled factors must be adjacent. In such an expression as .; = -./'■" V" =iy = -i it might be supposed permissible to write also (&) (c) since in either case the correct result is obtained. This arises, however, from the fact that both the fractions in the first mem- ber of (6) are equal to A\ and therefore may be permuted so as to read kk = ^^yi^ = ^^ = — 1 . The process of (c) is, how- GEOMETRIC MULTIPLICATIOISr AND DIVISION, 47 ever, illegitimate, and the result is correct, not because the process is so, but because the factors are in this case commu- tative. 34. Since the act of tension is independent of that of version, and their order is immaterial, xi . yj = xy . ij = yx . ij = zk ... (32) , where x and y are any two scalars and xy = z. Hence the com- mutative principle applies to tensors. If then a, f3, y are in the direction of i, j and k respectively, and a, &, c are theu' tensors, a/3 = TaT/? . ij = ab . k, ay = TaTy . ik = — ac , j, etc., or, the product of any two rectangular vectors is the product of their tensors and a unit vector at right angles to their plane. So also a _ Ta . I _ Ta £ _ _ <^' 7. a Ta . i a . , -J, etc., y Ty . k c or, the quotient of two rectangular vectors is the quotient of their tensors times a unit vector at right angles to their plane. 35. If, as above, a = ai, then aa = ai . ai, a' = cv i", a^ = -a- (33). Hence, tlie, square of any vector is minus the square of its tensor. Since Ta = a is the ratio of the lengths of a and TJa, the square of any vector is the square of the corresponding line, regarded as a length or distance only, with its sign changed. If ai = a and bi = (3, a/3 = abi- = — ab. 48 QUATERNIONS. 36, That the multiplication of rectangular vectors is a dis- tributive operation maj' be seen directl}' from Fig. 32 by ob- '^ servina: that Fig. 3-2. iU + l-)=iJ + ik (34), i being perpentlicular to and in front of the plane of the paper. 37. Exercises in the transformations of ?*, j, k 1. J(-0-- = k. 2. Ji-k) = 3. k(-j)-- = i. 4. k(-i) = 5. -kU)-- = i. G. (-k)i = 7. (-J)k- = — i. 8. (-J)(-k) = 9. (-./)(- -0 = •10. (-0(-i) = 11. J __ k. 12. i 13. -A- 14. A- 15. J -k JC. 17. i 18. 19. 20. k _ 21. k _ 22. 1 A-_ 23. ijk_ kji 24. i j k _ k ' j ' i 25. Is it correct to write, in general, the product of anj* frac- tions, as - . -, in the form — ? ./ J _ JJ 26. State whether ^^— . - = — - is correct or not, and why. k i ki 27. i^fk' = -{ijJcy. GEOIMETRIC MULTIPLICATION AND DIVISION. 49 38. Resuming Equation (15), a Ta / , , • , \ q—~ = — {cos^ dz = — vdtjj. (a) GEOMETRIC JNIULTIPLICATIOX AND DIVISION. 51 Multiplj'ing (a) b^- V— 1, and adding the result to (6), dz + dv . V^= {—v+z -s/^l) d<)>, dz + dv . V^ = {v V^^+ z) V^ d(l>, whence d{z + v^/-l) /— - ——=-= dcj> . V- 1, (c) z + v V — 1 which may be writteH "' 2 + t; V— 1= e'>^~'; Ol- eosa + sua (/). V—l= e*^^^, (cZ) whence cos HI (/) + sin m^ . V — 1 = e'"''^ ^~^. (e) But we have from (d) (cos (ji + sin (^ . V^ ) "» = €'»'> ^~i, (/) and therefore, from (e) and (/), (cos + sin 7?i<^ . V— 1 (37), which is the well-known formula of De ]Moi\Te. This formula may be made the basis of a s^'stem of analytical trigonometry. Thus, for example, to deduce the formulae for the sine and cosine of the sum of two angles, we have from (f?) cos cji + sin cj)V —I Ji-Z^i cos 6 + sin (9 V — 1 = &■ Multipl^-ing member b}- member, cos <^ cos + cos <^ sin ^ . V — 1 + cos ^ sin ^ . V— 1 — sinsin^ = e^^ + ^^^^- (S') But from De Moivre's formula cos ((/. + ^) + sin {cj> + e)\/^l= 6^'^ + ^^^'- ('0 52 QUATERNIONS, Equating the first mombcrs of (f/) and (h), since in any equa- tion between real and imaginary (quantities these are separately equal in the two members, we have cos [0 -\- cf)) = cos 6 cos <^ — sin ^ sin ^. sin {6 + ^) = sin cos <^ + cos 6 sin ^. These formulae, while they ma^- be of course demonstrated independently of De Moivre's formula, are here deduced from imaginary- expi'cssions. It would therefore appear that these expressions admit of a logical interpretation. If any positive quantity 7)i be multiplied bj- (V— 1)- the re- sult \s — m. That is, in accordance with the geometrical inter- pretation of the minus sign, we ma^* regard the above factor ( V— 1)" as having turned the linear representative of m about the origin through an angle of 180? If, instead of multiplying m by ( V — 1 ) ", we multipl}- it by V— 1, we ma}' infer from analogy that the line m has been turned through an angle of 90° about the origin. If, too, we ob- serve that each of the four expres- sions Fi!?. 33. }' -<•* "t v^ \ x'i -w 'II, 1-^' 1 \ \ ^ m, m V-1, -1 — ?u V— 1 is obtained from the preceding by multi])hing b}' the factor V — 1 , they may be regarded as denoting in order a distance m on the co-ordi- nate axes OX, OY, OX,' OY' (Fig. 33), V— 1 being, as a factor, a versor turning a line left-handed through a quadrant. These expressions therefore locate a point on the axes, both as to dis- tance and direction from the origin. Since ever}- imaginar}- expression can be reduced to the form ± a ±b V — 1, we ma}', in accordance with the above interpre- tation of V — 1, regard such an expression as defining the posi- tion of a point, out of the axes. Thus oa = a (Fig. 34) and GEOMETRIC MULTIPLICATION AND DIVISION. 53 AP = &, laid off at a at right angles to oa since h is multiplied b}' V — 1 ; so that in passing over oa and ap in succession we reach the point p. It is also evident that such an expression Implicitl}' fixes the position of p b}' polar co-ordinates, since Va- -f b"^ = op and tan poa = - . In like manner a — 6 + a V — 1 would locate a point p^ oa' having a length = a, but laid off perpendicular to oa, since V — 1 is a factor, and a'p'= — h. As before, we have implicitl}^ op'= Va^-f-&" and , , a tan p OA = — . T^'l Fig '.34. / o A Furthermore, if we operate on the first expression, a+6V — 1, which fixes the point p, with V— 1, we obtain the second, — 6 + aV — 1, or V— 1 as a factor turns op thi'ough 90° so as to make it coincide with opI As an operator, therefore, we may regard V— 1, like i, J, Zj, as a quadrantal versor, tmniing a line through a quadrant in a positive direction. Algebraically it denotes an impossible operation. (In Algebra quantities are laid off on the same line in two opposite directions, + and — . It was because quan- tities are so estimated onl}- in Algebra that Su- W. Hamilton called it the Science of Pure Time, since time can be estimated onl}- into the future or the past.) But it is unreal or imaginary only in an algebraic sense. If the restrictions imposed by Al- gebra are removed, b}' enlarging our idea of quantit}' and at the same time modifj'ing the operations to which it is subjected, this imaginary character disappears. In applying the old nomen- clature to these new modifications, it will be seen that the prin- ciple of permanence is observed, i.e., the new meaning of terms is an extension of the old ; and when the new complex quantities reduce to those of Algebra, the new operations become identical with the old. If now we operate upon a-i-6 V — 1, 54 QUATERNIONS. which, if we regard a = oa (Fig. 35) and ^V — 1= ap as vectors, is equivalent to or, with the Fig. 35. expression cos<^ + sin sin <^ + V — 1 (a sin 4> + h cosff)). Draw OX' so that X'OX=cf>; also pa" and ai. perpendicular and as par- allel to ox: Then a cos — b sin <^ = ol — a"l = oa" a sin <^ + 6 cos (f> = la 4- sp = a"p. -'X' Make oa'= oa" and lav off a'p'= a"p perpendicular to OX, since it has V — 1 as a factor ; then (a cos <^ — & sin <^) + V — 1 (a sin + b cos (f>) = o\'+ a'p'= op,' and p'op = . But the formulae for passing from a set of rectangular axes OX, Y, to another rectangular set OX', Y', are X = x'cos , y = y'coscf> — x'sincf>, in which A''OX'= , .i; = oa, y = xv, x'= o\',' y'=vx',' or OA = OK + KA, AP = NP — a"k, a"k being perpendicular and a"x parallel to OX. Hence the effect of the operator has been to turn op left- handed through an angle , which is equivalent to turning the axes rio-ht-handed thi-ousrh the same angle. GEOMETRIC MULTIPLICATION AND DIVISION. 55 + 1,-1 and V — 1 are particular cases of the general versor cos ^ + sin <^ . V — 1 , nameh', when cf> is 0° 180° and 90° respectively, + 1 preserv- ing, — 1 reversing and V— 1 semi-inverting the line operated upon. "We may now see the meaning of De Moivre's formula (coscf) -f sinc^ . V— 1)™ = cosmcf) + sin mcfi . V — 1. As operators, the first member turns a line through an angle ^ successively m times, while the second member turns it through m times this angle once, producing the same result. The expressions cos . (3, DA = c (sin (fi . 13) = sin . €/5, 6 being a unit vector perpendicular to the plane aob, as in the figin-e. Hence a = cos (f> . (3 -{- sin (f> • «/?• (^0 Now when a and (3 are luiit vectors, we have bj' definition ^ . f3 = (cos 4>-\- (. sin (f>)(3 = a ; or, comparing with (a) , (cos ^ + csiu 4>)^ = cos . a) . cos . a — sin ^ . ca = yS, or, by the distributive principle, (cos (^ — sin <^ . e) a = (3. GEOirETEIC ISrULTIPLICATIOX AND DIYISIOX. 57 Using the two memlDers of this equation as multipliers on the corresponding members of («) (cos ^ — sin (^ . e) aa = /? (cos (^ . ^ + siu <^ . e/?) , or, since a-= — 1, — cos <;?) + € sin ^ = /5a (38). If a and /? are not unit vectors, /Sa = T/3Ta(-cos<^4-esin<^) . . . (39). Operating with each member of (a) on yS, ap = (cos <^ . /3 + sin (^ . e/3)y3 = cos^ . /3"+ sin<^ . e/3' = — cos^ — esin<^ (40), or, if a and /3 are not unit vectors, a^=TaT;S(— COS(/) — esinc^) . . . (41). Tlie product of any tivo vectors is, therefore, a quaternion, ■which, as before, ma3' be regarded either as the sum of a scalar and a vector or the product of a tensor and a versor. In gen- eral notation aft = Saf3 +Yal3 = Hq +Tq .... (42), al3 = Tq.Vq (43). TJie scalar of the product [Sa/? = — TaT^S cos ^] is the product of the tensors and the cosine of the supplement of the contained angle. Tlie vector of the prodixt [Tay8 = — TaT/3 sin . c] has for its tensor [TVa/5= TaT/3 sin ] the product of the tensors and the sine of the contained angle, and for a versor [tJVa/5 = — c] a unit vector at right angles to their plane such that rotation about it as an axis is positive or left-handed. 68 QUATERNIONS. Representing the tensors of a and fi hy a and 6, we have, as in Art. 38, from Equation (41), Fig. 36. Tqz=ab Vq — — cos (^ — € sin , and consequently a change in the direction of rotation. Hence UY^a=€ = -rVa/5-\ uv^ =. = -rv? [ ft a ) . . (45). Vector multiplication is not therefore in general commutative. 2d. If the vectors are unit vectors, a ft . (46), GEOMETRIC MULTIPLICATIOX AND DIVISION. 59 the product being expressed also by a quotient. This is of course always possible, as appears from (a), (6), (c) and (t?), and the transformation may be efiected thus : ^ = -^ = ^(eosc^-.sin<^), [Eq. (.31)] a la la — (3a = T,8Ta (cos c/) - e siu <^) ; or /3a = T/3Ta ( - cos (^ + e siu 4,) . 3d. If ^ = 0, then in either (a) and (6) or (c) and (d) the vector part of q becomes zero, and the quaternion de- grades to a scalar. When <^ = the vectors are parallel, and aB = — TaTB = — ab, as in Ai't. 35; also - = — = -, as in /3 T/3 b\ Art. 8. If at the same time a and (3 are unit vectors - = - =1 /3 a [or = aa ~^ = — a- = 1] and a^ = a- = — 1 , as in Arts. 33 and 28. If then q he any quaternion and \q = 0, the vectors oftchich q is the quotient or jiroduct are parallel. 4th. If c/) = 90° then in either (a) and (&) or (c) and {d) the scalar part of q becomes zero, and the quaternion degi-ades to a vector ; and either the product or quotient of two rectangu- lar vectors is therefore a vector at right angles to their plane, a/3 reducing to — abe and - to -c, as in Art. 3-4. If at the ' ^ (3 b same time a and /3 are unit vectors, a/3 = — e and - = e, as in Art. 27. ^ If then q he any quaternion and Sg = 0, the vectors ofzchich q is the quotient or product are perpendicular to each other. 5th. If an equation involves scalars and vectors, the vector terms having been so reduced as to contain no scalar parts, then since the scalar terms are purely numerical and independent of the others, the sums of the scalars and vectors in each member are separately equal. Thus if X + aa -j- b/3 = d + y -{- a'a + (&'- b")/3 ) then C (47), x = d + y and aa + h/3 = a'a + (&'- b")(3 ) 60 QUATERNIONS. which might also be written (i\j't. 38) S (x + aa + 6/3) = S [rf + 2/ + a'a + (b'- b")^] , y(.v. + aa + b/3)=\[cl + y + a'a + {b'- &")/3]. Gth. ^ being the quotient which operates on a to produce ^, a we have by definition p j/=(3 (48). 7th. TVa^, or ab sin<^, is the area of a paraUelogram whose sides are equal in length to a and b and parallel to a and (3. Sa/3, or —ab cos cf), is numerically the area of a i)arallelogi'am whose sides are a and 6, and angle ab is the complement of <^. 8th. Since the scalar s^-mbol S indicates the operation of taking the scalar terms, Sa = (49), and, for a similar reason, Ya = a (50) . Again, since a- is a scalar, V(a2)=0 (51), 8{a^) = -a- (52). T(a^) ma}- be wTitten T . a^, as also S(a^) = S . a^, but these forms must be distinguished from (Va)- and (Sa)^, which latter are also sometimes written Va and S'a. 9th. Comparing (a) and (6), Sa/3=S/3a (53), and Ta/3 = -T/3a (54). Adding and subtracting (a) and (b) , we have also a/3 + /3a=2Saft (55), a/3-(3a=:2\a/3 (56). GEOMETRIC MULTIPLICATION AND DIVISION. 61 10th. a/3 . /3a = (Sa/3 + Va/3) (Sa/3 -Ya(3) [Eqs. (53) and (54)] = {Sa/Sy- Sa/3Va/3 + Sa/3\a/3 - (Va/?)^. HsucG a/3. /3a = (Sa/3)2-(Va,8)- .... (57), or, from Equation (44), a^ ./3a = (Ta/3)2 (58). 42. Powers of Vectors. The symbol i", m being a positive whole number, has been seen (Art. 28) to represent a quadrantal versor used m times as an operator ; the exponent denoting the number of times i is used as a quadrantal versor. B}' an extension of this meaning of the exponent, «™ would naturally represent a versor which, 1 as a factor, produces the — th part of a quacbantal rotation. Thus /^ produces a rotation through one-third, and i^ through three-fifths of a quadrant, respectively. With the additional meaning attached to the negative exponent (Art. 32), as indi- cating a reversal in the direction of rotation, we may in general define i', where i is any vector-unit and t any scalar exponent, as the representative of a versor lohich ivoukl cause any right line in a plane j)erj)endlcular to i to revolve in that ^jlane through an angle t X 90° the direction of rotation depending upon the sign of t. Hence ever}^ such power of a unit vector is a versor, and, conversel}', every versor may he represented as such a poicer. ^ 2./> Since the angle (^) of the versor is ^ x -, we have t = — , and any versor cos cji-\- e sin (f) may be expressed 2« cos (j!) + e sin ^ = e"^ (59), and cosc/) — esin^ = e"-^ .... (00), the vector base being the unit vector about which rotation takes place, and the exponent the fractional part of a quadi'ant tlu'ough which rotation occurs. 62 QUATERNIONS. The operation of which *- is the agent is one-half that of which i is the agent, and therefore two operations with the former is eqnivalent to one with the latter ; or, as in Algebra, iW^=i=i'^^'^ (Gl), or, emplo3'ing the other versor form, if a, yS, y are complanar nuit vectors so that a . 2^ - = cos ^ + e sm <^ = e - , then since we have /? 2d - = cos 6 + e sin 6 = e~i 7 a /3 a (cos (^ + e sin (f>) (cos 9-{- e sin 0) = cos 4> cos 9 + t"^ sin (^ sin 6 + €(sin<^cos^ + cos ^ sin 6) = cos (<^ + $)+ £sin ((/) + ^) . The second member is the U— , its angle being (+0), and ma}' be therefore expressed as the power of a unit vector, and 2(0+0) wntten e — tt — the factors, or written e — tt — ; this exponent is the sum of the exponents of 20 20 2(0 + 6) €^e~ = €~^^~ (62). This is evidently an abridged form of notation to which the algebraic Imu of indices is applicable. Since £^= — 1 and therefore €''=1 ; if £'=—1, t must be an odd multiple of 2, and if £'=+1, t must be an even multiple of 2. ^ In either case the coefficient of tt in ^ = -tt is a whole num- ber, and cos <^ ± € sin + cm)/3^ which, for p = a, involves l)ut two equations x + zn = a + cn, y-\-zm = h -\-cm. GEOMETKIC MULTIPLICATION AND DIVISION. 65 Resuming the quadrinomial form of g, when the component vectors are at right angles, we have q = zv + xi + yj+ zk Sg = 10 Yq = xi + yJ-\- zk (70). Since (TYq)- = - (YqY = x^ -{- if + z\ we have TVg = Var + y- + z^ ] VYq. Yq xi + yj-Jr zk (71). TVg Var + ?/2 + 2-' J Also, since (Art. 41, 10th.) (Tqy = (Sg)^ - {YqY = vr + .^^ + f + z\ Ug SUg TVUg Tg = Vif- + .x-2 + 2/2 + ^2 g w + xi + 2/J+zfc Tg Vtc^ + of' + ?/- + z^ Sg _ ?o Tg Ay.j(;2 _j_ 3^2 _^ 2/^ + 2^ TVg^ I a.-^ 4-^24- ^2 Tg \^(;2_,_^.2_^^2_^^2 J (72). 44. The plane of a quaternion has been alreach' defined as the plane of the vectors or a plane parallel to them. The axis of a quaternion is the vector perpendicular to its plane, and its angle is that included between two co-initial vectors parallel to those of the quaternion. If this angle is 90° the quaternion is called a Right Quaternion. Any two quaternions having a common plane, or parallel planes, are said to be Complanar. If their planes intersect, the}' are Diplanar. If the planes of several quaternions intersect in, or are parallel to, a common line, they are said to be CoUinear. It follows that the axes of coUinear quaternions are complanar, being perpendicular to the common line. Complanar quaternions are alwaA's coUinear, and 66 QUATERNIONS. complanar axes coiTospoud to oollincar quaternions, but the lat- ter ma}' of course be tliplanar. O ' A O "c Let — - and be any two quaternions. If coinijlanar, they o'b o"d - 1 I ^ J ma}' be made to have a common plane ; and, if diplanar, their planes will intersect. In the former case let oe be au}' line of their common plane, or, in the latter, the line of intersection of their planes. Now, without changing the ratios of their vec- tor lengths, the planes, or the angles of the given quaternions, two lines, of and og, may always be found, one in each plane, or iu their common plane, such that with oe we shall have O A OF , O C _ OG O'b ~~ OE o"d ~~ OE and, therefore, any two quaternions, considered as geometric fractions, can be reduced to a common denominator ; or, in the above case o'a o"c _ of og _ of + og O'U o"d oe OE ~ OE Moreover, a line on, in the plane ao'b, may always be found such that o'a _ OE o'li on and therefore o"c o'a_og oe og o"d o'b ~~ OE on ~ oh' and o'a ^ o"c _ of ^ OG of oe_of o'b ' o"d oe * OE oe og og 45. Reciprocal of a Quaternion. The reciprocal of a scahir is another scalar with the same sign, so that, as in -tUgebra, if x be au^' scalar, its reciprocal is GEOMETRIC MULTIPLICATION AND DIVISION. 67 The reciprocal of a vector lias been defined (Art. 33) , so that, if a be any vector, - = a"^ = Ua. a Ta The reciprocal of a quaternion has also been defined (Art. 26) ; thus ^ being any quaternion. tt q Fig. 37. is its reciprocal. The onl}- difference between the quotients - and ^ (Fig. 37) is that, as opera- ^ a tors, one causes ^ to coincide with a, while the other causes a to coincide with /3. A quaternion and its recipro- cal have, therefore, a common plane and equal angles as to magnitude, but opposite in direction ; that is, ^>^- their axes are opposite. Or Since Z t = Z.q (1 and axis — axis q. and the 20Toduct of tivo reciprocal quaternions is equal to positive unity, and eacJi is equal to the quotient of unity by the other ; we have, therefore, as in Algebra, _g=l and ^' = 7, and no g 1 '1 new symbol is necessary for the reciprocal. - is, however, sometimes written Rg, R being a general symbol of operation, namely, that of taking the reciprocal. It follows from the above that T 1 1 Tg ^3), 68 QUATERNIONS. or, the tensors of reciprocal qtiaternions are reciprocals of each other ; while the versors differ only in the reversal of the angle. If then Ta we shall have <7 = - = — (cos ^''Sa jiig'ite have, therefore, a common plane and tensor, as also, in the ordi- nary' sense, equal angles ; but their axes are opposite ; or 1 and If then we shall have Z K7 = Z ry = Z TK7 = 1q = \ T? axis K7 = — axis q = axis fn a Ta / , , ■ , \ q = - = — (cos d> + c sm - - ~~~ ' -1, or, their qxiotlent \s negative nnity. If then we shall have a Ta / , , • 1 \ q =- = — (cos (f) + € Sin T(a'+y) unless a' = .Ty' anda;>0, it follows that 1q + Tq'>'i{q + q') unless q-=xq' and.r>0. Hence, in general, T2g is not equal to 2Tg. Moreover, since U2g is a function of the tensors under the 2 sign, while 2U7 is independent of the tensors, Vlq is not equal to 2U<^. This also appears from the representation of ver- sors by spherical arcs (Fig. 38). Hence, in the addition and subtraction of quaternions, T and U o?'e not, in general, dis- tributive symbols. 51. Multiplication of Quaternions. 1- Let q = Sq+^q, r=hr+^r be any two quaternions. Then p = qr = SryS/- + Sq\r + SrVry +\qYr. The last member, being the sum of a scalar and a vector, is a quaternion. Hence, the product of two quaternions is a quater- nion, and p = SjJ +Vi) = ^qr +yqr, in which Sgr = S^Sr + S . YvVr .... (90), and \qr = ^q\r + ^ryq+y .\qyr . . . (91). GEOMETRIC MTJLTIPLTCATION AND DIVISION. 75 If we multiply ^.b}' r, Ave obtain Srg = SrSg + S . YrYq, Yrq = Hr\q + SfyVr +V . YrYq. But, Equation (53), S . YrYq = S . YqYr. ... Srq = fiqr (92). But, Equation (54), Y.YqYr = -Y .YrYq, and therefore the products qr and rq are not equal. Hence, quaternion multi'plication is not in general commutative. If, howGA'er, q and r are complanar, Yq and Yr are parallel, and V . YqYr = ; in which case qr = rq. Conversely, if qr — rq, q and r are complanar. Since Reciprocal, Conjugate and Opposite quaternions are complanar, they are commutative, or qKq = Kq , q q - = -q = qq-^ = q-^q . (93). 2. It has been shown (Art. 44) that any two quaternions q, q', can be reduced to the forms " and Z having a common a a denominator, or to the forms " and 1. Hence o a "We have then , y /S y a y q' '.q = l:^ = L =1 a a a [5 p g /8 T^ Ta T^ Ta Ta ^ * ^ ry (3 VI3 Ua U/3 Ua'Ucc ^'^'^"i (94). 76 QTJATEBNIONS. In a similar mauucr ^ -• • (95). V{r/q) = uI=Vq'\]q 6 Hence the tensor of the product (or quotient) of any two qua- ternioiis is the j)roduct (or quotient) of their tensors, and the ver- sor of the j)roduct (or quotient) is the x)roduct (or quotient) of their versors. In foct, tensors being commutative, we have, in general, Tng = nT(7 (96), nry = TUq . Vliq = HTg . liVq, .'. Unry = nUg (97). 3. The multiplication and division of tensors being purely arithmetical operations, we proceed to the cori'esponding opera- tions on the versors. It has been shown (Art. 41) that an}'^ two versors 5, q', may be reduced to the forms B on , y' oc' /T-,. on\ A, B, c', being the vertices of a spherical triangle on a unit sphere. Then ,A = t.^ = l' = 2£'. fS a a OX If we represent the versors q' and q liy the vector arcs i;c' and An, then the versor ^, the product of q'q, will be rcprc- « y' sented b}- the arc ac' ; moreover if q" = - represent anv divi- (3 . a - dcnd and q= - an}- divisor, then q ^ a ' (3^ ^~ oa* the versor of the product q'q being BC' + All = AC', GEOMETRIC MULTIPLICATION AXD DIVISION. 77 Fi-. 39. aud the versor of the quotient i_ AC'— AB = BC'; and, as in the addition aud subtraction of quatcruious, the pro- cess consisted in an algebraic addition and subtraction of scalars but a geometric addition and sul)trac- tion of vectors, so the multiphcation and division of quaternions is reduced to the corresponding arithmetical ope- rations on the tensors and the geome- trical multiplication and division of the versors, the latter being con- structed by means of representative arcs and the rules of spherical addition and subtraction. 4. The representation of a versor by the arc of a great circle on a unit sphere illustrates the non-commutative character of quaternion multiplication. For, ab and ba' (Fig. 39) being equal arcs on the same great circle, as versors and similarly Now if then AB = BA CB = BC g = a'/3 li and y = — and 7 the versors qr and rq being represented b}- the arcs ca' and ac' respectively. These arcs, though equal in length, are not in the same plane, and therefore the versors rq aud qr are not equal. Constructing these versors, by spherical addition we should have BC' + AB = AC', AB + BC' = ba' + CB = ca', a change in the oi'der giving unequal results. 78 QUATERNIONS. Hence, unless ac' and c.v' lie on the same gi'cat circle, In which case q and r are complanar, quaternion multiplication is not commutative. 5. Other results, hereafter to he ohtained symliolically, may be readily proved by means of spherical arcs, as follows : If AB (Fig. 30) represents the versor of g = -, a b = ba repre- 1 "^ sents the versor of Yiq or -. The spherical sum of ab + ba '^ 1 being zero, the effect of the versors in the products qKq and q- is to annul each other. Hence, if the vectors are not unit vectors, " qKq = ILq . q = {TqY (98), Again, from ,1=1, =1. ab -f BC' = Ca', we have qr = -, y and the versor of K {qr) will therefore be represented b}- a'c. But a'c = BC + a'b, whence lL{qr) = JLrlLq (99), or, the conjugate of the product of tioo quaternions is the irroduct of their conjugates in inverted order. 6. The product or quotient of complanar quaternions is readilv derived from the foregoing explanation of versor products and quotients as dependent uj^on a geometric composition of rota- tions. For, disregarding the tensors, the vector arcs which represent the versors, since the latter are complanar, will lie on the same great circle, and the processes which for diplanar ver- sors were geometric now become algebraic. Thus for a qq'=q'q= ^'d=-, p a a GEOMETEIC MULTIPLICATIO:^: AND DIVISIOX. 79 and, Fig. 39, ba' + ab = ab + ba' = aa' ; also for (/"= - aud fj'= -, and ^_^.^_^ a_a' q' a * a a /3 jS' BA + aa' = ba'. Fis. 39. The product or quotient of an}- two complauar quaternions is therefore obtained by multiplying or dividing their tensors and adding or subtracting their angles. Thus 2jq = Ti) . Tq [cos (cf> -{- 0) -{- esin(c^ + ^)]. If J9 = q, f/ = {Tqf (cos 2 + esin<^) (cos^ + csin^) = cos^ cos 6 + e(sin-\-6)= cos cos ^ — sin ^ sin 6, sin { -\- 6) = s'm (fi cosO + cos0 sin^. 10. To find, the sine and cosine of the difference of two angles. Let the angle between y and a (Fig. 43) be ij/. Then y a y' GEOMETRIC IMTJLTIPLICATIOJf AXD DI\^ISION. in which 87 cos(i// — 6) — esiu(t/^ — 0), y' - = cosO + esin^, a _ = COS ij/ — e sin ij/, and, as in the preceding example, cos(i/^ — d)z= cosO cost// + sin^ sini/', sin ({j/ — 0) = cos 6 sivnj/ — sin d cos i{/. 11. If a straight line intersect txvo other straight lines so as to make the alternate angles equal, the two lines are parallel. Let a and y (Fig. 44) be unit vectors along ab and cd, and yS a nnit vector along ac. Then whence af3 = — cos 6 -\- e sin $, Py = — cos 6 — i. sin 9 ; a.p -/3y=2 Va/3, Fig. 44. and therefore, Eqnation (56), y = a. If a = AB,' then a(3 = cos (9 — €sin^, (By = — cosO — csin^, a/3-(3y=2^a(3; •■• y = — a- [Eq. (55)] 12. If a parallelogram he described on the diagoncds of any p>arallelogram, the area of the former is twice that of the latter. Let a and /? represent the sides as vectors ; then the diagonals are a + ^ and a — (3, and V(a + ^) (a - (3) =Y(/3a - a/S) = 2Y/3a, since Ya^=Y/3"- = and -Va^ = V/3a. 88 QUATERNIONS. / But, from the order of the factors, UV(a + /3)(a-^)=UV)8a, hence TV(a + ^)(a-/?) = 2Tyy3a, which is the proposition (Art. 41, 7). 13. Par allelocj rams on the same base and between the same parallels are equal. We have (Fig. 45) Fig. 45. Fip:. 46. BE = BA + AE = BA + .KBC. Operating with V . bc X V(bc . be) = V(bc . ba), since Va'BC-= 0. BC . BE sin EBC = BC . BA Sin ABC, which is also true when tlie bases are equal, l)ut not co-incident. 14. If , from any point in the j)/a«e of a parallelogram^ per- peiidiculars are let fall on the diag- onal and the tico sides that contain it, the product of the diagonal and its j^^n^c^dicular is equal to the su7n, or difference, of the products of the sides and their respective per- pendicidars, as the j)oint lies loith- out or within the parallelogram. Let OA = a, OB = /?, OP = p (Fig. 4G) . Then Vap+V/3p=V(a + /3)p. But UYap = UV/3/3= UV (a + p)p. Hence TVap + TV/?p = TV(a + p)p. GEOMETEIC MULTIPLICATION AND DIVISION. 89 For p' = op', we have V\ap'= - VY(3p'= ± UY(a + /?)p'; .-. TYap'^T\(3p'= TY{a + ft)p: 15. If, on any two sides of a iriangle, as ac, ab (Fig. 47). any two exterior parallelograms, as acfg, abde, he constructed, and the sides ed, gf, produced to meet in ir, then ivill the sum of the areas of the parcdlelograms he equal to that ivhose sides are equal and parallel to cb and ah. Let AE = a, AB = /?, AC = y and AG = 8. Then AH = AE + EH = a — X[B. Operating with x V . /? V(AH.^)=ya;S. We have also Operating with x V . y (a) AH = AG + GH = 8 - yy. V(AH.y)=V8y. (&) Hence, from (a) and (&), VAH(/3-y)=Va/?-V8y, Y(aH . CB)=Va/3-VSy=Tay3+Yy8. These vectors have a common versor ; whence the proposition. If one of the parallelograms, as ad', be interior, then ae'= — a and ah' = — a — x'^ = 8 + y'y, and V(AH'./3) = -Va/3, V(An'.y)=V8y; .-. VAn'(/3-y) = -Va^-VSy=Y/?a-VSy. 90 QUATERNIONS. But in this case rV(AH' . cr) = - UT/?a = - UTSy, and the area of the parallelogram on aii', cb, is the area of af minus the area of ad'. 1 G. To find the angle behoeen the diagonals of a parallelogram. Let AD = Bc = a (Fig. 48) , Fig. 48. and BA = CD = /?, d and d' being the tensors of the diagonals. Tlien AC . DU = - (a - ^) (a + ^) = -a?-{a(i-Pa) + l? = - a- - 2 Va;8 + ;8-. Taking the sealars cos DOC . dd'= a- — b-. Taking the vectors sin DOC . dd'=2absm0, since UV(ac . db) = - UVa^. . + ^ 4. ^ 2o&sin^ .'. tan doc = — tan ^ = — ; —. or — b- 17. TJie sum of the sqxiares on the diagonals of a parallelo- gram equals the sum of the squares on the sides. In Fig. 48 BD^ = (a +(3)-= a' + 2 Saft +(i\ ca2 = (y8 - a)- = /3- - 2 Sa/3 + a? ; .-. CA2+BD-=2a- + 2/32, or BD^ + CA- = BA- + AD- + DC" + CB". 18. The sum of the squares of the diagonals of any quadri- lateral is twice the sum of the squares of the lines joining the middle points of the opposite sides. GEOMETEIC MULTIPLICATION AND DIVISION. 91 Let AB = a, AD = /3, DC = y (Fig. 49). For the squares of the diagonals, we have Fig. 49. (/? + y)-" + (/3-a)^ and for the bisecting hues Whence the proposition readil}' follows. 19. The sum of the squares of the sides of any quadrilateral exceeds the sum of the squares on the diagonals by four times the square of the line joining the middle points of the diagonals. Let AB = a, AC = /?, AD = y (Fig. 50). The squares of the sides as vectors are or a^ + (/3-a)-^ + (y-/3)^ + yS or 2(a2 + /3- + y2) - 2 S^a - 2 Sy/?. The squares of the diagonals are ^^ + (y-a)^ /32 + y2 + a- - 2 Sya. The former sum exceeds the latter by aH /S^ + y- - 2 S/3a - 2 Sy^ + 2 Sya, (a-i8 + y)S which may be put under the form a + y (SV or by 9-2 QFATERNIONS. But "lJU = AG, and — '' = s>a 2 2 we obtain Substituting these values, 4(ao + sa)-, or 4so^, wliich is also true of the vector lengths. 20. In any quadrilateral^ if the lines joining the middle points of opj)osite sides are at right angles, the diagonals are equal. With the notation of Fig. 49, we have FK.GII=[H7-a)+/?]Ka+y). But, by condition, S(FE . OIl).= }^ _ ^V ^ + ^ = 0. ^ ^44^2^2 Whence FiL'. 51. or or AC- = BD^, AC = BD. 21. In any quadrilateral jirism. the sum of the squares of the edges exceeds the sum of the squares of the diagonals by eight times the square of the line joining the points of intersection of the two 2xdrs of diagonals. Let OA = a, OB = /?, oc = y, CD = S (Fig. 51). For the sura of the squares of the edges we have 2[a-" + /5- + (8 -a)-+2/+(S -/?)=], 2[2 a- + 2^- 4- 2 / + 2 82 _ 2 S8a - 2 S^^]. (a) The sum of the squares of the diagonals is {y-h^y + {y-8yi-{y + a-(3y- + (y + (3-ay, 2(a- + ^= + S- + 2/-2Sa^). (6) GEOMETKIC MULTIPLICATION AND DIVISION, 93 The vectors to the intersections of the diagonals are U^ + y) and |(y + a + ^), and the vector joining these points is Squaring and multiplying by eight, we have 2la^-\-(3- + S- + 2 Sa/? - 2 SaS - 2 S^8], which added to (6) gives (a) . 22. In any tetraedron, if two pairs of opposite edges are at right angles, respectively, the third pair will be at right angles. Let OA = a, OB = /?, oc = 7 (Fig. 52) . The conditions give Sa(^-y) = 0, S/3(a-y) = 0. Subtracting the first of these equa- tions from the second Sy(a-^) = 0, which is the proposition. 23. To find the relations betiveen the edges, plane angles and areas of a tetraedron. With the notation of Fig. 52, we have or CA . CB=(a — y)(^ — y), CA . CB = a/3 — ay — yy8 + y^. (a) Representing the tensors of ca and cb by m and n, and taking the scalars of (a) , S(CA . cb) = Sa/3 - Say - Sy^ + y^ whence (? — ac cos Aoc — he cos boc = mn cos acb — ah cos aob, 94 QUATERNIONS. which is the relation between the edges and their included angles. Taking the vectors of (a), and squaring, [y(CA . en)]' = (Va/S)- -ya/3Vuy -Ya^Vy/S -\ay\a/3 \ .j, + (Yay)'-' + VayVy/g - \y(3\a/3 -f Vy/3Vay + (Vy/?)^ i ^ "'' But {\aft\yp +\yftYa(3) = - 2 S . Va/3Vy^ (Eq. 55) = 2TVn/3TVy/? COSB, in which b is the angle between the planes aou, boc. Also - (VaySVay +VayVa/3) = - 2 S . Va/3Vay = 2 TYa/STYay COS A, and YayYy/8 + Vy/SYay = 2 S . VayTy^ = - 2 TVayTVyyScos (180°- c) = 2TVayTVy/3cosc, in which a, b and c are the angles opposite the edges bc, ac and ab re- spectively. Hence (b) becomes - [TV(CA . Cb)]-= - (TYafSy-- (Tyay)2 -iT\y(3f + 2 TYa/S T Vay COS A + 2 TYa/STYy/B cos B + 2TVayTVy;8c0SC. But (Art. 41, 7th) TV(cA . cb) = 2 area acb, and similarl}- for the others. Hence, dividing by —4, (areaABc)^= (area aob)- + (ai-ea aoc)- + (area boc)- — 2 area aob area aoc cos a — 2 area aob area boc cos b — 2 area aoc area boc cose. which is the relation between the plane faces and their included angles. GEOMETRIC IVnrLTIPLICATIOX AXD DIVISION. If the angles are right angles, then (area ABc)- = (area aob)^ + (area aoc)^ + (areaBOc)- 95 24. To inscribe a circle in a given triangle. Let a, p, y (Fig. 53) be unit vec- tors along the sides. Then, Ait. 16, the angle-bisectors are - 2/ (y + ") , Now «^(^ + 7) = cy-2/(y + a), Operating with V . (y + a) x ^ - <^^«y Vy^+Va/3+Vay' Hence A0 = .T(/3 + y) = '^^^ or, since a, /?, y are unit vectors, c sin B Fig. 53. (/5 + y), AG = sin A 4- sin b + sin c (/S + y). Squaring, to find the length of ao, we have, since (/3 + y)2 = — 2(1 + cos a), — AO- = AO = c sm b sni A + sin b + sni c c sin B sin A + sin B + sinc c sin B sin A + sin b + sin c 2(1 + cos A), V2(1 + cosa), 2 cos ^ A. 25. If tangents he draivn at the veHices of a triavgle inscribed in a circle, their intersections ivith the opposite sides of the triangle will lie in a straight line. 96 QUATERNIONS. Let o be the center of the cu-cle (Fig, 54) whose radius is r, and OA = a, OB = (i, oc = y. Since oa and ap are at right angles, S(oa . ap) = 0. But ap = AB + BP = AB + V/BC =z f3 — a-\-l/{y — (3) ; Fiff. 54. hence, substituting this vahio above, Sa[/3-a + .'/(y-^)]=0, and z/ = - Say — iia/3 Therefore OP = OB + DP = ^ + .'/BC = f3- f + ^f^^ (y - P) huy — »ap _ (r+Say)/3-(/-" + Sa/3)y ^ Say — Sa/3 Similarly, or, by a C3X-lic change of vectors, on- (>-' + Sa^)Y-(>"+S;8y)a Sa/? - S/?y ' ^,,^ (r + S/3y)a-(r + Say)/? S^y — Say "WTience (Say - Sa/3)0P + (Sa/3 - S/?y)oQ + (S/3y - Say)oii = 0. But also (Say - S:i(3) + (Sa^ - S/3y) + (S/3y - Say) = 0. Hence p, q and k are collinear. 26. The sum of the angles of a triangle is two rigid angles. GEOMETEIC MULTIPLICATION AXD DIVISI0:N-. 97 Let a, yS, y be unit vectors along bc, ca and ab (Fig. 55). Then (Ai-t. 42) 20 a iS - = e " , y ^ ^^ But /8" e", e I ^ 7 ^ - 2<;) 29 2i// 1 = e~^ e^ e ~ = €"£"£"= 6" ;(0 + 9 + r^) Hence -(0 + ^ + i/^) = an even multiple of 2 (Art. 42) , as 2 7i, as we go round the triangle n times. In taking the arithmetical sum, or passing once round, we take the first even multiple of 2, or - (c/, + ^ 4- ^) = 4 ; .-. cjb + e + i/.=27r, and the sum of the interior angles is Stt— 27r = -, or two right angles. 27. The angles at the base of an isosceles triangle are equal to each other. Let a and /3 (Fig. 5G) be the vector sides f^=- ^• of the triangle, and Ta ^ T,3. Then, if the proposition be true, = K or a-fS 13 -a a(a- ^)-'=Kf3{(3 - a)-'={l3 - a)-% a(/3-a) = {a-f3)f3; 2 C2 which is true, since Ta = T/j. 98 QUATERNIONS. 28, To find a point on the base of a triavrjle s^ich tJiat, if lines he draivn through it parallel to and limited by the sides, they will be equal. Draw DE (Fig. 57) and df parallel to the sides. From similar triaugles, if AE = XAC, AE FB AB — AF whence Now AC AB AB AF 1 — .T = AB AD = AF + FD, or, smee fd = ae, = ( 1 — X) AB + X\C. But, since fd is to be equal to ed, (1 — a-) Tab = .tTac = y ; .-. (1 — a:)TABUAB = ?/UAB, xTacUac = 2/Uac, and therefore AD = 2/(Uab + Uac), and D is on the angle-bisector, 29. If any line be draivn through the middle point of a line joining two jparallels, it is bisected at that point. 30. If the diagonal of a parallelogram is an angle-bisector, the parallelogram is a rhombus. 31, In an}' triangle the sum of the squares of the lines gii, ke, df (Fig, 58) is three times the sum of the squares of the sides of the triangle. 32. Tlie sum of the angles about tico right liiies tchich intersect is four right angles. Fig. 58. GEOMETRIC MULTIPLICATIOK AND DIVISION. 99 33. If the sides of any polygon be produced so as to form one angle at each vertex^ the sum of the angles is four right angles. 34. Find the eight roots of tinity (Art. 39). ^ 35. The square of the medial to any side of a triangle is one- half the sum of the squares of the sides tvhich contain it, minus one-fourth the square of the third side. 55. Product of two or raore Vectors. 1. Let q = a/3, r = y. Then, since Sqr = Srg, Sa/3y = SyayS. Let q = ya, r = /3. Then Sqr = ^rq = Sya/3 = SySya ; .-. SaySy = S/3ya = Sya/3 (108), or, the scalar of the 2'>^'odiict of three vectors is the same if the cyclical order is not changed. This ma}' also he shown b}" means of the associative law of vector multiplication as follows : ay5y = (a^)y = (Sa/5 +Ta/3)y. Taking the scalars Sa/3y=S(Sa^+Ta/3)y = S(Va^ . y), since S(Sa/3 . y) = 0, = S . yTa^ ; introducing the term S . ySa^ = 0, = S . yYa/3 + S . ySa;3 = S.y(Sa/3+Va^) = Sy(a^) = Sya;8. 100 QUATERNIONS. In a similar manner Sa/3y=S.a(S/3y+Ty3y) = S . a\Py = S(y/3y.a) = S(V/3y + S/3y)a = S^ya, and, as before, 2. Again Sa,(3y=S/3ya=Sya/?. S«/3y = S.a(S/3y+V/3y) = S . aypy = - S . aVy/3 = _Sa(Vy^ + Sy/?); .♦. Sa^y^-Say^ (109), or, a change in the cyclical order of three vectors changes the sign of the scalar of their product. 3. Kesuming a,5y = a(/3y) and taking the vectors, Ta^y = T. a(S,^y+T^y) = aS/3y +V . aV/?y. VyiSa = V(Sy^+Vy/3)a = Y. aSy^-V.aTy^ = V. aSy;3+V. aV^y = y.a(Sy/? + y/3y) = aSySy + Y . aVySy ; .-. Ta/3y=Vy/3a (110), or, the vector of the product of three vectors is the same as the vector of their jyroduct in inverted order. 4. Geometrical interpretation of Sa/Sy. Let a, /3, y be unit vectors along the three adjacent edges oa, OB, oc (Fig. 59) of an}- parallelopiped, 6 being the angle be- Also GEOMETRIC MULTIPLICATION AND DIVISION. 101 tween a and /3, find 6' the angle made b}' y with the plane aob. Then a/3 — — cohO + e sin6', e being a vector perpendicular to the plane aob. Oi:)erating with x S . y Sa/?y = S ( — cos 6 + e sin ^) y = S(sin^ . ey). But Sey = — cos of the angle between e and y = — sin^' ; Fi"-. 59. Sa/Sy sinO sinO'. 1/ j-'-^-' Now, if a, ;8, y represent as vectors the edges oa, ob, oc, whose lengths are a, h^ c, Sa/3y = - TaT/iTy sin Q sin Q' = — «6csin^sin^' But ab sin 6 = area of tlie parallelogram whose sides are a and h, and csin^' = perpendicular from c on the plane aob. Hence — Sa/3y = volume of a parallelopiped tvJiose edges are a, b and c, drawn parallel to a, /3 and y. Cor. 1. Whatever the order of the vectors, the volume is the same ; hence, as already shown, ± Sa/?y = ± S/Sya = ± Sya^ = ip Say/3, etc. Cor 2. If Sa;8y = 0, neither a, ^, nor y being zero, then either ^=0, or 0' = 0, and tlie vectors are complanar. Cor. 3. Conversely, if a, ^, y are complanar, Sa^y = 0. Cor. 4. The volume of the triangular pyramid of which the edges are oc, ob, oa, is — ^ Sa/3y. 5. We have seen that when a, (3 and y are complanar, Sa/8y=0, and therefore a/5y is a vector. To find this vector, suppose a 102 QUATERNIONS. triangle constinictod whose sides ab, bc, ca have the directions of a, /3 and y respective!}', a vector not being changed b}- motion parallel to itself. Since the tensor of the vector sought is the prod- uct of the tensors of a, /8 and y, we have to find U(ab . bc . ca), i.e., its direction. Circumscribe on the triangle abc a circle and- draw a tangent at a, represented b^- t'at. Since the angles tab and BCA are equal, we have CA AT atJ whence TJ(bc . ca) = U(ab . at') [= U(ba . at)]. Introducing Uab x TJ(aB . BC . ca) = U(aB . AB . at') [ = U(aB . BA . AT)], or, since U(ab . ba) = — (U . ab)-=1, U(aB . BC . ca) = — U . at' = U . AT. Hence, if a, b, c are ax^y three non-colUnear points in a plane, or if a, /?, y are the sides of a triangle joining them, in order (in either direction, since Va^y = Vy/3a) , a/3y, /3ya, ya/3 are the vector tangents to the circumscribing circle at the angles of the triangle. Again, if a, b, c are any three points in a plane, not in a straight line, and a and ft are two vectors along the two succes- sive sides AB, bc of the triangle which the}- determine, and cd a vector drawn from c parallel to y, intersecting the circumscribed circle at d, then is da parallel to Va^y = 8. For 8 = a^y = a^^y = aft'^ft'^y = - {TftYaft-^ = - (T^)^^y, whence U . ^^^, which turns /? parallel to — a, turns y into a direction 8 = da, the opposite angles of an inscribed quadi'ilateral being supplementary. GEOMETEIC MULTIPLICATION AND DIVISION. 103 If y have a direction siieli that CD crosses ab, or the quacM- lateral is a crossed one, it is evident on construction of the figure that „, , , "^ U8'= VajSy = U(ad) = - U8. Hence tlie continued product of the three successive vector sides of a quadrilateral inscribed in a circle is parallel to the fourth side, its direction being towards or from the initial point as the quadrilateral is uncrossed or crossed ; and, converse^, no plane quadrilateral can satisf}' the above formula ± US = Utt/?y, unless A, B, c and d are con-circular. The continued product of the four successive sides of an inscribed quadrilateral is a scalar, for ^..0,^0 a/3yS = (a/3y) 8 = ±S' = Td\ Since the product of two vectors is a quaternion whose axis is perpendicular to their plane, while the product of a quaternion b}^ a vector perpendicular to its axis is another vector perpen- dicular to its axis, and so on, it follows that the continued product of any even number of complanar vectors is generally a quaternion whose axis is perpendicular to their plane, while the product of any odd number of complanar vectors is a vector in the same plane. Hence the formulae Sa=0, Sa/3y=0, Sa/SySo" = 0, etc., for complanar vectors. If, however, the given vectors are parallel to the sides of a pol^'gon ABC MN inscribed in a circle, then TJ(ab . bc . CD MN . na)= U(ab . BC . ca) TJ(ac . CD . da) X U(aM . MN . Na). But each of the products U(ab . bc . ca) is equal to U . at, at being the tangent to the circle at a. Hence TJ(ab . bc . CD MN . na) = (TJ . at)", which reduces, according as n is even or odd, to ± 1 or ± U . at. Hence the product of the vectors wiU be a scalar or a vector 104 QUATERNIONS. according as their number is even or odd, and in the latter case this vector is parallel to the tangent at a. If the vectors are not complanar, but parallel to the successive sides of a gauche polygon inscribed in a sphere, the polygon ma}' be divided as above into triangles, for each of which the product of the three successive sides is a vector tangent to the circumscribing circle, all these vectors Ij'ing in the tangent plane to the sphere at the initial point. If the number of sides is even, their product will be a quaternion whose axis is perpendicular to the tangent plane, i.e., lies in the direction of the radius of the sphere to the initial point ; if odd, the product is a vector in the tangent plane. Hence, if a, b, c and d are four given points, not in a plane, AB = a, BC = /S, CD = y being given vectors, and p any other point such that dp = o-, pa = p, if p lies on the surface of a sphere through the four given points, we have the necessary and sufficient condition a/3y(— asina') = — sin e sin & sin a . a. Substituting in (a) , — siua . a'=cose sin& . /8'-fcos6sinc.y'— sincsin6sinA.a. (b) Operating with x S . y'~S — sina . S-, = cose siu6 S^ +cos6 sine S—,— sine sin&slnAS-:) y y y y in which S — , = cos b' = — cos B, y S — = — COSA, y S — , = 0, since a and y' are at right angles. Hence sin a cos B = cos& sine — cose sin 6 cos a, GEOMETEIC SnjLTIPLICATIOlSr AilSTD DIVISION. 115 and in the same manner, or hj a cyclic permutation of the letters, sin 6 cos c = cos c sin a — cos a sin c cos b , sin c cos A = cos a sin b — cos b sin a cos c. 9. Operating on Equation (6) of the last example with X y . y'-^ instead of X S . y'-\ — sinaV But V- 7 B' ' = coscsin&V — + COS& sine V— , — sine sin& siuAV— ,• = jSsinb' = ySsiuB, = — a sin a' = — a sin A, = 0. Fig. 63. Substituting these values — sin a sin b . /3 = — cos c sin b sin a , a — sin c sin b sin a . V Operating with X a~ ^ , and substituting for 7' - = cos c + y sm c, we obtain or — sin a sin b cos c — sin a sin b sin c . y ' = — cos c sin b sin a — sin c sin b sin a . y'. Equating the scalar or vector parts, we have in either case sin a sinn = sin a sin&, sina : sin& : : siuA : siuB. The formulae of the preceding examples have all been deduced from the equation - = The product as well as the quotient may also be employed, as follows : 116 QUATERNIONS. 10. Assuming the A^ector product Ya;QV/?y, and taking the vector part, we have [Equaliuii (117)], V . \a(3\/3y = - fSSa/Sy. (o ) But V . Ya/^Y/Sy = y(7'sino) (a' sin o.)= sine sinrt sins . f3, and, Art. 55, 4, Sa/8y = — sine sin^,' 6' being the angle made by oc with the plane of c. Substituting ill (c)^ sine sino sinii . (3= sine sin^' . (3, or sin 6' = sino sine. B3- permutation, from («), ^---U' y , \ya\ afS = - aSya(3 = - aSa/?y, T^-^ii or c siu6 sine siuA . a = sine sin^' . a, .'. sin^' = sin 6 sin a. Equating these values of sin^' we have, as in Example 9, sin a : sin & : : sin A : sin B. 11. Let p„, Pj, 2^c represent the arcs drawn from the vertices of ABC perpendicular to tlie opposite sides. Resuming Equation (a) of the preceding example, and taking the tensors, TV . Ya(3\/3y = Sa/3y = sinc sin/)„ = Sf3ya = sin a sinp„, = S/a/3 = sin b sin^9j, GEOMETRIC MULTIPLICATION AND DIVISION. 117 and, taking the tensor of V . Ya/SY^y from the last example, sine sin a sinB = sin a s'mj^a = sin 6 sinj^j = sine sinp^, or sin Pa = sine sinB, sin c sin a . sni 2h = sin b , sin 6 sinpc = sin a sinB. 12. SJioio that if abc, a'b'c' be two tri-rectangular triangles on the surface of a sphere^ cosaa' = cosbb'coscc' — cosb'c cosbc^ the triangles being lettered in the same order. Let a, j3, y, a', (3', y' be the vectors to the vertices. These being at right angles, in each triangle, we have cos Aa' = - Saa' = - S . YfSyYfS'y', or. Equation (122), cos aa' = S^^'Syy' - ^/3'yS/3y' = COS bb' cos cc' — COS b'c cos BC' [The vectors of Equation (122) are arbitrary, but we may divide both members by the tensor of the product of the vectors, so that S(VUay8VUyS) =SUaSSU^y - SUaySU/3S, for the unit sphere.] 13. Let ABCD be a spherical quadrilateral whose sides are AB = a, BC = b, CD = c, DA = d, the vectors to the poles of these arcs being a, /3J y', 8' respectively-. Then Va/8 = a' sin a, VyS = y'sinc. 118 QUATERNIONS. From Equation (122), S . YafiJyB = SaSS^y - SayS/?S, or sin a sin c Sa'y' = ( — cos da) ( — cos bc) — ( — cos db) ( — cos ac) . But Sa'y' = — cosL, L being the angle formed by the arcs ab and CD where they meet, the arcs being estimated in the directions indicated by tlie order of their terminal lettei's. Hence siuAB sin CD cosl = cosac cosbd — cos ad cosbc, a formula due to Gauss. 14. Retaining the above notation, abcd being still a spherical quadrilateral, denote the angles at the intersections of the arcs ab and cd, ac and db, ad and bc, by L, m and n respectively. Then, from Equation (125), S[Ta/3Vy8 + VayYS/3 -|- Ya8T/3y] = 0, we have identically siuAB sin CD cosl + sin ac sIubd cosm + sin ad siuBC cosn = 0. Were the points a, b, c, d on the same great circle, the angles L, M and N would be zero, and the above reduces to sinAB sin CD + sin ac siuBD + sin ad siuBC = 0, and for a line oaJ perpendicular to oa and in the same plane, dropping the accent, we have cosAB sin CD + cos AC siuBD + cos AD sinBC = 0, which are the results of Example 5 of this article. GEOMETRIC MULTIPLICATION AISTD DIVISION. 119 58. General Formulae. 1. We have seen, Equation (86), that SS = 2S and T2 = 2Y ; but (Art. 50, 4) that 2T is not equal to T2, nor 2U to US. We have also seen. Equations (9G) and (97), that Tn = nT and Un = nU; but Sn is not equal to ns, nor Vn to HV : for, 1st, sn is independent of the factors under the 11 sign, provided the product remains the same, while ELS is dependent upon them ; and, 2d, because (Art. 55, 5) nV is not necessarily a vector. 2. Resuming Equation (92), Srg = Sgr, and, since r is arbitrary, writing rs for ?•, we have, by the asso- ciative law (Art. 52), S(rs)g = Sg(rs), Sr(sg)=S(sg)r, .-. Srsg=Ssgr=Sgrs . . . . (126), a formula which may evidently be extended. Hence, tJie scalar of the product of any number of quaternions is the same, so long as the cyclical order is maintained. 3. Let J)-, g? '% s be four quaternions, such that qr=ps. (a) Operating with Kg x , Kg . qr = (Kg . q)r = (gKg)?* = Kg . ps, since conjugate quaternions are commutative. Hence (Tg)^r = Kg . ps, or Kg . ps 1 Operating on (a) with xKr, we have qr . Kr = ps , Kr, Rg.j?s = -»ps • • • (l^'^)' 120 QUATERNIONS. or q{Try- = p«Kr, psKr 1 .*. 9 = ^' = i^«R'-=/>s- . . . (128). Hence, in any eq^iation of the jjroduds of two quaternions, the first factor of one member may he removed by loriting its con- jxigate as the first factor of the second member, and dividing the latter by the square of the tensor, or simply by introducing the reciprocal as the frst factor in the second member. )iy substi- tuting the word last for first, the above rule will appl}' to the second transformation. 4. Resuming, for facilit}' of reference, the equations 9 = 1 = ^ (cos<^ + £sin<^) = Tq . Vq = Sq + Yg, (A) 1 /S T^ fy-i = - = - = ^(cos<^-esiu - £ sin 4>) = Sq- \q, (C) we observe directly that Sq = S{Tq.Vq) = Tq.SVq . . . (129), Tg = TVry . V\q = 1q . Tllry . . . (130) , TVg = Try . TYU5 = TVKg .... (131). 5. It has been alread}' shown (Art. 54, Fig. 40) that (Ta)2-|-(Tj8)-=(Ty)-, and (Art. 54, Fig. 42) that Ta=Ty.cos<^, T/3 = Ty . sin <^ ; and therefore (Ty)=^COS=^<^ + (Ty)=^ sin-0 = (Ty)S or sin-^ + cos- = l. Hence, from Equations (44), (SU(/)2 + (TTLVy)- = l .... (132). GEOMETRIC MULTIPLICATION AND DIVISION. 121 This important formula might have been written at once by assuming the above well-known relation of Plane Trigonometry. 6. From Equations (129) and (131), we may write Equa- tion (132) under the form (Sry)- + (T\V7)- = (Tg)2 (133), or, from Equation (107), {iiqr-{\qr = {Tqy = (^qy-h{T\qy . (134), since e^ = — 1 . 7. Comparing (A), (B) and (C), SUg = SU- = SUKg . . . (135), 1 T VUg = T VU - = T VUKg . . (136), and from Equations (129) and (135), 1 S(/ = Tg. SUg = Tg. SU- = Tg. SUKg. . (137). 8. Since Tg = TKg, we have Tg.TKg = (Tg)2 (138), and Tg being a positive scalar, KTg = TKg (139). As exercises in the transformation of these and the following S}Tnbolical equations, some of the results alread}^ obtained will be deduced anew. Thus, to prove that T(gg') = TgTg', whence T.g2 = (Tg)2, we have (Tgg')-=(gg')K(gg') Equation (107) = gg'Kg'Kg Equation (99) = g(g'Kg')Kg = (Tg')2gKg = (Tg')2(Tg)2, .-. Tgg' =TgTg: 122 QUATERNIONS. 9. Siihstitnting for Sq and T\q their values from Equations (79) and (131) {SKqy- + {T\Kqy = {iiqy- + {'r\qy- . . (uo). 10. Resuming from Art. 51, 1, the expressions Yrq = Sr\q + SryYr -\- V . \rYq, (a) \qr = HqYr + ^r\q + V . YqYr, (6) Sgr = S(/Sr + S . VgVr, (c) we have, hy adding and subtracting, V(//- — \rq = 2 V . \q\r And, if (/= r, from (a) and (c), \qy \ (142), V. q- = 2Sq\q S.f/ = (Sg)= + (V. whence g2 = (Sg)2 + 2S(/V5 + (Yg)2. . . (143). Dividing Equations (142) by (Tg)^ SU.r/ = (SU5)^+(VUg)M .144. TU . fy- = 2 SUg . VUg ) ' ' * since, evidently, S.r/ = (Tg)^SU.f/) ,145. V.fy- = (Tr/)2VU.r/j Again, substituting in the second of Equations (142) the value of {\qy from Equation (134), we have S.q' = 2{Sqy-{Tqy (146), and dividing by (Tfy)- SU.f/=2(SUry)2-l (147). Substituting (Sg)- from the same equation S.^- = 2(Vry)2 + (Try)2 (148). GEOMETRIC MULTIPLICATION AND DIVISION. 123 Equations (146) and (148) may be written (S + T)g2=2(Sfi)2 and (S - T)r/= 2(Vg)'. Introducing in (a), or (6), tlie condition that q and r are complanar, we have, after substituting versors, yUgr = VUfiSUr + VUrSUg, since, under the condition, V(VU5VUr) = 0. Taking the tensors, since q and r are complanar, TVUgr=TVUgSUr + SUgTVUr . . . (149), and, interpreting, Art. 51, 6, sia(^ + ^) = sin^ cos-^ (150), the reciprocal of the jyrodiict of tioo quaternions being equal to the product of their reciprocals in inverted order. This formula may be extended by the Associative principle, hy a process siniihu- to that employed in the deduction of Equation (126), so that if 11' represent the product of tlie same factors as those of n, in reverse order, {Uq)-' = U'q-' ..... (157). GEOMETRIC iniLTIPLICATION AND DIVISION. 125 The equation 'Kpq = 'KqE.p may be deduced without reference to spherical arcs. For, by Art. 44, any two quaternions can be reduced to the forms g = — , P = -,■> whence a p PQ. = ^^ or 2^Q . a = y, P/3 = y, and therefore Kp . y = Kp . i?/? = (Kp . p)[i = (Tjjy-fS. Now (KgK2))y = Kg(Tp)2/3 = (Ti;)-Kg . (3 = {TjyyKq . qa = (Ti3)"(Tg)-a = (Tjjg)-a = Kj3g , pq . a = Kjjg . y .•. K^jg = KgKj;, • which, by the Associative law, gives Kn = n'K (158). 14. Show that K(-g) = - Kg. 15. Show that T(p + qf = (P + q) (Kp + Kg) = (Tp)2 + (Tg)2 + 2S.i7Kg = {T2:>y + (Tg)2 + 2 TpTgSU . pKq = (Tj) + Tg) 2 - 2 TijTg ( 1 - SU . jMq) , and therefore that T{p-\-q) cannot be greater than the sum or less than the difference of Tj? and Tg. 16. Show that gUVg"' = TVg - SgUVg. 59. Applications to Plane Trigonometry. 1. For formulae involving 2 0, let g = Tg(cos2^ + esin2^). Then . _ Vg = g' = Vlg (cos (9 + e sin (9) . 126 QUATERNIONS. From Equation (142) , S . r/ = (S7)- + (ygY, we then have Sry = (Sr/)2 + (Yr/)^ or, dividing out Tq, and, interpreting, cos 26 = COS" ^ — sin^ 6. Again, from Equation (147), SU . q- = 2(SU(/)-— 1, SU9 = 2(SU^')--1; whence cos 2^ = 2eos-^ — 1. Again, from Equation (142), V . q^=2Sq\q, \q = 2Sq'\q', or, dividing out Tq and c, TYVq = 2 SUry'TVUfy' ; whence sin 2 6 = 2eos^ sin^. 2. Resmning Equations (149) and (150), TVU^r = TVUrySUr + SlVyTVUr, SVqr = SU^SUr - TVU^TVUr, which have already' been interpreted as the sine and cosine of the sum of two angles, and wi'iting for r = T/-(cos<^ + esin<^), r~' = — (cos<^ — esinf, r = -y/pt'^ (Art. 58, 3), T\Vp + TYl]t = 2fil](VpF')TY\]{Vpt) . (IGl), or sinx + sin?/ = 2cos|^(a; — y) sin|^(a; + y). Similarly, by subtracting tlie same equations, TVUgr - TYVqr'^ = 2 SUgTVUr, TYVp-TYVt = 2H\]{-Vpt)TYV{Vpr') . (162), or sin X — sin y =2 cos ^ {x -\- y) sin^ {x — y) . 4. From Equations (150) and (160), by addition and sub- traction, we obtain, in a similar manner, SUi) + SUi = 2SU(ViT0SU(Vy^0 • • • (163), and SVp - SU^ = - 2 T VU ( Vpt) T vu ( Vpr') , whence cos a; + cos y = 2 cos ^(x-\-y) cos ^(x — y), cosy — cosa; = 2 sin -1^(0; + y) sin^{x — y), 5. Resuming Equation (152), TY Vg = Vi(Tfy-Sg) , it may be put under the form 2(TVUV^)- = l-SUg, or 2sin2i^ = l-cos^. and, in a similar manner, from Equation (151), 2(SUVg)' = SUg+l, or 2cosH^ = l+cos^. 128 QUATERNIONS. 6. From Equation (142) (TV:S)./ = ^MI^^ 2TVr/ (SrjY 2 (TV : S)7 1- [(TV: 8)9]^ or . o^ 2tan^ tan 2 ^ = — . l-tan^e And, in a similar manner, cot-^-1 cot 2 6 = 2cot^ 7. From Equations (90) and (91), q and /• being complanar, Sfyj- = SqSr + S . \q\r = SqUr - T\qT\r, T\qr = SqT\r + S^-TVr/, we have, by division, (TV:S)gr = 5^IX!:+^^:^ '^ ^^ SryS/- - TV7TVy• (TV:S)r+(TV:S)9 or Also or tan(^ + <^) = (TV: 8)^/-' = tan(^-<^) = 1-(TV: S)y(TV: S)r tan<^ + tan^ 1— tan^ tan^ (TV: S)r/- (TV: S)r 1 + (TV: S)ry(TV: S)r tan $ — tan cf> l + tan^ tan<^ 8. Adding and subtracting (TV:S)p = :^, (TV:S)< = f , GEOMETRIC IVIULTIPLICATION AND DIVISION. 129 we have (TV:S)i9 ± (TV:S)« _ TTpS^ ± T\7Sp ^ TYUpSU^ ± TYOSUp ^ Hence, from Equations (l-iO) and (159), (TV : S)p ± (TY : S) ^ = "--^' , or , , , sin (x ± y) tan X ± tan u = ^^ ^ • cosx cosy By a similar process, , , , sin (y ± x) cot a.* ± cot?/ = —_ — ■- — : — '-' sinx smy 9. From Equations (161) and (1G3) ,- TYUp + TYm TYUV/^i = — 7=^-' ^ 9 HJJ-\/nf.-^ whence or 2 su Vi^r Jp + SI 2SuVi9r TYU« + TYU« (TVU : SU) yp = (TY : S) Vi^^ = ',.... > ^ SUiJ + su« -' 9STTa/)-i^-1 SUi^ + SUf COSX- + COS?/ And, in a similar manner, from Equations (162) and (163), (TY.b)Vi3« _ ^^^._^^^^ . or ^ 1 . . sin X — sin ?/ tani(a;-2/) = _ ^. coscc + cos?/ 130 QUATERNIONS. 10. Similar formulae ma}' be deduced for functions of other ratios of an angle. Thus, from Equation (90), writing rs for ?-, and making q ~r = s all complanar, we have, by Equation (142), s.(/ = {sqy-3Sq{Ty(jy-, or cos 36 = cos'^O — 3 cos sin- 6, or, under the more familiar form, cos3^ = 4cos='6-3cos^. CHAPTER III. Applications to Loci. 60. Any vector, as p, may be resolved into three component vectors parallel to any three given vectors, as a, y8, y, no two of which are parallel, and which are not parallel to an}' one plane. Thus p = Xa + y[i+Zy (164) refers to anj- point in space. If the A'ariable scalars x^ y, z are functions of two independ- ent variable scalars, as t and w, p is the vector to a surface, which, if the functions are linear, will be a plane. We ma}', therefore, write P = <^(^«) (165) as the general equation of a surface. If x^ y and z are functions of one independent variable scalar, as ^, p is the vector to a curve, which, if the functions are linear, becomes a right line. We may, therefore, write P = <^(0 (166) as the general equation of a curve in space. If a, y8, y are complanar, we may replace either two of the vectors in Equation (164) by a single vector, in which case p = <^(i) contains but two variable scalars, functions of f, and is the equation of a plane curve, or of a straight line if the func- tions are linear. The essential characteristic of the various equations of a straight line is that they are linear, and involve, explicitly or implicitly, one indeterminate scalar. 131 132 QUATERNIONS. 61. Assuming p = xa + yf3, (a) in which x and y are variable scalars, functions of a single vari- able and indoiiondcnt scalar, as t, as the general form of the equation of a plane curve, by substituting in an}- particular case the known functions x= f{t), y =f'{t)^ or x=f"{y), we ma}' avail ourselves of the Cartesian forms and apply to the resulting function in p the reasoning of the Quaternion method. For example, suppose a and /3 are unit vectors along the axis and directrix of a parabola, the origin being taken at the focus. In this case we have the Cartesian relation f=2px+p\ (6) or, substituting in (a), '2p as the vector equation of the parabola. Or, again, a and /8 being any given vectors parallel to a diam- eter and tangent at its vertex, P = |'« + f/3 (c) is the vector equation of a pai'abola, in terms of a single inde- pendent scalar t. 62. Let/(.r) be any scalar function as, for example, f{x) = x^. Then d[f(x):\=2xdxr=[f'(x)](lx. If, however, f((/) be a function of a quaternion q, as, for example, in the above case, . A = n [/(g + Ag) -/(g)], and we pass to the limit, writing dp for nAp^ and dq for ?iAg, we have 7 limit f[q + '^]-f{q) a formula for the differential of a single explicit function of a single variable. ltQ=F{q,r, ), clQ =l!=l^ niF(q + n-hlq, r + n-hlr, )-F{q, r, )] (1G8). In these formulae, dq, dr, are any assumed variables, no reference having been made to their magnitudes, and n any positive whole number conceived so as to tend to infinit}'. To show that these differentials need not be small, as also the ap- plication of the formula to the differentiation of ordinary- scalar quantities, let 136 then whence, as usual, QUATERNIONS. (l/ + Ay) = {x-{-Axy; Ay=2xAx + {Axy, or, n being a positive whole number, n Ay =2x11 Ax -\- n~^{n A x)-. If, now, the differences A y and A x tend together to zero, while n increases and tends to infinit}' in such a manner that nAx tends to some finite limit, as a, we have, for the other equimultiple n A ?/, n Ay = '2 xa + n~^ a-. But, since «, and therefore a^, is finite, n'^cr tends to zero, and, at the limit, nAy = 2xa. Hence the limits of the equi- multiples nAx and nAy are respectively a and 2xa, and 'dx = a, dy = 2xa hy definition; from which dy = 2 xdx. For a vector function we should write limit '^p = ,;= 'i '' ^-^^p + "~'^^) -•^(^)] and for a scalar function, p = <^ (<) , (1G9), (170), in which latter t and dt are independent and arbitrar}^ scalars. 64. As a further illustration of the definition, let p=cj>{t) APPLICATIONS TO LOCI. 137 Fig. G4. be the equation of any plane curve in space, and op = p (Fig. G4) a vector from the origin to a point p of the curve ; t being an}' arbitrary sca- lar representing time, for example ; so that its value, for an}' other point p' of the curve, represents the interval elapsed from an}- definite epoch to the time when the point generating the curve has reached p.' If p' be the vector to p' then p'~ P = pp'= Ap is strictly the finite difference between p and p', and, if the corre- sponding change in the, At, pp'= (p + A p) - p = A p= c^(; + A - <^(0 = A c/,(0 ; where op'= ^(^ + A ^) , and A Hs the interval from p to p! In ^A i, p would have reached some point as p", for which Op"= <^(i 4-i^A ^), on the supposition that pp" is described in ■| A i. On the basis of this closer approximation to the velocity at p, p would have been found at p", had this velocity remained unchanged, such that P2y'= 2 pp"= 2(op"- Op) = 2[(/>(« + A-A - c/.(0]. For a closer approximation to the vector described in A ^ with the velocity at p, suppose at the end of ^At the point is at p'", for which op"'= (^(i + |-Ai). Under this supposition, the vec- tor described in A t would have been Pp"'= 3pp"'= 3 (op"'- Op) = 3[<^(< + iA - -f') = n (177). 70. Right line through two given points. If 0A = a (Fig. 65), OB = /3 are the vectors to the given points, and p the variable vector to an}- Fi^. 65. point R of the line whose equation is re- R quired, we have and AR = x\n = x{ft — a), OR = OA + AR, or, for the required equation, p = a+.r(/3 — a) (178), which, if one of the points, as a, coincides with the origin, becomes p = x^, as before. "We have seen, Art. 55, that if Sa^y = 0, a, /3 and y are com- planar. Replacing y by the varia1)le vector p, Saf3p=0 (179) is tJie equation of a plane., since it expresses the condition that p is complanar with a and /?. If we have also Sayp = 0, the two equations, taken together, represent the line of intersection of these two planes. These equations may be obtained from the line p = xa bv ope- rating with S(Vay8) X and S(Vay)x ; or, conversel}-, to find the equation of the line in terms of known quantities, having given Sa/3p = 0, Sayp = 0, APPLICATIONS TO LOCI. 147 write these latter under the form S.pya^ = 0, S.pVay=0, whence it appears that p is perpendicular to both Ya/3 and Yay, and is consequeutl}' parallel to the axis of their product ; therefore "^ p = yY . Ta/3Yay = l/(ySa/?a-aSa^y) [Eq. (112)] = — 2/aSa^y, or, putting — ySajSy = x, p = Xa. 71. Right line perpendicular to a given line. 1. Let 8 = CD (Fig. Q6) be a vector through the origin. To find the equation of dc through its extremity -pig. 66. and perpendicular to it. Now p —8 is a d k c vector along dr, and therefore b}' condition SS(p-8) = 0. Whence SSp = -(T8)-, or ^Sp = c, a constant (180). In order that p, p — 8 and S be complanar, we must have S.Sp(p-8) = 0, or S . (V8p) (p - S) = 0. 2. p — 8, being perpendicular to both 8 and YSp, will be parallel to the axis of their product, or to V . 8Y8p. Hence, if y = GO be a vector to any point c, in the plane of od and dr, the equation of a right line through a given point c, perpendicular to a given line od, wUl be p = y + a;V.8VSy (181). 148 QUATERNIONS. 3. If the perpendicular is to pass through the origin, then, from Equation (180), SSp = (182), or, in another form, from Equation (181), y being parallel to V . 8V3y, p = y\.8\8y (183). 4. The student will find it useful to translate the Quaternion into the Cartesian forms. Thus, from Equation (180) , if rod= 6, S8/3 = -T8Tpcos^, whence, if r and d represent the tensors, rd cos 6 = d-, or ?• = cos 6^ the polar equation of a right line. 5. Equation (181), of a line through a given point and per- pendicular to a given line through the origin, ma}' be otherwise obtained, as follows : Let y and 8, as before, be vectors to the point and along the given line, respectiveh', and ft a vector along the required per- pendicular, whose equation will then be p = y + a-/?. (a) To eliminate /S we have the conditions S8(3 = 0, since 8 and (3 are perpendicular to each other, and Sy8)8 = 0, since y, 8 and /5 are complanar. But T8y is perpendicular to this plane, and therefore V . SV8y is parallel to /3 ; hence, substitut- ing in (a), p = y + x\ , 8V8y, or simpl}^ p = y + x6\oy. APPLICATIONS TO LOCI. 149 K SyS/?^ 0, y, S and /S are not complanar, and the problem is indeterminate ; which also appears from (a) , b}' operating with X S . 8, whence, since S/S8 = 0, Sp8 = Sy8, a result which is independent of /3. and an infinite number of lines satisf\' the condition. 6. If the line to which the perpendicular is ch-awn does not pass through the origin, let P = f3-i-xa (a) be its equation. Then, if p be the vector to the foot of the per- pendicular, we have Sa(p — y) = 0, or Sa(.ra + iS-y) = 0, (6) because the line is peq^endicular to (a) , or its parallel a. Hence, from (&), Xa = a~^ Sa(y — y8) , or, for the perpendicular p — y, p - y = .ra + /5 - y = a'^ Sa(y - /5) - a-^a(y - /?) ^_a-iYa(y-/?). Its length is evidently TT[tJa.(y-/3)] (184). 7. This perpendicular is the shortest distance from the point to the line. The problem may, therefore, be stated thus : to find the shortest distance from c to the line p = xa-\- (3. p being the vector from c to any point of the given liue. this vector is (3 + Xa — y, and, in order that its length be a minimum, dT(^ + a;a-y) = = T(^ + xa - y)dT(l3 + xa - y) = - S[ (/8 + Xa - y) a]dx = 0, 150 QUATERNIONS. or S(/? + a-a-y)a = 0, that is, the line must be perpendicular to p = .ra + /?. 8. If the perpendicular distance from the origin to p = (3 + xa is required, p, being as before the vector to the foot of the per- pendicular, coincides with it ; hence, y being zero, and 8 repre- senting this value of p, 8 = xa + ^. Operating with X S . 8, since SaS = 0, - {Td)- = i>(38. Hence or ^•> S,'3S S.^T8U8 1 J — — — — , T8 T8 T3 = S./3US (185). 72. "We are to observe that the foregoing equations of a right line are, as remarked in Art. GO, all linear functions invqjving, explicitly' or implicitly, a single real and independent variable scalar. Such is evidently the case for such equations as p = xa, [Eq. (173)] p=:/3 + xa, [Eq. (175)] p = a-\-x{(3-a). [Eq. (178)] So also for the implicit forms, as Vap = [Eq. (174)] ; em- ploying the trinomial forms a = ai -\-hJ + ck, p = xi + >/J + zk, we have ap = (bz — cy) i + {ex — az)j + (cuj — bx) k — (ax + bij -\- cz) . Whence Va/j = (bz — cy) i + {ex — az)j + {ay — bx) k = 0; .-. bz = ey, cx = az^ ay = bx, in which x and y are functions of z. APPLICATIONS TO LOCI. 151 The Plane. 73. Equation of a plane. 1. If, in the equation S . 8/3 = 0, ■which denotes that /3 is per- pendicular to 8, we replace /3 b}- tlie variable vector p, S . 8p = (186) is the equation of a plane througli the origin perpendicular to 8. 2. Or, let 8 = CD (Fig. 66) be the vector- Fig. 66 (6is). perpendicular on the plane, and dr any line d r c of the plane. / -^ Then or SS(p-S) = 0, SS/) = 8- = -(TS)2, SSp = c, a constant (187) is the general equation of a plane perpendicular to S. Here dr is any line of the plane ; and, if \8p = e. Sep = an indeterminate quantity (188), If the plane pass througli the origin, we have, as liefore, SSp = 0. Conversel}', if SSp = c is the equation of a plane, 8 is a vector perpendicular to the plane. 3. The equation of a plane through the origin perpendicular to 8 may also be written in terms of an}' two of its vectors, as y and (3 ; P = xf3 + yy. Both of these indeterminate vectors may be eUminated by operating with S . 8 X , whence SSp = as before ; or one may be eliminated by operating with V . )8 X , whence T/Sp = z8, 162 QTJATERNIOXS. from which we may again derive SSp = In* operating with Y . 8 X ; for V . STySp = \z^- = = pSS/3-/3SSp, [Eq. (Ill)] whence, since SS^ = 0, SSp = 0. 4. The equation of a plane through a point b, for which OB = ;8, and perpendicular to 8, is S8(p-^) = (189). 5. Having the equation of a plane, SSp = c, to find its dis- tance from the origin, or the length of p when it coincides with 8, we have p = xS ; hence S8/3 = c = SxS- = xB^, or which, in p = a;8, gives or x = c p = C Tp = c "t8 (190). 74. To find the equation of a plane through the origin, making equal angles with three given lines. Let a, ft, y be unit vectors along the lines. The equation of the plane will be of the form S8p = 0. By condition, Sa8 = S^8 = SyS = T8 sin <^ = x, 4> ^ing the common angle made b}' the lines with the plane. Hence . , X sm d) = — -. ^ TS APPLICATIONS TO LOCI. 153 To eliminate S, we have, from Equation (118), 8Sa(3y = YaySSyS + T/SyHaS + TyaS/88, and, by condition, 8Sa/5y = X (VafS + \/3y + Tya) . Tlie vector represented by the parenthesis is, then, the per- pendicular on the plane, whose equation, therefore, is Sp(Ya/3 + y/3y + yya)=0 . . . . (191), and the sine of the angle <^ is So^y T(ya;8+Vy8y + Vya)' 75. Equation of a plane through three given points. Let a, /?, y be vectors to the given points ; then are the lines joining these points, as (a — ^), (/S — y), lines of the plane. If p is the variable vector to any point of the plane, p — a is also a line of the plane. Hence S(p-a)(a-^)(^-y) = 0, or S(pa^ - pay - p/5- + p^y - a"/? + a'y + a/^" - a^Sy) = 0. But S(-p^2) = 0, S(-a2/3) = 0, etc., S(— pay) = Spya = S . pVya, Spa^ = S . pTa^, etc. , lienc6 S.p(Va^ + T/Sy + Vya)-Sa/3y=0 . . (192), which, by making the vector-parenthesis = 8, may be written under the form Sp8 - Sa^y = 0, 154 QUATERNIONS. in which 8 is along the perpendicular from the origin on the phme. "When p coincides with this perpendicular, p = x8, and, from the above equation, x8- = SaySy, or, for the vector-perpendicular, p = .T8 = 8-iSa/Sy= ^'^^V Va^ + V^y+Vya 76. "We observe again, from inspection of the equations of a plane, that, as remarked m Art. 60, the}' are linear and func- tions of two indetermmate scalars. Thus, for a plane through the origin SSp = 0, [Eq. (186)] emplo3ing the trinomial forms 8=ai-\-bJ-\-c]c and p=xi-\-i/j-{-zk, we obtain 8p = (bz - cy) i + {ex - az)j + {ay — bx)k — {ax + by + cz) , the last term of which is the scalar part ; hence ax + hy -f cz = 0, the equation of a plane through the origin o, perpendicular to a Inie from o to (a, 6, c), which may be written /(x, y, z) = ; or as a function of two indetormiiuites. In the same way, from an inspection of the other forms, p = xa + yji, [Art. 73,3] P = 8 + .ra + 2//3, SS/3 - c' = ax + hy + cz-c' = 0, [Eq. (1«7)] we observe the)' are linear functions of two indeterminate scalars. 77. Exercises and. Problems on the Right Line and Plane. 1. /8 and y being vectors along tico given lines icJiich intersect at the point a, to ivhich the vector is oa = a, to ivrite the equation of a line perpendicular to eaph of the two given lines at their intersection. APPLICATIONS TO LOCI. 155 T/3y is a vector in the direction of the required hne, whose equation, therefore, is p = a + x\(Sy (193). If a = oa' be a vector to any other point, as a^ then is p = a' -\- X\^y the equation of a hne through a given point perpendicular to a given plane ; the latter being given by two of its hnes. 2. a and (3 being vectors to two given points, a and b, and S8p = c the equation of a given ■plane, to fiyid the equation of a plane through a and b perpendicular to the given plane. 8, p — a and a — /? are lines of the required plane, hence S(p-a)(a-/3)8 = 0, or Sp(a-/3)S + Sa/3S = (194) is the required equation. 3. oc = y being a vector to a given p>oint c, and p = a-\- x/3, p = a'-\-x'fS' the equations oftioo given lines, to ivrite the equation of a jilane through c parallel to the two given lines. If lines be drawn through the given point parallel to the given lines, they will lie in the required plane. As vectors, /5 and /3' are such lines, and p — y is also a line of the plane. Hence S^^'(p-y)=0 (195) is the requii'ed equation. If y = a, or a', it is the equation of a plane through one line parallel to the other. Or, if y is inde- terminate, it is the general equation of a plane parallel to two given lines. Otherwise : the equation of a j^lane through the extremity' of y parallel to two given lines, whose directions are given hy a and yS, is evidently p = y -\-xa + 1//3. 4. To find the distance between two points. a and /3 being vectors to the points, y = /3-a. Squaring c^ = 6- + a- — 2 ab cos c. 156 QUATERNIONS. 5. A iilane being given by tivo of its li7ies, /3 and y, to icrite the equation of a rigid line through x perpendicular to the plane. Let OA = a. Draw two lines through a parallel to /8 and y. Then p = a + x-V/3y (190). If the plane is given b}- the equation S8p = c, then p = a + a;S (197). G. Find the length of the perpendicular from a to the j^lane, in the preceding example. Operating on Equation (197) with S . 8 x SSp = SSa + x5" = c, or xh^ = c- S8a ; .-. 3;8 =S-^(c-S8a) (198). 7. S8(p — y3) = 0, Equation (189), being the equation of a plane through b perpendicular to 8, to find the distance from a jjo'int c to the plane. Let y = oc. The perpendicular on the plane from c, being parallel to 8, will have for its equation p = y + xS. To find X, operate with S . 8 x , whence S8p = SSy + xh\ or. from the equation of the plane, S8y + a;82 = SS/8; .-. x8 = -S-'HHy-(3), and xT8 = TS-^SS(y-^) = S[U8 . (y-^)]. 8. Write the equation of a plane through the parallels p = a + xp, p = a'+ xfi. APPLICATIONS TO LOCI. 157 9. Write the equation of a plane tJirough the line p = a + Cf/S perpendicular to the plane SSp = 0. 10. Given the direction of a vector-perpendicular to a plane, to find its length so that the plane may meet three given planes in a point. Let 8 be the given vector-perpendicular, and Sap = a, S^p = &, Syp = c the equations of the given planes. If the equation of the plane be written SSp = X, then X must haA'e such a value that one value of p shall satisfy the equations of all four of the planes. From Equation (118) we have pSa^y = Va^Syp + Y^ySap + VyaS^p = cYa/8 + a\liy + h\ya. Operating with S . S x , to introduce x, icSa/3y = cSSa/3 + aSS/3y + &SSya. 11. To find the shortest distance between tiuo given right lines. Let the lines be given by the equations p = a + o:/3, (a) p = a' + x'/3[ (b) The equation of a plane through either line, as (6), parallel to the other (a), is [Equation (195)] S^;8'(p-a') = 0. (c) Y(3(3' is a vector-perpendicular to this plane. Hence, if ?/V/8/3' be the shortest vector distance between the lines, we have, since a — a'— yY/3(i' is a vector complanar with (3 and (3', S(3ft'ia-a'-y\/3f3') = 0, 158 QUATERNIONS. or S(S/Si8' + V^/3') (a -«' - yV/?/?') = 0, whence -2/(V/3/3')^ + S[V|8;Q'(a-a')]=0; or, dmdingby T(y^/3'), T(t/Y/?;8')=TS[(a-a')U(T/S^')]. . .(199), the sj'mbol T denoting thtit onl}- the positive numerical value of the sciiUir is taken. Otherwise : siuce the distance is to be a minimum, whence or S(p'-p)/? = and S(p'-p)/3' = 0, or the shortest distance is their common perpendicular, whose length may be found as above. 12. Give7i SSi/3 = (?! and SSop = do, the equations of two planes, to find the equation of their line of intersection. This equation will be of the form p z= 7u8i + ?i§2 + .tVSiSj. (a) To find m and ??, we have, from (a), S8,p = ??i8i- + ?;SSi82, SSjp = nh.£ + ?uS8iS2, from which we obtain SS,p-?(S8,82 SS2p-«82^ m = 71 = But K S8182 S8i8oS8]p — 8i'S82p (SS182)-- 8,^82^ = (V8,8o)^ (?, 88, 80-^2 81- •*• "- (¥8182)^' APPLICATIONS TO LOCI. 159 And similarly m= " Substituting these values in (a) which is the equation of the required line, a less useful form than those of the two simple conditions of Art. 70. If the two planes pass through the origin, then also does their hue of intersection ; and since ever}' line in one plane is perpen- dicular to 81, and ever}' line in the other to So, VSiSo is a line along the intersection, as in (o), and the equation becomes P = .tV8i82 (200). 13. The planes being given as in Equation (189), S8(p-/S) = 0, (a) SS'(p-iS') = 0, (&) to Jin d the line of intersection. The vector p to an}- point of the line must satisfy both («) and (h). This vector may be decomposed into three vectors parallel to 8, 8' and T88' which are given, and not complanar, by Equation (118) ; whence pS . 8S'V88' = Sp8T(S' . T88') + Sp8'V(V88' . 8) + S(pT88') VS8', or, from (a) and "(6), - p(TV8S')2 = SSy8T(S' . VS8') + S8';8'V(V8S' . 8) + S88'pT88', or, since 835 'p is the only indeterminate scalar, putting it equal to X, we have - p(TV88')2 = SS/SV(8' . V88') + SS'^'V(V88' . 8) + a;T88: If the planes pass through the origin, in which case (S and /?' are zero, we have, as before, p =z x\88'. 160 QUATERNIONS. 14. To tcrite the equation of a X)lane through the origin and the line of intersection of S8(p-;8) = 0, (a) S8'(p-)8') = 0. (6) If p is such that SSp = S8/?, and also SS'p = SS'^', then both the above equations will be satisfied. Hence, from (a) and {h) S8pSS'/3'-SS;8SS'p = 0, which is also a plane through the origin. This equation ma}' also be written S[(8S8'/3'-8'SS/3)p] = 0, which shows that 8S8'/S'-8'SS^ is a vector-perpendicular to the plane, and therefore to the line of intersection of (a) and (&) . 15. To find the equation of condition that four j^oints lie in a plane. If the vectors to the four points be a, /8, y, 8, then, to meet the condition, 8 — a, 8 — /3, 8 — y must be complanar, and therefore S(S-a)(8-/3)(8-y) = 0, whence S8;8y + SaSy -f Sa^8 = SaySy . . . (201), which is the equation of condition. Or, X and y being indeterminate, we have also or 8 + (a;-l)a + (z/-'^')/?-Z/7 = 0, and l+(x-l) + (y-a;)-y = 0. APPLICATIONS TO LOCI. ' 161 Or, in general, (202), aa + bj3-{-cy-{-(l8 = 0] a-\-b + c-j-d = 0) are the sufficient conditions of eomplanarit}'. These conditions are analogous to Equation (9). 16. Given the three planes of a triedral, to find the equations of planes through the edges perpendicidar to the opposite faces ^ and to shoiv that they intersect in a right line. Taking the vertex as the initial point, let Sap = 0, (a) S^p = 0, (6) Syp = (c) be the equations of the plane faces. Then Ta/? is a vector par- allel to the intersection of (a) and (6), and V . yVa/3 is a vector perpendicular to the required plane through their common edge. Hence the equation of this plane is S(pY.7Va^) = 0. (a') Similarl}', or by a cyclic change of vectors, S(pV.aV/3y) = 0, (6') S(pV./8Vya) = (c') are the equations of the other two planes. If from their common point of intersection normals are drawn to the planes, then are V . yYa^S, V . aY/?y and V . ^Vya vector lines parallel to them ; but, Equation (123), V(yVa/3 + aV/3y + ySVya) = 0. Hence these vectors are complanar, and the planes therefore intersect in a right line. Otherwise: from Equation (111) V(aY/?y) = ySa)8-/3Say; 162 QUATERNIONS, hence, from (h'), S{pyiaft - pfiSay) = Sa/SHpy - SaySp^ = 0. Similarly, or b\- cyclic permutation, S/SySpa - HfSaSpy = 0, SyaSp/? - Sy/3iipa = 0. But the sum of these three equations is identical!}' zero, either two giving the third b}' subtraction or addition. 17. To find the locus of a point ichich divides all right lines terminating in tioo given lines into segments ichich have a com- mon ratio. Let DA and d'b (Fig. C7) be the two given lines, a and yS unit vectors parallel to them, BA any line terminating in the given lines, and r a point such that RA = ?«Bu. Assume dd', a perpendicular to both the given lines, o, its middle point, as the origin, and od = 8, od' = — 8, ou = p. Then OA = p + UA = 8 + Xa. ou = p + KB = — 8 + y(i. Adding 2 p + UA + iiB = .Ta + y^. (a) But RA -j- RB = RA = ( — p -f 6 4- Xa) , m m which substituted in (a) gives p — h-xa = m{yp - p — ^)y (6) whence, since SS/3 = SSa = 0, S8p(m + l) = 8-(l-m) = c, 4 or the locus is a plane perpendicular to dd'. APPLICATIONS TO LOCI. 163 If the given ratio is unit}', or br = ra, then m = 1 and S8/D = 0, ' and the locus is a plane through o perpendicular to dd'. If a and /S are parallel, then (b) becomes p — 8 = m {x'a — /3 — 8) , whence SSp{m-\-\) = {l-m)S', a right line perpendicular to dd'. If at the same time m = l, SS/D = and p = x''a, a right line through the origin parallel to the given lines. IS. If the sxi7ns of the perpendicidars fro7)i tivo given points on two given planes are equals the sum of the perpendiculars from any p>oint of the line joining them is the same. Let A and b be the given points, oa = a, ob = /?, and SS/3=(Z, SS'p = d' be the equations of the planes ; 8 and 8' being unit vectors, so that a;8 and ?/8' are the vector-perpendiculars from a on the planes. Then X = Sa8 — cZ, y = Sa8'- d\ and a; + ?/=Sa(8 + 8')-((Z + cr). Similarly .T'+y=S^(8+8')-(cZ + (0. But, by condition, Sa(8 + 8') = 8^(8 + 8'), or S(/3-a)(8 + S')=0. (a) The vector from o to any other point of the line ab is a + 2 (/3 — a) ; whence, for this point, cc"+y' = S[a + z(/3-a)](8+8')-(rf + fO, for which point, smce (a) remains true, the sum therefore is unchan2;ed. 1G4 QUATERNIONS. 19. To find the locus of the middle poi7its of the elements of an hyperbolic paraboloid. Let the equations of the plane director and rectilinear direc- trices be SSp = 0, p = a + x[i and p = a' + .t'/8I Also, let CM = yiA be the vector to the middle point of an ele- ment so chosen that the vectors to the extremities are a + x/3 and a' + x'[i'. Then, since m is the middle point, 2/x = a + Xy8 + a'+x'^'. (a) The vector element is - x'fS' -a' + a-\-xp, and, being parallel to the plane director, SS(-a' + aH-.r/?-.r'/3') = 0. This is a scalar equation between known quantities from which we ma}' find x' in terms of x ; substituting this value in (a) , we have an equation of the form ^ = ai + .T/3i, or the locus is a right line. 20. If from any three points on the line of intersection of tivo planes, lines be draivn, one in each plane, the triangles formed by their intersections are sections of the same pyramid. The Circle and Sphere. 78. Er/uations of the circle. The equation of the circle may be written under various forms. If a and /3 are vector-radii at right angles to each other, and Ta = T/?, we ma}- write p = cos^ . a + sin^ . ;8 .... (203), in terms of a single variable scalar 6. APPLICATIONS TO LOCI. 165 If a and (3 are unit vectors along the radii, or, since x^ -(- 1/-= ?•-, Tlie initial point being at the center, T^=l p- = -r (204), (205) are evidentl}' all equations of the circle. If o (Fig. 68) be any initial point, c the center, to which the vector oc = y, p the variable vector to any point p, cp = a, then whence p — y=a, (p-y)^ = -7-'. .(206), the vector equation of the circle whose radius is r. If Ty = c, it ma}' be put under the form q /D2-2Spy = c--?'2 (207). If the initial point is on the circumference, we still have (p — y)- = — ?" ; but -f = — 7~, hence p--2Spy = (208), or, since in this case Spy = Spa, p2_2Spa = (209). 79. Equations of the sphere. This surface may be conveniently treated of in connection with the circle ; for, since nothing in the previous article restricts the lines to one«plane, the equations there deduced for the circle, are also applicable to the sphere. 166 QUATERNIONS. Another convenient form of the equation of a sphere is (Fig. 68) T(p-y) = Ta (210), the center being at the extremit}- of y and Tu the radius. 80. Tangent line and 2^1 cine. A vector along the tangent being dp, we have, from Equation (203), dp = — sin^ . a + cos^ . /?, and for the tangent Hne Tr — p + xdp, 7r=cos^ . a + sin^ . /? + .r[-sin^ . a + cos^ . ^] (211), where tt is any vector to the tangent hne at tlie point corre- sponding to 6. From the above we have directl}- Spdp = 0, v^ or the tangent is perpendicular to the radius vector drawn to the point of tangency. By means of this property we ma}' rB write the equation of the tangent as follows : let tt be the vector to any point of the tangent, as b (Fig. 69), c being the initial point and p the vector to P, the point of tangenc}'. Then Sp(,r-p) = 0, S/JTT = — >- (212), S- = l P are the equations of a tangent. Since nothing restricts the line to one plane, they are also the equations of the ttingent plane to a sphere. APPLICATIONS TO LOCI. 167 The above well-known propert}' may also be obtained by differentiating Tp = Ta ; whence, Art. 67, 2, Spdp = 0, and therefore p is perpendicular to the tangent line or plane. 81. Chords of contact. In Fig. 69 let cb = /3 be the vector to a given point. The equation of the tangent bp must be satisfied for this point ; hence, from Equation 212, S/Sp = - 7-2, or S/3o- = -?-2 (213), which is equally- true of the other point of tangency p,' and being the equation of a right line, it is that of the chord of contact ppI And for the reason previously given, it is also the equation of the plane of the cu'cle of contact of the tangent cone to the sphere, the vertex of the cone being at b. 82, Exercises and Problems on the Circle and the Sphere. In the following problems the various equations of the plane, line, circle and sphere are emploj-ed to familiarize the student with their use. Other equations than those selected in any special problem might have been used, leading sometimes more directly to the desired result. It will be found a useful exercise to assume forms other than those chosen, as also to transform the equations themselves and interpret the results. Thus, for example, the equation of the circle (209), p^-2^pa = may be transformed into Sp(p-2a) = 0, 168 QUATEKNIONS. which gives immediately (Fig- 7U) the property- of the circle, that the angle inscribed in a semi-circle is a right angle. Obviousl}', this includes the case of chords drawn from an}- point in a sphere to the extremities of a diameter, and the above equation is a statement of the prop- osition that, p being a variable vector, the locus of the vertex of a right angle, whose sides pass through the extremities of a and ^ — a, is a sphere. Again, with the origin at the center, we have (Fig. 71), (p + a) + (a-p)=2a, and, operating with x S . (p — a) , S(p + a)(p-a) = 0; .'. p is a right angle. This also follows from Tp=Ttt, whence p-=a- and S(p+a)(p— a) = 0. Again, from Tp = Ta, T(p + a)(p-a)=2TTap. The first member is the rectangle of the chords pd, pd' (Fig. 71), and the second member is 2 CD , OP sinDOP. Hence the rectangle on the chords drawn from an}- point of a circle to the extremities of a diameter is four times the area of a triangle whose sides are p and a. Also, from Tp = Ta, and for au}- other point P^ = -A ••• S(p'+p)(p'-p) = 0. But p'— p is a vector along the secant, and p'-{- p is a vector along the angle-bisector ; uow when the secant becomes a tan- APPLICATIONS TO LOCI. 169 gent, the angle-bisector becomes the radius ; therefore the radius to the point of contact is perpendicular to the tangent. 1 . The angle at the center of a circle is double that at the cir- cumference standing on the same cax. Tp = Ta, We have and therefore, Art. 56, 18, p = (p + a)-ia(/3+a), whence the proposition. 2. In any circle, the square of the tangent equals the product of the secant and its external segment. Fig. 69 (bis). In Fig. 69 we have p CB =CP +PB, .'. CB-=CP-4-PB^, or PB-=CB-'— CP^ = CB^— CD^, as lines, = BD . bd'. 3. Tlie right line joining the points of intersection of two circles is perpendicular to the line joining their centers. Let (Fig. 72) cc' = a, cp = p, cp' = p', and r, r' be the radii of the circles. Then also Hence or (p_a)- = -r'-; {p'-ay = -r'^. Spa = Sp a, Sa(p-p') = 0; hence pp' and cc' are at right angles. 170 QUATERNIONS. 4. A chord is drawn paruUel to the diameter of a circle; the radii to the extremities of the chord make equal angles with the diameter. If p and p' be the vector-radii, 2 a the vector-diameter, then xa = the vector-chord, and (p'-Xa)- = -'t^, (p-j-a;a)- = -)-, whence the proposition. 5. If ABC is a triangle inscribed in a circle, shoiv that the vector of the product of the three sides in order is parallel to the tangent at the initial p>oint. [Compare Ai't. 55.] If AB = /3, CA = y, and o is the center of the cii'cle, then -T(AB . BC . Ca) = V . /3(^ + y)y c and B being points of tlie circumference satisfying p- — 2Spa=0 [Eq. (209)], substituting and operating with S . aX S . aV (ab . BC . Ca) = 2 Sa/3Say - 2 Sa^Say = 0. Hence y(AB . bo . ca) is perpendicular to a, or parallel to the tangent at a. 6. The sum of the squares of the lines from any x)oint on a diameter of a circle to the extremities of a 2^arallel chord is equal to the sum of the squares of the segments of the diameter. Let pp' (Fig. 73) be the chord parallel to the diameter dd' Fiji. 73. o the given point, and c the center of the circle. Let cp = p, cp' = p, oc = a, op = ft and op' = /?! Then ^^' op2 = -^^ = -(a= + 2Sap+p2), .Ov'2 = -/3'-=-{a'+2fiap'+p") ; . • . OP- + OP'- = 2 OC^ + 2 DC- - 2 (Sap -I- Sap'). APPLICATIONS TO LOCI. 171 But S(p-p')(p + p') = S(p+p>-a=0. Therefore Sap + Sap' = 0, and op^ + op'- = DO- + od'^. 7. To find the intersection of a plane and a sphere. Let p- = — ^-^ be the equation of the sphere, 8 a vector-perpen- dicular from its center on the plane and TS = d. Then, if /3 be a vector of the plane, P = ^ + IB. Substituting in the equation of the sphere, since S(38= 0, Tve have the equation of a circle whose radius is V?*^ — rf^, and which is real so long as d < r. 8, To find the intersection of two spheres. Let the equations of the given spheres be (Eq. 207) p2 - 2 Spy = c- - 7-2, p--2Spy'=c'--r'% Subtracting, we have 2 Sp(y — y') = a constant. The intersection is therefore a circle whose plane is perpen- dicular to y— y' the vector-line joining the centers of the spheres. Assuming (Eq. 210) T(p-y) = Ta and T(p-y') = Ta,' show that 2 Sp (y — y ') = a constant, as above. 172 QUATERNIONS. 9. Tlie planes of intersection of three spheres intersect i?i a right line. Lot y', y". y'" be the vector-lines to the centers of the spheres, and their equations p--2Spy'=c', p'-2Spy" = c", p^-2Spy"'=c"'. The equations of the planes of intersection are, from the pre- ceding problem, 2Sp{y'-y") =c"-c\ (a) 2Sp{y'-y"')=c"'-c', (b) 2Sp(y"-y"')=c"'-c". (C) Now, if p be so taken as to satisfy (n) and (&), we shall obtain their line of intersection. But if p satisfies (a) and (6), it will also satisfy their difference, which is (c) ; the planes there- fore intersect in a right line. 10. To find the locus of the intersections of perpendiculars from aJixedx>oint ujwn lines through another fixed point. Let p and v' be the points, rp' = a, and 8 a vector-perpen- dicular on any line thi'ough pj as p = a -f- xft. Then ^ = a + 7//5, and operating with S . 8 x h- = SSa, which is the equation of a circle (Eq. 209) whose diameter is rv'. 11. From a fixed point p, lines are draicn to j^oi^its, as p', p", of a given right line. Required the locus of a point o on these lines, such that pp' . po = ?u^. Let the variable vector PO=p ; then rv'=xp. By the condition T(pp' . po)= ?u^, or T{xp . p)= m^; APPLICATIONS TO LOCI. 173 If 8 be the vector-perpendicular from p on the given line, and T8 = d, SS(a;p-8) = 0, or ajSSp = — c?- ; ..P=-S8p, hence the locus is a circle through p. 12. If through any 2'>oint cJiords be draion to a circle^ to find the locus of the intersection of the pairs of tangents draion at the 2)oints of section of the chords and circle. Let the point a be given b}' the vector oa = a, o being the initial point taken at the center of the circle. Let p' = or be the vector to one point of intersection r. The locus of r is required. The equation of the chord of contact is (Eq. 213) Sp'o- = — 7", which, since the chord passes through a, may be written Sp'a = — r-, where a is a constant vector. The locus is therefore a straight line perpendicular to oa (Eq. 180). 13. To find the locus of the feet ofperjoendicidars drcavn through a given point to planes passing through another given point. Let the initial point be taken at the origin of perpendiculars, a the vector to the point through which the planes are passed, and 8 a perpendicular. Then SS(8-a)=0, or 82_Sa8=0 is true for any perpendicular. Hence the locus is a sphere whose diameter is the line joining the given points. 174 QUATERNIONS. Otherwise : if the origin be taken at the point common to the planes, and the equation of one of the planes is SSp = 0, then the vector-perpendicular is (Eq. 198) S-'SSa, and, if p be the vector to its foot, p = a — 8~^ S8a, or p — a = — 8~' S8a, ■whence (p-ay= 8-\s8ay, and Sap — a" = — 8~"(S8a)'. Adding the last two equations p' — Sap = 0, or T(p--U) = T^a, which is the equation of a sphere whose radius is T- and center CI is at the extremity of -, or whose diameter is the line joining the points. 14. To find the locus of a point p wliich divides any line os draicn from a given point to a given plane, so that OP . OS = m, a constant. Let OP = p and os = o- ; also let SSo- = c be the equation of the plane. We have, by condition, TpTa- = m, and Up = Ucr ; .-. Tcr = ^, Tp and ??iUp "^ mp APPLICATIONS TO LOCI. 175 Substituting in the equation of the plane mSSp + cp^ = 0, which is the equation of a sphere passing through o and having — for a diameter, OD 15. To find (lie locus of a point the ratio ofwJiose distances from tivo given points is constant. Let o and a be the two given points, oa = a, or = p, r being a point of the locus. Then, by condition, if m be the given ratio, T(p — a) = mTp, __ whence p^— 2 Spa + a- = ni^p^i (1 — m")p^ = 2 Sap — a? = 2Sap-J^a^ 1 — nr or 2 _ 2 Sap _| a? _ m^o? , 1 — m- (1 — ni')- (1 — m-)- 1— my 1— m^ which is the equation of a sphere whose radius is T^; 5 a, and ^ 1 — m" whose center c is on the line oa, so that oc = ; a. (Eq. 210) . 1—m- 16. Given two j^oints a and b, to find the locus ofv lohen AP^ + BP- = 0P^ o being the origin, let oa = a, ob = ^, op=p. Then, by condition, p2 = (p-a)2 + (p-/3)S whence p2_2Sp(a + /3) = -(a--f^=^), [p-(a + ^)]-=2 Sa;8, T[p-(a + ^)] = V-2Sa^, 176 QUATERNIONS. which is the equation of a sphere whose center is at the extremity of (a + )8), if Sa/3 is negative, or the angle aob acute. If this angle is obtuse, thei'e is no point satisfying the condition. If AOB = 90°, the locus is a point. 83. Exercises in the transformation and interpretation of elementary symbolic forms. 1 . From the equation derive in succession the equations T(p + a) = T(p-a) and T^-^= 1, p — a and state what locus they represent. 2. From the equation K-+-=0 a a derive sj'mbolicalh' the equations ap + pa = 0, S- = 0, SU- = 0, (U^)=-l, and TVU- = 1, and interpret them as the equations of the same locus. 3 . Transform to the forms and interpret. p — a S = P pa S-=l and SU-=T-. 4. Transform S^ — ^ = to S-=S-' 'ind interjoret. a a 5. Transform (p -/5)- = (p - a)^ to T (p -^) = T (p -a), and interpret. 6. "What locus is represented by K = ? APPLICATIONS TO LOCI. 177 7. What by Q'= - 1 ? By Q'= - a^ ? 8. Whatby U^=U^? Up = U/3? U^=l? P /^. 9. U- = -U-? 10. (U-) =u-? 11. V ^ = 0? V- = V-? 12. V-=0? 13. -K- = fr? a a 14. su- = su-? su- = -su-? (su-)=(su-)? 15. Tp = l? 16. Transform (p — a)- = a- to T(p — a) = Ta, and interpret. 17. Under what other form may we write (p — ay=(j3— aY? Of what locus is it the equation ? 18. p^ + a- = 0? p2 + 1 = ? Translate the latter into Car- tesian coordinates, b}' means of the trinomial form, and so detei'- mine the locus anew. 19. T(p-^) = T(/3-a)? 20. Compare SU - = T- and S- = 1 with the forms of Ex. 3. pa p 21 . What locus is represented by S/5p + p- = when T/3 = 1 ? 22. (S^Y-f'v^'l = l? 23. (T-V=-l? 24. Sliow that V . Ya^Yap = is the equation of a plane. What plane? [Eq. (112)]. 178 QUATERNIONS. The Conic Sections. Cartesian Forms. 84. TJte Parabola. Resuming tlie general form of the equation of a plane curve p = xa-\- y(3, from the relation y- = 2p.r, we obtain P = ^^a + y(3 (214) for the vector equation of the parabola when the vertex is the initial point. If tlie latter is taken anywhere on the curve, from the relation y- = 'Ij^'x, we obtain P = ^« + 2//? (215); and if the initial point is at the focus, then y"^ = 2px -\- p- gives P = ^{y--ir)a + y(B .... (216); or again, in terms of a single scalar t, P = \o. + tp (217). In Equations (214), (21.5) and (21 G), a and /? are ?n^, t is imaginary, and no tangent can be drawn ; in this case (a) becomes P=[^ +aja + n^, the point being within the curve. 180 QUATERNIONS. 86. Examples on the parabola. 1 . The intercept of the tangent on the diameter is equal to the Fig. 74. abscissa of the x>oint of contact. Since the tangent is parallel to the vector ta-\- ft, or to an}' multiple of it, it is parallel to t-a + tft or to /- t- -a + i/3 + -a, that is, to (Fig. 74) But OP + ox. TP = TO + OP ; TO = ox. 2. If from any point on a di- ameter 2)roduced , tangents be draicn, the chord of contact is piarallel to the tangent at the vertex of the diameter. If t' and <" correspond to the points of tangenc}', we have for the A-ector-chord of contact P'-P which is parallel to ^la + t'ft-^^a-t"(i. P + f'+t" or, from Equation (6), Art. 85, to /3 + na^ which is independent of m. 3. To find the locus of the extremity of the diagoncd of a rect- angle whose sides are two chords draicn from the vertex. Let OP and op' be the chords. Then OP = P = lj^a + 7jft, or'=p'='fa-y% 2p («) (&) * APPLICATIONS TO LOCI. 181 The vector-diagonal w' is p + p' or which may be put uuder tlie form of the equation of the parabola 2 till' hy adding and subtractmg -^ a, giving But, by condition, S/ap' = 0. Hence, from (a) and (&), Sa/3 being zero, yy'-^^=^^ ••• 2/i/'=(2p)^ (^) which in (c) gives Changing the origin to the extremity of 4j3a, . {y-iir 2p ■a + {y-y')fi. Hence the locus is a similar parabola whose vertex is at a distance of twice the parameter of the given parabola from its vertex. Moreover, from (f?), xx' = {2py. Hence the parameter is a mean propoHional between the ordinates and the abscissas of the extremities of chords at right angles. 4. If tangents be drawn at the vertices of an inscribed triangle, the sides of the triangle produced ivill intersect the tangents in three points of a right line. Let opp' (Fig. 74) be the inscribed triangle, and one of the vertices, as o, the initial point. Then, for the points p and p' respective!}' , we have P = ^'^ + '/5' 182 QUATERNIONS. Let TTi, 7r2, TTg be the vectors to the points of iutorseetiou ; then t- Also TTi = OP + PSi = - a + //? + X{ta + (3) . ■7ri = x'or'=x'(— a + t'f3]; t- r't'- ^ •Zit'-t'' Hence ' 2tt'-t'\2 ^j 2t-t'\2 V In a similar manner But Also '==iFb(l°+^ TTg = OP + .?/pp' = ov+y(p'— p) = U + //? + ^ L_±a + {t'-t)/3 tt' t + t' Hence Now -'^-':^,_ 2«z:_«^,_ ^V3=/^-Wy8V t Also 2t-t' 2t'-t t-—t'- = 0. t t' W Hence tti, ttj and TTg terminate in a straight line. APPLICATIONS TO LOCI. 183 5. The jyfincqyal tangent is tangent to all circles described on the radii vectores as diameters. Fig. 75. Let AP = p (Fig. 75) , a and /3 being unit vectors along the axis d and principal tangent. Tlien, if tlie circle cut the tangent in x, and TC be drawn to the center, T(tc) = T(fc) = T(^FP) ; .'. TC- = i(p — ma)". Also TC = TA + AF + FC = — ;^/3 + i?ia + i (p — '>na) , TC- = \^—z(3-\-Vla-j-^{p — ma)Y. Equating these values of tc", we have, since fi/3a= 0, 2-/3" — z^f3p + mfiap = 0, ■^y-f- 0, which o-ives but one value for z. 6. To find the length of the curve. It has been seen (Art. 62) that, if p = (f)(t) be the equation of a plane curve, the differential coefficient is the tangent to the curve. Hence, if this be denoted by p'=(f>'(t), Tp'dt is an element of the curve whose length will be found b}- integrating Tp' with reference to the scalar variable involved between proper limits ; or -..=jrv. For the parabola P=^«+2/^» 22) 184 QUATERNIONS. we have 7. To Jin (I the area of the curve. AVith the notation of the previous example, twice the area swept over by the radius vector will be nieasured hy (Art. 41, 7) TYpp'dt. The area will then he found by integrating T\pp' with reference to the involved scalar between proper limits and taking one-half the result ; or A - A, = I- C'lYpp'. For the parabola A A or, since ayS = 90°, V(|^« + !//i)(?«+/J .J From the origin, where y^ = 0, to an}' point whose ordinate is y, the area of the sector swept over b}' p is -; — y^ = -^xy ; adding the area ^xy of the triangle, which, with the sector, makes up the total area of the half curve, we have ^xy, or two-thirds that of the circumscribing rectangle. The origin ma}' be changed to an}' point in the plane of the curve, to which the vector is y, by substituting the value p = y + p^ in the equation of the curve, Pi being the new radius vector ; we ma}' thus find any sector area limited by two positions of pi, the vertex of the sector being at the new origin. Thus, transferring to an origin on the principal tangent, distant b from the vertex, p = hp -\- pi\ which, in the equation of the parabola, gives Pi = ^« + (y-&)A P^=ja + li; Fig. 7G. APPLICATIONS TO LOCI. 185 integrating, as before, between the limits y = h and ?/ = 0, 2n. Relations between three intersecting tangents to the Parabo- la. ["Am. Journal of Math.," vol. i. p. 379. M. L. Holman and E. A. Engler.] Let pi, p2i Ps he the vectors to the three points of tangenc}^ Pi, P2, Ps [Fig. 76], and TTj, TTg, TTs the vectors to Si, S2, S3, the points of intersection of the tan gents. Resuming Equa- tion (216), where the focus is the initial point, and a and (i are unit vectors along the axis and the directrix, 2p («). Since p' = — (Tp)", and Sa/3 = 0, we have for the three points Pi, P2, P3 Tpi = — (z/r+ir) 22) Tp2 = i-(H+p^) •22) 22) (&). The vector alons; the tangent is |«+^, 186 and therefore QUATERNIONS. ^i = P2 + P2S1 = —Oji -iy')o. + y2^ + z ( -a + (A, 7^1 = P3 + P3S1 = — {yi -p^)o- + Vzl^ + yy-o. + ft] ; whence, equating the coefficients of a and /8, 2 = 1(2/3-2/2), ■it^ = 10/2-^3), the cj'clic permutation of the sulo- whence, substituting, and b; scripts, 7^1= — (2/3 2/2- 2i) T2 = — (2/12/3- 7^3 = 7- (2/2 2/1- 2i9 From (6) TpiTp,= 4p' f^^Tp3 = J^ and from (c) (Tm)= = (T.,)^ = (T:r3)^ = 4p2 _1_ 4p2 _1_ 1 4ir and from (d) and (e) (Ttts (Ttt^ (Ttt, J '^')a + ^(y2 + y3)/3 >')a + ^(2/3 + 2/i)/? '^')a + ^(2/i+2/2)^ 2/i'+P')(2/2Hir) y2'+p')(2/3'+F) 2//+i>')(2/i^+/) 2/2'+i>=')(2//+F) 2/3'+p')(2/i'+i>') - yi+p-){yi+p-) (c). (f?), (e), J 2 = Tp,Tp/ 2=T/D,Tp3 2 = Tp2T/33. (/). APPLICATIONS TO LOCI. 187 From (c) , it appears that the distance of the x>oint of intersec- tion of two tangents from the axis is the arithmetical mean of the ordinates to their points of contact. From (/) , that the distance from the focus to the point of intersection of tico tangents is a mean proportional to the radii vectores to the points of contact. 1st. If p2 becomes a multiple of /3, P2 = T- {yi -ir) a. + 2/2/5 = 2/5 ; 2p .'. z = y.2 = ±p. Or, the piarameter is the double ordinate tnrough the focus, or tivice the distance from the focus to the directrix. Fig. 77, 2cl. If pi is the multiple of pa (Fig. 77), then po — /d^ is a focal chord, and ^p2 — Pll or, from (a). ^(2/2' -i^')a+ 2/2/5 2p) = ^(yi--2r)a + y^fi; 2p whence yi-p" 2/2' 188 QUATERNIONS. or and yiy2+2r = 0. (rj) From (a) and (c) s-3pi = - — (z/22/1 - p-) — (.'/i- - ;>') - K^i + !/2)yi 2i> -22) = -^.(y' + p-) (yiy^ + jr) = ; (h) or, a line from the focus to the intersection of the tangents at the extremities of a focal chord is perpendicular to the focal chord. The vectors along the tangents are pi — TTg and p., — TTg, and, from (h), S(pi — Ta) (P2 — ■^n) = Sp, po + 7^3" = 0? or, the tangents at the extremities of the foocd chord are perpen- dicular to each other. Since, from (<;), we have !/i!/2 = -iA ^z = — (yi ?/2 - 2^') a + H^/i + 2/2) y3 •2p or, the tangents at the extremities of a foccd chord intersect on the directrix. 3d. If P2 becomes a multiple of a (Fig. 78), 7/, = 0, and from (c) -3 = ^ (!/22/i - p-)o. + ^{yi + y-dl^ 2p 2 "'"2^' • ^^ or, the subtangent is bisected at the vertex. APPLICATIONS TO LOCI. 189 Also -3-pi = -fa + |/?-(^^^a + y,yS 2p 2p 2 Operating with S . ttj x 4 4 or, a perjiendicular from the focus on the tangent intersects it on the tangent at the vertex. Fig. TS. Again, since tt^ is parallel to the normal at Pi, the latter maj be written, from (<), whence or = a;('-|a+|/5) = ^a + 2/,/?; V Vi a; = 2, 2 = -i; 190 QUATERNIONS. hence, the subnormal is constant ; and the normal is twice the pei'pendicular on the tangent from the focus. The normal at Pj may be written or ■whence, from (h), x = 2, and z'=^(!h' + p') = Tp,; or, the distance from (he foot of the normal to the focus equals the radius vector to the point of contact, or the distance from the 2'>oint of contact to the directrix, or the distance from the focus to the foot of the tangent. The portion of the tangent from its foot to the point of con- tact may be written za + pi, in which z has just been found. Hence Za + pi=— (?/i- + p") a + — (!/i- - p-) a + 2/1/3, •2p -Ip or the portion of tlie tangent from the foot of the focal perpendicu- lar to the point of contact is - T3 + pi = |a - I /3 + ^^(2/f - 2r)a + yi/?, or -^3 + pi = |La + |/3, (k) or, comparing (j) and (k), the tangent is bisected by the foccd perpendicular, and hence the angles between the tangent and the axis and the tangent and the radius vector are equal, and the tangent bisects the angle between the diameter and radius vector to the point of contact. APPLICATIONS TO LOCI. 191 (Tc) is also the perpendicular from the focus on the normal, and shows that the locus of the foot of the x)erpendicular from the focus on the normcd is a xKinihola, ivhose vertex is at the focus of the given 2^cirabola and ivhose parameter is one-fourth that of the given parabola. 88. The Ellipse. 1 . Substituting in the general equation p = xa + yfB the value of y from the equation of the ellipse referred to center and axes a^y^ + b-xr = a^l/, we have p=xa + mi{a--xy^(3 .... (220), 72 in which m = — and a and (3 are unit vectors along the axes. a- For unit vectors along conjugate diameters, the equation of the ellipse becomes p = xa + m'i{a"'-x-)hft .... (221). Again, if 4> be the eccentric angle, the equation of the ellipse ma}' be written in terms of a single scalar variable, p = cos<^ . a + sin<^ . /? .... (222). 2. From Eq. (220) we have, for the vector along the tangent, 1/0 o\ _i /-J ni X >3 mx n a. — m'iicr — XT) ^Xf3 = a ———=p = a (d Vm Va" — XT y = X(ya — mx/S) ; hence, for the equation of the tangent line, TT = xa + ?//3 + X (ya -mxp) . . . ( 223) ; or, more simply, from Eq. (222), the vector-tangent is — sin ^ • a + cos (fi , /5, 192 QUATERNIONS. aud the equation of the tangent is 7r = cos<^ . a + sin^ . j3 -\-x(— sin<^ . a + cos<^ . (3), (224). Since — sin<^ . a + cos<^ . ^ is along tlie tangent, cos^ . a + sin<^ . (i aud — siu<^ . a + cos<^ . yS are vectors along conjugate diameters. 89. Examples on the Ellipse. 1. The area of the ]}arallelogram formed by tangents draiai through the vertices of any pair of conjugate diameters is constant. "We have directl}'' TY[2(cos<^ . a + sin^ . (i) 2(— sin<^ . a4-cos<^ . /3)] = 4 TVa^ = a constant; namely, the rectangle on the axes. 2. Tlie sum of the squares of conjugate diameters is constant, and equal to the sum of the squares on the axes. For, since Sa^ = 0, (COS^ . a -f- f^'incj) . ft)- +{ — s\n(f> . a-|-COS(^ . /?)" = a- + /3". 3. The eccentric angles of the vertices of conjugate diameters differ by 90? The vector tangent at the extremit}' of p = cos . a + sin ^ . (3 (o) is — sin <^ . a + cos <}> . /3. This is also a vector along the diameter conjugate to p. and is seen to be the value of p when in (a) we write ^ + 90° for eft. 4. TJie eccentric angle of the extremity of equal conjugate diam- eters is 45° and the diameters full upon the diagonals of the rectangle on the axes. APPLICATIONS TO LOCI. 193 5. Tlie line joining the X)oints of contact of tangents is parallel to the line joining the extremities of parallel diam- eters. 6. Tangents at right angles to each other intersect in the cir- cumference of a circle. 7. If ayi ordinate pd to the major axis be produced to meet the circumscribed circle in q, then QD : PD : : a : 6. 8. If an ordinate pd to the minor axis meets the inscribed circle in Q, then QD : PD : : & : a. 9. Any semi-diameter is a mean proportional between the dis- tances from the center to the points ichere it meets the ordinate of any point and the tangent at that point. For the point p (Fig. 82) we have p = cos ^ . a + sin cfi , (S. Also OT = a'OP = OQ + QT = .t(cos^ . a + sin(;6 . /3) = COS^' . a + siu' . (B -\-t{— siu^' . a -j- COS^' . (3). Eliminating t, ^ COS{ , (3) Eliminating f', x' = cos(<^ — cj>'), or ox = cos(^ — — <^o) • From Eq. (220) , we obtain p'= a — m'i (a- — x^) -2.r/3, a ^ a- — .X" v«" — •1' ^ « J'»-i; a L e'-x- ■ -yi — -, ,, sjcv^ — x^ ^ a- ■'o which may be expanded and integrated ; giving for the enthe curve 27ra 1 etc. 1, V 2.2 2.2.4.4 2.2.4.4.6.6 / a converging series. If e = 0, we have, for the cu'cle, 27rr. 11. To find the area of the ellipse. With the notation of Ex. 7, Art. 86, TVpp'= TY(cos . a + sin ^ . ft){— sin <^ . a + cos^ . /3) = TT(cos-<^ . a/3 - sin- . a = sec . a -f tan . fB -j- ?/ (tan sec ^ . a + sec- <}> , (3)', 1 •'• •'^ = — :r-> sec- . a) = a^, or CT . CN = a-. 10. If the tangent at any point p meet the conjugate axis at t', and pn' be the ordinate to the conjugate axis, then ct' . cn' = b-, c being the center and 6 the semi-conjugate axis. APPLICATIONS TO LOCI, 199 92. The preceding examples on tlie conic sections involve directly the Cartesian forms. A method will now be briefl}' indicated peculiar to Quaternion analysis and independent of these forms. 1. The general form of an equation of the first degree, or as it ma}^ be called from analogy, a linear equation in quaternions, is aqh + a'qb'+ a"qb"+ = c, or ^aqb = c, (a) in which q is an unknown quaternion, entering once, as a factor onlj', in each term, and a, h, a', h', , c are given quaternions. It may evidently be written 2Sag& + Vfaqb = Sc + Vc, whence 2Sag& = Sc, (5) SVagft = Vc. (c) But ^aqh = S(/6a = S(/S6a + S . XqYba, and Taqb = V(Sa + Va) {Sq + Vg) (Sb + V6) = V . Sg(Sa + Va) (S6 + Yb) + V(SaVgS& + SaVgV6 + V«VgS& + YaYqJb) = SqYab + V(SaS6 - SaV& + S&Va) Yq + V . VaVgV& + V . YaYbYq - V . YaYbYq [Eq. (116)] = SqYab + V(SaS& - SaV& + SbYa - YaYb)Yq + 2 VaS . YqYb = SgVa& + V . a{Kb)Yq + 2 VoS . YqYb. We have therefore, from (b) and (c), Sc = Sg2S6a + S . Vg2V6a, Vc = Sg2Va6 + 2V . a (K6) Vg + 2 SVoS . VgV6, or, writing 2Sa& = d, 2Va& = S, 2V&a = 8,' Sg = i«, Vg = p, 200 QUATERNIONS. we obtain Sc = tod + SpS; Yc = icB + 2V . a{Kb)p + 2 2VoS . pYb. "We ma}' now eliminate lo between these equations, obtaining Yc .d-Sc .8 = d2Ya(K&)p - 88/38'+ rf22YaS . pY6 which involves onlv the vector of the unknown (luaternion 7, and which, since Y and 2 are commutative, ma^- be written under the general form y = \i'p + 2,5Sap, in which y, a, a, , /8. /3J are known vectors, r a known quaternion, but p an unknown vector. This equation is the general form of a linear vector equation. The second member, being a linear function of p, may be written Yrp + 2/3Sap = <^p = y .... (229), where (ftp designates an}' linear function of p. If we define the inverse function (f>~^ b}' the equation ~\4'p) = P-^ the determination of p is made to depend upon that of <^~^ 2. Without entering upon the solution of linear equations, it is evident on inspection that the function ^ is distributive as regards addition, so that ip + p'+ ) = p + V+ . . . (230). Also that, a being any scalar, cf>ap = acf>p (231), and dc}>p = cf,dp (232). 3. Furthermore, if we operate upon the form p = 1 (235). 2. B}- inspection of the value of p it is seen that, when p coincides with either axis, p and (ftp coincide. Operating on (j>p with S . o- x ,-we have g , _o^S|rp + SaaSap ^'^ a\l-e-) ' ArPLiCATiOiSrs to loci. 205 operating on <^o- = — ^ — with S . p x , we have a\l— e-) ^ , ^ _ ft-Spcr + SpaSacr . ^^ a\l-e') ' hence Sp(/)o- = So-<^p (236), and ^ is self-conjugate. 3. Diflferentiating Equation (235), we have Hclpcjip -j- SpcZ^p = 0, Sdp<^p + Sp<^dp = 0, [Eq. (232)] Sp 1, that is, when 1 —e- = ;, giving the corresponding equations a^y- — Irx- = — a- b^, (i-^yy'— b-xx' = — a^b-. 95. Examples. 1. To find the locus of the middle points of parallel chords. Let ^ be a vector along one of the chords, as rq (Fig. 82) , the lepgth of the chord being 2?/, and let y be the vector to its middle point ; then p = y + ?//3 -'^iifi p = y — y(^ are vectors to points of the ellipse, and S(y+2/^)<^(y+?/^) = l, S(y-2/^)<^(y-?/^) = l; whence, expanding, subtracting, and appl3'ing Equation (236), Sy/3=0, APPLICATIONS TO LOCI. 207 the equation of a straight line through the origin. Since fi = 0, In the latter, (3 is perpendicular to the nomtial ^a at the ex- tremity of a, and is therefore parallel to the tangent at that point; hence this is the equation of the diameter bisecting all chords parallel to a. Therefore, diameters which satisfy the equation Sac^/? = are conjugate diameters. 3. Supplementary chords. Let pp' (Fig. 82) and dd' be conjugate diameters, and the chords PD, pd' be drawn. Then, with the above notation, and DP = a — ^, D'p=a-|-/S, S(a-f;8)0(a-/3) = S(a+/S)(c^a-<^^) = S(ac/>a - acf>l3 + ftcj>a - y8<^y8). 208 But QUATERNIONS. Sa<^a = 1 , H/3/3 = 1 , Sa<^/3 = S/3(^a ; Hence, if dp is parallel to a diameter, pd' is parallel to its conjugate. 4. Iftico tangents be drawn to the ellipse, the diameter 2'>cirallel to the chord of contact and the diameter through the intersection of the tangents are conjugate. Fig. 82. Let TQ (Fig. 82) and tr he the tangents at the extremities of the chord parallel to /3, and or = tt. Then OQ = .Ta + ?//?, OR = Xa ■+■ 7/'/?. From the equation of the tangent Sttc^/j = 1 , we have S7r<^(.Ta + .V/S)=l, ii7r{xa-\-y'l3)=l. Expanding and subtracting Sttc^/S = 0. Hence, Ex. 2, tt and /3, or op and on, are conjugate. The locus of T for parallel chords is the diameter conjugate to the chord through the center. APPLICATIONS TO LOCI. 209 5. If qoq' (Fig. 82) he a diameter and qr a chord of contact^ then is q'r parallel to ot. RQ being parallel to jB, and oq' = — oq, we have RQ = 2 yfS, rq' = y(3 — xa — xa — ?//3 ; whence, directl}' rq' = — 2.Ta ; as also Srq^rq' = 0, rq and rq' being supplementaiy chords. 6. The points in ivhich any two parallel tangents as q'tJ qt (Fig. 82) are intersected by a third tangent, as ttJ lie on conju- gate diameters. The equation of rt' is Sttc^p = 1, and that of q't' is S7r'm (a - /?) = 1 , (a) S?i(a + ^)<^n(a+/S)=l. (b) Now, from (a) , since SyS^/? = SacjSa = 1 and S/3^a = Sa<^/3 = 0, 2m-=l. 29t-=l; Similarly, from (b), Also DP : d'p : : T(a - ^) : T(a + ^) ::T??i(a — /3) : Tii(a + /3) : : OE : OF. 1 1 . TJie diameters along the diagonals of the parallelogram on the axes are conjugate; and the same is true of diameters along the diagonals of any jKirallelogram lohose sides are the tangents at the extremities of conjugate diameters. 12. Diameters parallel to the sides of an inscribed parallelo- gram are conjugate. Fiff. S3. K Let the sides of the parallelogram (Fig. 83) be pp' = a, PQ = /?, and let Then OP OP = p, OQ = p. '-p+a, OQ' = p' + a, p'-p = ff. 212 QUATERNIONS. From the equation of the ellipse, S/3<^p= 1 , we have for q' and p' S(p'+a)<^(p'+a)=l, S(p+a)<^(p4-a)=l; whence, since Sp4>p = Sp' 4>p = 1 , 2Sa<^p'+Sa<;f)a = 0, 2 Sap + Sa^a = 0. S/3 <^a — S/j^a = 0, Subtracting or 13. The rectangle of the perpendiculars from the foci on the tangent is constant, and equal to the square of the semi-conjugate axis. Fi-. S3. Let the tangent be drawn at r (Fig. 83) and or = p. Tlien <}>p is parallel to the normal at r, that is, to the perpendiculars FD, f'd! Hence, of being a, OD ^ X (j>p — a, OD = a + Xc^tp, which, since d and d' are on the tangent, in Sttc^/d = 1 give S(.7; <^p — a) <^p = 1 , S(a + Xcf)p)(l>p= 1, or X'(cf>py=\ -hSacf^p, X (p 4>p FD X f'd' = T f,,f_rrl -(Sac^p)^ But iM' f 0?p + aSapY _ a^cv^p-) + 2 ff^(Sap)^ + a^ Sap)^ or, substituting a'p' from Equation (23-4) and a- = — a^e^, _ (Sap)2-fl* Also l-(Sa<^p)2=l ft" Sap + a' Sap a*-(Sap)^ • •. FD X f'd' = T ft^l -e-) ft^-(Sap)- ft'^(l-e^) a'' (Sap)- — a -^=a-(l~e2) = 52 14. The foot of the perpendicular from the focus on the tangent is in the circumference of the circle described on the major axis. To prove this we have to show that the line od (Fig. 83) is equal to a. Now CD = a + Xp{\—^a4p) ="+ (#)= from the preceding example. Hence 2Sa<^p(l -Sa<^p) , (1-Sa(^p)2 (oD)- = a2 4-: {^pY = — a^e^ — a^(l — e-) = — a^ ; {i>py ft^-(Sa.p)^ft'^(l-e') a* (Sap)^— a* 214 QUATERNIONS. The Parabola. 96. 1. Resuming Equation (233) and making e = l, the equation of tlie parabola is a- p- = {a- - SapY (238), which ma}' be written p2 + 2Sap-a-nSap)^ _, '> — '■J a" or „ p + 2 a — a~^aSap~l Sp ^ , L =1; _ a in which, if we put p — a~' Sap 9P = 2 ' a we have for the equation of the paral)ola Sp(p + 2 «-')=! (239), and, as in the case of the elHpse, S(T(f>p = Sp(t>ar (240) . Operating on <^p b}^ S . a x , we obtain Sa<^p = (241); hence, (ftp is a perpendicular to the axis. Operating on ^p b}' S . p X Sp<^p = e^I1^4^^'=a^(<^p)^. . . (242). a 2. Ditforentiatiug Equation (239), we have 2Sp<^dp + 2Sf?pa-i = 0. For an}' point of the tangent Hue to which the vector is tt, ~ — p + xdp, APPLICATIONS TO LOCI. 215 from which, substituting dp in the above, Sp(7r-p) +S(7r-p)a-i = 0, S(pp + a~^ is in the direction of the normcd. 4. If o- be a vector to an}' point of tlie normal, the equation of the normal will be a = p + X(cf,p + a-') (244). 5. The Cartesian form of Equation (239) is obtained b}' making p = xi + yj, a = FO (Fig. 80) = -jn ; xpi ^i + yJ whence, Equation (239) becomes yi-zl=i- F P ••• y- = '2px-\-i?, the equation of the parabola referred to the focus. 216 QUATERNIONS. 97. Examples. 1. Tlte tiublaityent is bisected at (he vertex. rig. SI. \ \ ^ ^\ \ \ ^^ -^x / '■ \ ^ y / / 1 \ j j ! i \ \ T () A V 1 \ .M N gives («) We have (Fig. 84) ft = .ra, which in the equation of the tangent S7r(<^p + a-^) + Spa-i=l Sxa(p + a~^) + Sa~*/3 = 1. But Sa<^/3 = ; hence rc + Sa-V= 1 ; multipl3'ing by a SCa + aStt" p =: a, (.r — i)a = a — ^-a — aSa~'p = ^a — aSa"Vi AT = — AF — aSa"'p. But the value of <^/j gives a* ^/j = p — a" Sap ; APPLICATIONS TO LOCI. 217 and, since ^p is a vector along bip and a~^^ap a vector along fm, from p = FM + MP we have FM = a~' Sap = aSa~^p, (6) - MP = a^(jip; (c) .'. AT = — AF — FJI = — AM, or, as lines, AT = AM. 2. The distances from the focus to the point of contact and the intersection of the tangent ivith the axis are equal. or (Fig. 84), a;a = a — aSa ^p, (FT)2 = (a-aSa-V)' = (a-a-^Sap)2 (a^ — Sap)^ [Eq.(238)] =p^ .'. FP = FT. 3. The siiJmormal is constant and equcd to hcdf the parameter. The A^ector-normal being (/)p + a~^ (Art. 96, 3), we have (Fig. 84) PN = 2(^p+a ^); but PN = PM + JIN = - a-<^p + xa ; [Ex. 1, (e) ] Z =■ — a" = X'a', or X' = — 1 , Xa=- — a ; or, the distances mn and fo are equal, and the subnormal = p., a constant. 4. The perpendicular from the focus on the tangent intersects it on the tangent at the vertex, and aq = ^mp (Fig. 84). 218 QUATERNIONS. Since (Ex. 2) fp = ft = rn, fd is perpciulicular to rr or par- allel to px. Otherwise : NP = - 2;(<^a + a-') = a-(^p + a"') (Ex. 3) = a-(f)p 4- a = MP + FO, [Ex. 1, (c)] . But ^ FI) = FQ = Tf (i'p + ^ a = ^ d-cf>p + FA ; .'. ^a-p = AQ = ^MP. 5. To find the locus of the. intersection of the perpejicUcukw from the vertex on the tangent and the diameter ^yroduced through the x>oint of contact. Fig. 84. Let Fs = C7 (Fig. 84) be a vector to a point of the locus. Then FS = FA + AS = FP -f PS, p -\- a-^) = p + Xa. APPLICATIONS TO LOCI. 219 Operating with x S . cf)p. tlien, since Sa^p = [Eq. (241)], z{cf>py- = Sp<^p = a^(0p)^ ; [Eq. (242)] .•. Z ^ a", and (r= ^a + ct"((^p + a~') = l^a + orcjip, ov Operating with x S . a S(cr — |^a)a= 0, So■a = -f(Ta)^ or [Eq. (180)], the locus is a right line perpendicular to the axis and fp distant from the focus. 6. To find the locus of the intersection of the tangent and the perpendicidar from the vertex. If the origin be taken at the vertex, then since ^p + a~' is a vector along the normal, the equation of the locus will be 7r = .r(•(!— cos^), and the equation of the c^x-loid is p = {e- sin^)a + (1 - cose)(3. 2. The vector along the tangent is (1— cose)a + sin6 . /?, and the equation of the tangent is Tr = (e- sin^)a + (1 - cos^)/? + ^[(1- cos^)a + sin^ . y?]. 3. The vector from r to the lower extremit}- of the vertical diameter of the generating circle through p is PC = — (\ — cos6)f3 + sm6 . a, and, from the above expression, for the vector-tangent pt, S(pc . pt) = ; hence pc is perpendicular to the tangent, or the normal passes through the foot of the vertical diameter of the generating cir- cle for the point to which the normal is drawn, and the tangent passes through the other extremity. 4. If, through p, a line be drawn i)arallol to the base, intersecting the central generating circle in q, show that PQ = r(7r — ^) = arcQA, a being the upper extremitj- of the axis. APPLICATIONS TO LOCI. 223 5. TTith the notation of Ex. 6, Art. 86, p' = (1 — cos 6)a+ sin 6 . /?, p'2 = _ [ ( 1 - COS 6) ' + sin- ^] r, Tp' = r Vl — 2 cos 6 + cos- d + sin- (9 = ?-V2 — 2cos^ = 2rsin^e; /•" " s — So = I "2 r sin|^ = [4 r cos|^] _ = 8 r, the length of the entire cnrve. 6. With the notation of Ex. 7, Art. 86, TTpp'= TV[ (^ - sin 0) sin ^ . a^S + (1 - cos Oy-jSa'] = TY[(^ sin^ - sin-^ -(1- cos^)-]a^ = 9-(^sin^ + 2cos^-2). A -Ao = 7-2 fie sin 61 + 2 cos ^ - 2) = r^(sin^-^cos^ + 2sin^-2^) ^(3siu^-^cos^-2^) the whole area of the cun^e. = 3 Trt-^, 2 7T- 99. Elementary Applications to Mechanics. 1. If 6 be the magnitude of any force acting in a known di- rection, the force, as having magnitude and direction, may be represented bj' the vector symbol /3, which is independent of the point of application of the force. In order, completely, to define the force with reference to an}' origin o, the A-ector OA=a, to its point of application a, must also be given. For concur- ring forces, whose magnitudes are b', b',' , we have, for the resultant, (3 = 2y3,' which is true, whether the forces are compla- nar or not, and is the theorem of the polygon of forces extended. For two forces, /S = /?'+ /3" ; whence (S' = /3'- + /S"- -f 2 S/S'^", or 224 QUATERNIONS. h^ = &'2 ^_ 7/-2 ^ 2Z;'i" COS 5, which is the theorem of the parallelo- gram of force a. For an}' number of concurring forces, the con- dition of c'(iuilil)riuin will be 1(3'= 0. For a particle constrained to move on a phine curve whose equation is p=z (f){^t), dp being in the direction of tlie tangent, since the resultant of the eA traneous forces must be normal to the curve for equilibrium, we have Sdp1/3'=Sdp/3 = 0. (a) 2. If OA'=a; and (3' is a force acting at \', then T\a'/3'=a'b' sin^ is the numerical value of the moment of the couple f3' at a' and — /3' at o. Kepresenting, as usual, the couple bv its axis, its vector s^mViol will be \a'/3'. If — /S' act at some point other than the origin, as c' and oc'= y', the couple will be denoted by V(a'— y')/?.' From this vector representation of couples, it fol- lows that their comjjosition is a process of vector addition; hence the ref^idtant coiqyle is 2V(a'— y')y3J and, for equilibrium, 2V(a'— y ')/?'= 0. If the couples are in the same or parallel planes, their axes are parallel and T2 = 2T. Since a!—y' is independent of the origin, the moment of the couple is the same for all points. Since V(a'— y')/S'= Va'/3'— Vy'/?^ the moment of a coxiple is the cdrjehraic sum of the moments of its component forces. If the forces are concurring, and a' is the vector to their common point of application, 2Va'/S'= y2a'^'= Va'2/3' = Va'^, or the moment of the restdtant about any point is the sum of the moments of the component forces. "When the origin is on the resultant, a' coincides with /?' in direction, and Ta'/S = ; or the cdgehraic sum of the moments about any point of the resultant is zero. If a single force /3' acts at a( we ma}-, as usual, intro- duce two equal and opposite forces at the origin, or at any other point c' and thus replace /S'^. by /?'o and Ma'f^', or by /J'^. and V(a'— y')(S'. If ^ be a unit vector along an)' axis oz through the origin, then the moment of (3' acting at \', with reference to the axis oz, will be - S/8'a'^, or - S . O'^S'^: If /S' and C are in the same plane, in which case they either intersect or are parallel ; or, if the axis passes through a' there will be no moment: in these cases, a', /3' and C are complanar, and — S/3'a'{ = 0. APPLICATIONS TO LOCI. 225 3. If the forces are parallel, theli- resultant /? = S/3'= 2&'Uy8' = U/326' ; and, therefore, for equilibrium, 2T/3'= %h'= 0. The moment of a force with reference to any axis oz through the origin being — S^S'a'^, and the moment of the resultant being equal to the sum of the moments of the components, we have Sy8aC=2Sy8'a'^, which, for parallel forces, becomes S(2&' . U/? . a^) = S(U;826'a' . ^), which, being true for an}- axis, is satisfied for 2&' . a = -^b'a' ; 25'a' 26' (&) which is independent of U;8, and hence is the vector to the cen- ter of ixtrallel forces. When 26'= 0, the abote equations give /3 = and a = oo, the sj^stem reducing to a couple. For a sys- tem of particles whose weights are to', ^v',' , we have the vec- ^ f t tor to the center of gravity a = —. From this equation, 2^o' 2^o'(a — a') = ; whence, if the particles are equal, the smn of the vectors from the center of gravity to each particle is zero ; and, if unequal, and the length of each vector is increased propor- tionatety to the weight of each particle, their sum is zero. For ?o'2a' equal particles, a = -, or the center of gravity of a system of '2,10' equal x)o,rticles is the mean point (Art. 18) of the polyedron of tvhich the pxirticles are the vertices. For a continuous body whose weight is 20, volume i', and density d at the extremity of a, a = , in which 2 may be replaced b}' the integral sign if the density' is a known function of the volume. For a homo- geneous body, a = , which is applicable to lines, surfaces '^clv or solids, v representing a line, area or volume. Thus, for a plane cm've p= (^{f) = a,' civ = els = Idp = Tcf)'(t)dt and '-!—^ _. (c) CTcl>'{t)dt 226 QUATERNIONS. 4. General conditions of equilihrium of a solid hodij. Lot the forces ft', /?'' , act at the points a,' a" of a solid body, and oa' = aj OA" = a," Replacing cacli force l)y an equal one at the origin and a couple, the given s3-steni will be equiva- lent to a sjstem of concurring forces at the origin and a S3stem of couples. Hence, for equilibrium, 2/3'= 0, (d) 2ya'/3'=0. (e) Let i be the vector to any point x. Then, from (rf), V . ^2/8'= 0, and therefore, from (e), V . ^2/3'= SVa'/S' ; whence 2V/5'a'- SY/S't^ = 2y/3'(a'- c?) = 0. (/) Converseh', | being a vector to any point, the resultant couple, for equilibrium, is 2y(a'- ^)/3'= ; .-. 5ya'^'= (» and 1(3'= 0. Therefore (/) is the necessary and suliicient condition of equi- librium. This condition mav be othem-ise expressed by the principle of virtual moments. Let 8' h" be the displacements. Tiien the virtual moment of (i' is — S,8'3' ; and, for equilibrium, 2S/3'S'=(). This equation involves ((Z) and (e). Thus, if the displacement corresponds to a simple translation, S'=8"=8"' = etc. = a constant, and we maj- write 2S^'5' = SS2/3' = ; whence, since 8 is real, 2/3'= 0. Again, if the displacement corrcsi)onds to a rotation about an axis ^, C being a unit vector along the axis, a'= tHa' = t\»W+ na') = - ^W - ^^1 the last term being a vector perpendicular to the axis. For a rotation about this axis through an angle 0, this term becomes — C^ C^ia'= — C cos^ y^a'+ sin^ V^aJ and a' becomes a'l = - CSCa' - ^ COS ^ \Ca' + sin 6 YCal which, for an infmiteh' small displacement, APPLICATIONS TO LOCI. 227 Placing the scalar factor under the vector sign and writing ^ simply for Ot,, to denote the indefinitely short vector along oz, a'+8'=a'+\V; or, 8'= YCa! Hence 2Sy8'8'= SS/3'VCa'= S^2Va'/3' ; or, since C is not zero, 2Ya'^'= 0. 5. Illustrations. (1) Three concurrent forces, represented in magnitude and direction by the medials of any triangle, are in equilibrium. (See Ex. 2, Art. 17.) (2) If three concurring forces are in equilibrium, they are complanar. By condition, jS' -\- /3"+ (3'"— 0. Operating with S . /S'/5"x , we have S/3'/S"/5"'= 0. (3) In the preceding case, operating with V. /S'x, we have V/3'/3"+V/?'/?"'= ; whence, since the forces are complanar, TV/37i"=- TY f3'l3'," or b'b" sin(y8; /3") = bV" sm{/3', ft'") . A sim- ilar relation may be found for an}' two of the forces ; whence b':b":b"': : sin(/3;' /5"'): sm{f3', /3"'): sin (ft', ft"). (4) If two forces are represented in magnitude and position b}' two chords of a semicircle drawn from a point on the circum- fei'ence, the diameter through the point represents the resultant. (5) A weight, iv', rests on the arc of a vertical plane curve, and is connected, by a cord passing over a pulley, with another weight, ivi' Find the relation between the weights for equili- brium. (a) Let the curve be a parabola, and the pulley at the focus. Then, from Eq. (a) of this article, the equation of the curve be- ing p = — (y--2r)a + >jft, we have 228 QUATERNIONS. ill which r = radius vector. Hence w ?/ X y p pr r ' or, since r = rc + ^), iv'=w'.' Ilcnce, if the weights are equal, equilibrium will exist at all points of the curve. (b) Let the curve be a circle and the pullev at a distance m from the curve on the vertical diameter produced. "With the origin at the highest point of the circle, p = xa -f '^'2iix — a:^(i. Hence, r being the distance of the pulley from iv', <'t-'^+°) ('"'"-' r'>^. riv' K + m' (c) Let w be placed on the concave arc of a vertical circle, and acted upon b}" a repulsive force varying inversely as the square of the distance from the lowest point of the circle. To find the position of equilibrium. The origin being at the lowest point of the circle, and r the distance required, let ]) be the intensity of the force at a unit's distance ; then 4 will be its intensit}' for any distance ?•, and ^ / , n — x \ fxa + ?//? p \ n whence , — r = \i — ((?) Let w^ rest on a right line inclined at an angle b to the horizontal, and connected with h;" by a cord passing over a pul- ley at the upper end of the line. Find the relation between the weights. With the origin at the lower end of the line, its equa- tion is p = xa. If (3 is in the direction of iv', then Sa(i(j'/3-)-M;"a) = 0; .'. ic"=io'iiinO. (6) To find the center of gravity of three equal particles at the vertices of a triangle, a, b, c being the vertices, the vector APPLICATIONS TO LOCI. 229 from A to the center of gravity of the weights at A and b is ^AB = AD. The vector to the center of gravit}' of the three weights is ■J(ab + ac) = |-ab -(-a;DC = ^ab + a;( — |^ab + ac) ; .'. 0;=^, and the required point is the center of gravity of the triangle. (7) Find the center of gravit}' of the perimeter of a triangle. (8) Find the center of gravit}' of four equal particles at the vertices of a tetraedrou. (9) Show that the center of gi'avity of four equal particles at the angular points of an}' quadrilateral is at the middle point of the line joining the middle points of a pair of opposite sides. (10) The center of gravity of the triangle formed b}' joining the extremities of perpendiculars, erected outwards, at the mid- dle points of au}' triangle, and proportional to the corresponding sides, coincides with that of the original triangle. Let abc be the triangle, bc = 2 a, ca = 2/S and e a vector perpendicular to the plane of the triangle. Then, if m is the given ratio, b the initial point, and Ri, r^, Rg'the extremities of the perpendiculars to BC, CA, AB, respectivel}', BRi = a + mea, BR2 = 2 a + ^ + me/3, BRg = a -\- (3 — me (a +/S) ; .-. i(BRi + BRo + BR3) = i(4a+2/?) = i[2a+2(a + /3)]. (11) To find the center of gravity of a circular arc. The equation of the circle p = ?-(cos^ . a + sin^ . ^8), gives clp = r{—sine . a + cos^ . ^)d9; Ccl>{e)Tcf,'(0)cl6 Cr(cose . a + sin^ . ft)d9 CT'(6)cl6 Ccie For an arc of 90° integi-ating between the limits - and 0, tti = — (a + /3) , the distance from the center being — V2 ; which 230 QUATERNIONS. may be obtained directl}' also by integrating between the limits - and — -. For a semicircumfereucc or arc of 00° we have, in like manner, — and — . (12) If a, y8, y are the vector edges of any tetraedron, the origin being at the vertex, then p — a, /S — y, a — /? ai'e hnes of the base, p being an}' vector to its plane. Hence this plane is represented by S (p — a) (/J — y) (a — ^) = ; .-. Sp {\afS + Vya + V/iy) — Huf^y = U. If 8 be the vector perpendicular on the base, 8 = x(Yal3 + Vya + Y^y) = .. ,, J"^ ^^—^^ and, taking the tensors, T(y«^ + y/5y + Vya) = ^JLl^L = 2 area base. But Va/3 + V/?y + Vya + V/?a + Vy^ + Vay = 0, iu which the last terms are twice the vector areas of the plane fiices. The sum of the vector areas of all the faces is therefore zero. Since an}- pol3'edron may be divided into tetraedra by.plane sections, whose vector areas will have the same numerical coefficient, but have opi)osite signs two and two, the sum of the vector areas of any polyedron is zero. These vector areas represent the pres- sures on the faces of a polyedron immersed in a perfect fluid subjected to no external forces. For rotation, since the points of application of these pressures are the centers of gravity of the faces, to which the vectors are i(a+^ + y), i(/3 + a), Uy + fS), i(a + 7), we have the couples --JVS(a + /8+y)(Va/3+V^y + Vya)-f-(a + /3)V^a+(/3 4-y) yyi3 + (y + a)Vayj = -lV(aV/Sy + y8Vya + yVa)8), since aVa^ + aVy8a = 0, etc. But, Equation (123), this sum is zero. Hence there is no rotation. MISCELLANEOUS EXAMPLES. 231 100. Miscellaneous Examples. 1. lu Fig. 58, F, A aud k are colliuear. 2. lu Fig. 58, AD" — AE^ = AB- — AC^. 3. lu Fig. 13, if the lines from the vertices of the parallelo- gram through o aud p are augle-bisectors, omiip is a rectangle. 4. If the corresponding sides of two triangles are in the same ratio, the triangles are similar. 5. ^, a, y being the vector sides of a plane triangle, if (3—a-{-y, show that &-=c-— ca cos B+a6 cose. 6. The sides bc, ca, ab of a triangle are produced to d, e, f, so that CD = 7/iBC, AE = ?JCA, BF =pAB. Find the inter- sections Qi, Qo, Q3 of EB, FC ; FC, DA ; DA, EB. 7. In any right-angled triangle, four times the sum of the squares of the medials to the sides about the right angle is equal to five times the square of the hypothenuse. 8. If ABC be any triangle, m its mean point, and o any point in space, then AB-4- BC-+ OA- = 3(OA-+ OB-+ OC-) — (3 Om)^. 9. If ABCD be any quadrilateral, m its mean point, and o any point in space, tlieu AB-+ BC-+ C'D-+ DA- = 4(OA-+ OB-+OC--j-OD^) — (4om)^— AC^— BD-. 10. If ABC be any triangle, and c', b', a' the middle points of AB, AC, CB, then, o being any point in space, AB- + BC- + CA- = 4(0A- + 0B- + 0C-)— 4(ob'-+OC'--|-Oa'-). 11. If ABC be any triangle and 51 its mean point, then AB-+ BC--f CA- = 3 (aM-+ BM-4- CM^) . 12. Points p, Q, R, s are taken in the sides ab, bc, od, da of a parallelogram, so that ap = 7?iab, bq = ?nBC, etc. Show that PQRS is a parallelogram whose mean point coincides with that of abcd. 232 QUATERNIONS. 13. The sides of any quadrilateral are divided equably at p, q, R, s, and the points of division joined in succession. If PQKS is a parallelogram, the original quadrilateral is a parallelogram, 14. The middle points of the three diagonals of a complete quadrilateral arc coUinear. 15. If any quadrilateral be divided into two quadrilaterals by any cutting line, the centers of the three are collinear. 16. If a circle be described about the mean point of a paral- lelogram as a center, the sum of the squares of the lines drawn from any point in its circumference to the four angular points of the parallelogram is constant. 17. A quadrilateral possesses the following property : any point being taken, and four triangles formed by joining this point with the angular points of the figure, the centers of gi-avity of these triangles lie in the circumference of a circle. Prove that the diagonals of this quadrilateral are at right angles to each other. 18. The sum of the vector perpendiculars from a, b, c, on any line through their mean point is zero. 19. a, b, c are the three adjacent edges of a rectangular paral- lelopiped. Show that the area of the triangle formed by joining their extremities is ^V6"(r+ crcr+a-b-. 20. Given the co-ordinates of a, b, c, d referred to rectangular axes. Find the volume of the pyramid o— abcd, o being the origin. 21. Any plane through the middle points of two opposite edges of a tetraedron bisects the latter. 22. The chord of contact of two tangents to a circle drawn from the same point is perpendicular to the line joining that point with the center. 23. If two circles cut each other and from one point of section a diameter be drawn to each circle, the line joining then- extremities is parallel to the line joining their centers, and passes through the other point of section. MISCELLANEOUS EXAMPLES. 233 24. The square of the sum of the diameters of two circles, tan- gent at a common point, is equal to the sum of the squares of any two common chords through the point of tangency, at right angles to each other. 25. T is any point without a circle whose centre is c ; from t draw two tangents tp, tq, also any line cutting the circle in V, and pq in r ; draw cs perpendicular to tv. Then SR . ST = SV^. 26. If a series of circles, tangent at a common point, are cut by a fixed circle, the lines of section meet in a point. 27. In Ex. 26, the intersections of the pairs of tangents to the fixed circle, at the points of section, lie in a straight line. 28. If three given circles are cut by any circle, the lines of section form a triangle, the loci of whose angular points are right lines perpendicular to the lines joining the centers of the given circles. 29. The three loci of Ex. 28 meet in a point. 30. Given the base of an isosceles triangle, to find the locus of the vertex. 31. Find the locus of the center of a circle which passes through two given points. 32. Find the locus of the center of a sphere of given radius, tangent to a given sphere. 33. The locus of the point from which two circles subtend equal angles is a circle, or a right line. 34. Given the base of a triangle, and 7n times the square of one side plus w times the square of the other, to find the locus of the vertex. 35. Given the base and the sum of the squares of the sides of a triangle, to find the locus of the vertex. 36. In Ex. 35, given the difference of the squares, to find the locus. 234 QUATERNIONS. 37. OB and oa are any two lines, and mp is a line parallel to OB. Find the locus of the intersection of oq and bq drawn parallel to ap and op, respectively. 38. From a fixed point p, on the surface of a sphere, chords pp', pp", are drawn. Find the locus of a point o on these chords, such that pp'. po = m'. 39. A line of constant length moves witli its extremities on two straight lines at right angles to each other. Find the locus of its middle point. 40. Find the locus of a point such that if straight lines be drawn to it from the four corners of a square, the sum of their squares is constant. 41. Find the locus of a point the square of whose distance from a given point is proportional to its distance from a given line. 42. Find the locus of the feet of perpendiculars from the origin on planes cutting off pj'ramids of equal volume from three rectangular co-ordinate axes. 43. Given the base of a triangle and the ratio of the sides, to find the locus of the vertex. 44. Show that TapYp/? = (Ja/3)- is the equation of a hyperbola whose asymptotes are parallel to a and /8. 45. Find the point on an ellipse the tangent to which cuts off equal distances on the axes. 4G. A and b are two similar, similarly situated, and concentric ellipses ; c is a third ellipse similar to a and b, its center being on the circumference of b, and its axes parallel to those of A and b : show that the chord of intersection of A and B is parallel to the tangent to u at the center of c. Presswork by Ginn & Co., Boston. MATHEMATICS. 165 Wentworth & Hill's Exercises in Algebra. I. Exercise Manual. i2mo. Boards. 232 pages. Mailing price, 40 cts. ; Introduction price, 35 cts. — II. EXAMINATION Manual. i2mo. Boards. 159 pages. Mailing price, 40 cts. ; Introduction price, 35 cts. Both in one volume, 70 cts. Answers to both parts together, 25 cts. The first part (Exercise Manual) contains about 4500 problems c/assified and arranged according to the usual order of text-books in Algebra ; and the second part (Examination Manual) contains nearly 300 examination-papers, progressive in character, and well adapted to cultivate skill and rapidity in solving i^roblems. Wentworth & Hill's Exercises in Arithmetic. I. Exercise Manual. II. Examination Manual. i2mo. Boards. 148 pages. Mailing price, 40 cts.; Introduction price, 35 cts. Both in one volume, 70 cts. Anszvers to both parts together, 25 cts. The first part (Exercise Manual) contains problems for daily practice, classified and arranged in the common order; and the second part (Examination Manual) contains 300 examination-papers, progressive in character. The second part has already been issued? and the first part will be ready in August, 18S6. Analytic Geometry. By G. A. Wentworth. i2mo. Half morocco. 000 pp. Mailing price, $0.00; for Introduction, The aim of this work is to present the elementary parts of the subject in the best form for class-room use. The connection between a locus and its equation is made perfectly clear in the opening chapter. The exercises are well graded and designed to secure the best mental training. By adding a supplement to each chapter provision is made for a shorter or more extended course, as the time given to the subject will permit. The book is divided into chapters as follows : — Chapter I. Loci and their Equations. " ' II. The Straight Line. " III. The Circle. " IV. Different Systems of Co-ordinates, " V. The Parabola. " VI. The Ellipse. " VII. The Hyperbola. " VIII. The General Equation of the Second Degree. 1 SS MA THE MA TICS. Peirce's Three and Four Place Tables of Loga- rithmic and Trigonometric Functions. By James Mills Peikce, University Professor of Mathematics in Harvard University. Quarto. Cloth. Mailing Price, 45 cts. ; Introduction, 40 cts. Four-place tables require, in the long run, only half as much time "iS five-place tables, one-third as much time as six-place tables, and one-fourth as much as those of seven places. They are sufficient for the ordinary calculations of Surveying, Civil, Mechanical, and Mining Engineering, and Navigation ; for the work of the Physical or Chemical Laboratory, and even for many computations of Astron- omy. They are also especially suited to be used in teaching, as they illustrate principles as well as the larger tables, and with far less expenditure of time. The present compilation has been prepared with care, and is handsomely and clearly printed. Elements of the Differential Calculus. With Numerous Examples and Applications. Designed for Use as a College Text-Book. I5y W. E. Bvekly, Professor of Mathematics, Harvard University. 8vo. 273 pages. Mailing Price, $2.15 ; Intro- duction, S2.00. This book embodies the results of the author's experience in teaching the Calculus at Cornell and Harvard Universities, and is intended for a text-book, and not for an exhaustive treatise. Its peculiarities are the rigorous use of the Doctrine of Limits, as a foundation of the subject, and as preliminary to the adoption of the more direct and practically convenient infinitesimal notation and nomenclature ; the early introduction of a few simple formulas and methods for integrating ; a rather elaborate treatment of the use of infinitesimals in pure geometry ; and the attempt to excite and keep up the interest of the student by bringing in throughout the whole book, and not merely at the end, numerous applications to practical problems in geometry and mechanics. James Mills Peirce, Prof, of Math., Harvard Univ. (From the Har- vard Register') : In mathematics, as in other branches of study, the need is is general without being superficial; limited to leading topics, and yet with- in its limits; thorough, accurate, and practical ; adapted to the communica- now very much felt of teaching which ' tion of some degree of power, as well MATHEMATICS. 189 as knowledge, but free from details which are important only to the spe- cialist. Professor Byerly's Calculus appears to be designed to meet this want. . . . Such a plan leaves much room for the exercise of individual judgment ; and differences of opinion will undoubtedly exist in regard to one and another point of this book. But all teachers will agree that in selection, arrangement, and treatment, it is, on the whole, in a very high degree, wise, able, marked by a true scientific spirit, and calculated to develop the same spirit in the learner. . . . The book contains, perhaps, all of the integral calculus, as well as of the differential, that is necessary to the ordinary stu- dent. And with so much of this great scientific method, every thorough stu- dent of physics, and every general scholar who feels any interest in the relations of abstract thought, and is capable of grasping a mathematical idea, ought to be familiar. One who aspires to technical learning must sup- plement his mastery of the elements by the study of the comprehensive theoretical treatises. . . . But he who is thoroughly acquainted with the book before us has made a long stride into a sound and practical knowledge of the subject of the calculus. He has begun to be a real analyst. H. A. Ne-wi;on, Prof, of Math, in Yale Coll., New Haven : I have looked it through with care, and find the sub- ject very clearly and logically devel- oped. I am strongly inclined to use it in my class next year. S. Hart, recent Prof, of Math, in Trinity Coll., Conn. : The student can hardly fail, I think, to get from the book an exact, and, at the same time, a satis- factory explanation of the principles on which the Calculus is based; and the introduction of the simpler methods of integration, as they are needed, enables applications of those principles to be introduced in such a way as to be both interesting and instructive. Charles Kraus, Techniker, Pard- tibitz, Bohemia, Austria : Indem ich den Empfang Ihres Buches dankend bestaetige muss ich Ihnen, hoch geehr- ter Herr gestehen, dass mich dasselbe sehr erfreut hat, da es sich durch grosse Reichhaltigkeif,besonders klare Schreibweise und vorzuegliche Behand- lung des Stoffes auszeichnet, und er- weist sich dieses Werk als eine bedeut- ende Bereicherung der mathematischen Wissenschaft. De Volsoa Wood, Prof, of Math., Stevens' Inst., Hoboken, N.f. : To say, as I do, that it is a first-class work, is probably repeating what many have already said for it. I admire the rigid logical character of the work, and am gratified to see that so able a writer has shown explicitly the relation between Derivatives, Infinitesimals, and Differentials. The method of Limits is the true one on which to found the science of the calculus. The work is not only comprehensive, but no vague- ness is allowed in regard to definitions or fundamental principles. Del Kemper, Prof of Math., Hampden Sidney CoU., Fa. : My high estimate of it has been amply vindi- cated by its use in the class-room. R. H. Graves, Prof, of Math., Univ. of North Carolina : I have al- ready decided to use it with my next class ; it suits my purpose better than anv other book on the same subject with which I am acquainted. Edw. Brooks, Author of a Series of Math. : Its statements are clear and scholarly, and its methods thoroughly analytic and in the spirit of the latest mathematical thought. 190 MA THEM A TICS. Syllabus of a Course in Plane Trigonometry. By W. E. Byerly. 8vo. 8 pages. Mailing Price, lo cts. Syllabus of a Course in Plane Analytical Geom- elry. By W. E. Bykkly. 8vo. 12 pages, ilailing Price, 10 cts. Syllabus of a Course in Plane Analytic Geom- ctry (^Advauced Course.) By W. E. Byerly, Professor of Mathe- matics, Plarvard University. 8vo. 12 pages. Mailing Price, 10 cts. Syllabus of a Course in Analytical Geometry of Three Dimensioyis. By W. E. Byekly. Svo. 10 pages. Mailing Price, 10 cts. Syllabus of a Course on Modern Methods in Analytic Geometry. By W. E. Byekly. 8vo. 8 pages. Mailing Price, 10 cts. Syllabus of a Course in the Theory of Equations. By W. E. Byerly. 8vo. 8 pages. Mailing Price, 10 cts. Elements of the Integral Calculus. By W. E. Byerly, Professor of Mathematics in Harvard University. 8vo. 204 pages. Mailing Price, $2.15; Introduction, $2.00. This volume is a sequel to the author's treatise on the Differential Calculus (see page 134), and, like that, is written as a text-book. The last chapter, however, — a Key to the Solution of Differential Equations, — may prove of service to working mathematicians. H. A. Newton, Pro/, of Math., Yale Coll. : We shall use it in my optional class next term. Mathematical Visitor : The subject is presented very clearly. It is the first American treatise on the Cal- culus that we have seen which devotes any space to average and probability. \ Schoolmaster, London : The merits of this work are as marked as those of the Differential Calculus by the same author. Zion's Herald : A text-book every way worthy of the venerable University in which the author is an honored teacher. Cambridge in Massachusetts, like Cambridge in England, preserves its reputation for the breadth and strict- ness of its mathematical requisitions, and these form the spinal column of a liberal education. 192 MA THE MA TICS. Elements of the Differential and Integral Calculus. With Examples and Applications. By J. Mathematics in Madison University. 8vo. price, $1.95; Introduction price, ^1.80. M. Taylor, Professor of Cloth. 249 pp. Mailing The aim of this treatise is to present simply and concisely the fundamental problems of the Calculus, their solution, and more common applications. Its axiomatic datum is that the change of a variable, when not uniform, may be conceived as becoming uniform at any value of the variable. It employs the conception of rates, which affords finite differen- tials, and also the simplest and most natural view of the problem of the Differential Calculus. This problem of finding the relative rates of change of related variables is afterwards reduced to that of finding the limit of the ratio of their simultaneous increments ; and, in a final chapter, the latter problem is solved by the principles of infinitesimals. Many theorems are proved both by the method of rates and that of limits, and thus each is made to throw light upon the other. The chapter on differentiation is followed by one on direct integra- tion and its more important applications. Throughout the work there are numerous practical problems in Geometry and Mechanics, which serve to exhibit the power and use of the science, and to excite and keep alive the interest of the student. Judging from the author's experience in teaching the subject, it is believed that this elementary treatise so sets forth and illustrates the highly practical nature ef the Calculus, as to awaken a lively interest in many readers to whom a more abstract method of treat- ment would be distasteful. Oren Root, Jr., Prof, of Math., Ha^nilton Coll., N.Y.: In reading the manuscript I was impressed by the clearness of definition and demonstra- tion, the pertinence of illustration, and the happy union of exclusion and con- densation. It seems to me most admir- ably suited for use in college classes. I prove my regard by adopting this as our text-book on the calculus. C. M. Charrappin, S.J., St. Louis Univ. ; I have given the book a thorough examination, and I am satis- fied that it is the best work on the sub- ject I have seen. I mean the best work for what it was intended. — a text- book. I would like very much to in- troduce it in the University. (fa7t. 12, 1885.) 194 MA THEM A TICS. Metrical Geometry: An Elementary Treatise on Mensuration. By George Bruce Halsted, Ph.D., Prof. Mathema- tics, University of Texas, Austin. i2mo. Cloth. 246 pages. Mailing price, $l.lo; Introduction, $1.00. This work applies new principles and methods to simplify the measurement of lengths, angles, areas, and volumes. It is strictly demonstrative, but uses no Trigonometry, and is adapted to be taken up in connection with, or following any elementary Geometry. It treats of accessible and inaccessible straight lines, and of their inter- dependence when in triangles, circles, etc. ; also gives a more rigid rectification of the circumference, etc. It introduces the natural unit of angle, and deduces the ordinary and circular measure. Enlisting the auxiliary powers which modern geometers have recog- nized in notation, it binds up each theorem also in a self-e.xplanatory formula, and this throughout the whole book on a system which renders confusion impossible, and surprisingly facilitates acquire- ment, as has been tested with very large classes in Princeton College. In addition to all the common propositions about areas, a new method, applicable to any polygon, is introduced, which so simplifies and shortens all calculations, that it is destined to be universally adopted in surveying, etc. In addition to the circle, sector, segment, zone, annulus, etc., the parabola and ellipse are measured ; and be- sides the common broken and curved surfaces, the theorems of Pappus are demonstrated. Especial mention should be made of the treatment of solid angles, which is original, introducing for the first time, we think, the natural unit of solid angle, and making spherics easy. For solids, a single informing idea is fixed upon of such fecundity as to place within the reach of children results heretofore only given by Integral Calculus. Throughout, a hundred illustrative examples are worked out, and at the end are five hundred carefully arranged and indexed exercises, using the metric system. OPINIONS. Simon Newcomb, Nautical Al- manac Office, Washington, D.C.: iSm much interested in your Mensuration, and wish I had seen it in time to have Alexander MacFarlane, Exam- iner in Mathematics to the University of Edinburgh, Scotland : The method, figures, and examples appear excellent, some exercises suggested by it put into and I anticipate much benefit from its my Geometry. {Sept. 8, 1881.) I minute perusal. MA THEM A TICS. 195 Elementary Co-ordinate Geometry. By W. B. Smith, Professor of Physics, Missouri State University. l2mo. Cloth. 312 pp. Mailing price, ^2.15; for Introduction, 32.00. While in the study of Analytic Geometry either gain of knowledge or culture of mind may be sought, the latter object alone can justify placing it in a college curriculum. Yet the subject may be so pur- sued as to be of no great educational value. Mere calculation, or the solution of problems by algebraic processes, is a very inferior dis- cipline of reason. Even geometry is not the best discipline. In all thinking the real difficulty lies in forming clear notions of things. In doing this all the higher faculties are brought into play. It is this formation of concepts, therefore, that is the essential part of mental training. He who forms them clearly and accurately may be safely trusted to put them together correctly. Nearly every seeming mis- take in reasoniiig is really a mistake in conception. Such considerations have guided the composition of this book. Concepts have been introduced in abundance, and the proofs made to hinge directly upon them. Treated in this way the subject seems adapted, as hardly any other, to develop the power of thought. Some of the special features of the work are : — 1. Its SIZE is such it can be mastered in the time generally allowed. 2. The SCOPE is far wider than in any other American work. 3. The combination of small size and large scope has been secured through SUPERIOR METHODS, — 7>wdern, direct, and rapid. 4. Conspicuous among such methods is that of determinants, here presented, by the union of theory and practice, in its real power and beauty. 5. Confusion is shut out by a consistent and self-explaining NOTATION. 6. The ORDER OF development is natural, and leads without break or turn from the simplest to the most complex. The method is heuristic. 7. The student's grasp is strengthened by numerous exercises. 8. The work has been tested at every point in the class- room. 196 MA THEM A TICS. Determinants. The Theory of Determinants: an Elementary Treatise. By Paul H. Hani.s, U.S., Professor of Mathematics in the University of Colorado. Svo. Cloth, ooo pages. Mailing price, 3o.oo; for Introduction, $o.oo. This is a text-book for the use of students in colleges and tech- nical schools. The need of an American work on determinants has long been felt by all teachers and students who have extended their reading beyond the elements of mathematics. The importance of the subject is no longer overlooked. The shortness and elegance imparted to many otherwise tedious processes, by the introduction of determinants, recommend their use even in the more elementary branches, while the advanced student cannot dispense with a knowl- edge of these valuable instruments of research. Moreover, deter- minants are employed by all modern writers. This book is written especially for those who have had no previous knowledge of the subject, and is therefore adapted to self-instruction as well as to the needs of the class-room. To this end the subject is at first presented in a very siinple manner. As the reader ad- vances, less and less attention is given to details. Throughout the entire work it is the constant aim to arouse and enliven the reader's interest by first showing how the various concepts have arisen naturally, and by giving such applications as can be presented with- out exceeding the limits of the treatise. The work is sufficiently comprehensive to enable the stiident that has mastered the volume to use the determinant notation with ease, and to pursue his further reading in the modern higher algebra with pleasure and profit. In Chapter I. the evolution of a theory of determinants is touched upon, and it is shown how determinants are produced in the process of eliminating the variables from systems of simple equations with some further preliminary notions and definitions. In Chapter II. the most important properties of determinants are discussed. Numerous examples serve to fix and exemplify the prin- ciples deduced. Chapter III. comprises half the entire volume. It is the design of this chapter to familiarize the reader with the most important special forms that occur in application, and to enable him to realize the practical usefulness of determinants as instruments of research. \Ready yune i. MA THEM A TICS. 197 Examples of Differentia/ Equations. By George A. Osborne, Professor of Mathematics in the INIassachusetts Institute of Technology, Boston. i2mo. Cloth, viii + 50 pp. Mail- ing price, 60 cts.; for Introduction, 50 cts. Notwithstanding the importance of the study of Differential Equa- tions, either as a branch of pure mathematics, or as applied to Geometry or Physics, no American work on this subject has been published containing a classified series of examples. This book is intended to supply this want, and provides a series of nearly three hundred examples with answers systematically arranged and grouped under the different cases, and accompanied by concise rules for the solution of each case. It is hoped that the work will be found useful, not only in the study of this important subject, but also by way of reference to mathematical students generally whenever the solution of a differen- tial equation is required. Elements of t/ie Theory of tiie Newtonian Poten- tial Function. By B. O. Peirce, Assistant Professor of Mathematics and Physics, Harvard University. i2mo. Cloth. 154 pages. Mailing price, 5l-6o; for Introduction, $1.50. A knowledge of the properties of this function is essential for electrical engineers, for students of mathematical physics, and for all who desire more than an elementary knowledge of experimental physics. This book, based upon notes made for class-room use, was written because no book in English gave in simple form, for the use of students who know something of the calculus, so much of the theory of the potential function as is needed in the study of physics. Both matter and arrangement have been practically adapted to the end in view. Chapter I. The Attraction of Gravitation. II. The Newtonian Potential Function in the Case of Gravitation. III. The Newtonian Potential Function in the Case of Repulsive Forces. IV. Surface Distributions. Green's Theorem. V. Application of the Results of the Preceding Chapters to Electrostatics. Mathematics. Introd. Prices. Byerly Differential Calculus §2.00 lutegral Calculus 2.00 Ginn Addition Manual 15 Halsted Mensuration 1.00 Hardy Quaternions 2.00 Hill Geometry for Beginners 1.00 Sprague Rapid Addition 10 Taylor Elements of the Calculus 1.80 Wentworth . Grammar School Aritlunetic 75 Shorter Course in Algebra 1.00 Elements of Algebra 1.12 Complete Algebra 1.40 Plane Geometry 75 Plane and Solid Geometry 1.25 Plane and Solid Geometry, and Trigonometry 1.40 Plane Trigonometry and Tables. Paper. . .60 PI. and Sph. Trig., Surv., and Navigation . 1.12 PL and Sph. Trig., Surv., and Tables 1.25 Trigonometric Formulas 1.00 Wentworth & Hill : Practical Arithmetic 1.00 Abridged Practical Arithmetic 75 Exercises in Arithmetic Part I. Exercise Manual Part II. Examination Manual 35 Answers (to both Parts) 25 Exercises in Algebra 70 Part I. Exercise Mximial 35 Part II. Examination Manual 35 Answers (to both Parts) 25 Exercises in Geometry 70 Five-place Log. and Trig. Tables (7 Tables) .50 Five-place Log. and Trig. Tables (Cojh;)..£' J.) 1.00 Wentworth & Reed : First Steps in Number, Pupils' Edition .30 Teachers' Edition, complete .90 Parts I., II., and III. (separate), each .30 Wheeler Plane and Spherical Trig, and Tables 1.00 Copies sent to Teachers for examination, with a view to Introduction, on receipt of Introduction Price. GINN k COMPANY, Publisliers. BOSTON. NEW YORK. CHICA( UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. ^£C 2 3 1921 JAN 2 5 1992 tlWb UBKARY FEB 2 2 1332 EMS LIBRAE m^y /I (112- A^iiii-^^^^ 315 .77 slEX 198 L 1987 I 1988 9 1988 'ED BRARY 01 i Engineering g Mathematical Scjencis Library^* , ^(^ AUXIUARV jyL72 lltefilyo; California, Los a™ L 005 777 017 4 SOUTHERN BRANCH, UNIVERSITY OF CALIFORNIA, LIBRARY, tLOS ANGELES, CALIF.