None
Cylinder, right circular, 45; standing on curve, 65; circular tangent, to radic, 115; case of general quadric, 17; geodesic on, 137; torsal tangent planes of, 140; differential equation of, 149; related to astatics, 161. Cylindroid, 84, 165. Degraded, cases of quaternions, 296. Damped oscillations, 186. Deformation of surfaces, 145. Degraded, cases of quaternions, 9, 19; symbolic equations, 95, 275. Degree, see Order. Degrees of freedom, 204. Delta, Hamilton's operator V, 70,211. See Operator. Degrees of freedom, 204. Delta, Hamilton's operator V, 70,211. See Operator. Determinants and combinatorial functions, 21(1 305. Developable surface, 65; circumscribing confocals, 126; related to curve, 135, 139; generated by tangent planes along curve on surface, 142; of twisted cubic, 287; circumscribing quadrics, tangent-line, of two quadrics, circumscribing generalized con-focals, 286. Development of quaternion function, 79. So; of vector of curve in terms of arc, 134. Deviation from osculating curve, 152. Diaphragm, 217. Dielectric, 251. Difference of two points, 263. Differential, 63, 66; condition for perfect, 74, 80, 214; indeterminate, 87; of equation of surface, 142; equation of geodesic, 141, 152; of lines of curvature, 144, 147; of family of surfaces, 149; of curves traced on surfaces, 287. Differentiation, chap, vn., 62; general formula, 66; successive, 79; following moving point, 229. Diffusion of electromagnetic disturbance, 255. Dilatation in strain, 178. Direct and inverse similitude, 14. Directed area, 23; angle, 31; curvature, 132, 141; volume in hyperspace, 304. Discontinuity in integration, 216. Displacement Of a body, 18, chap. xn., It is: in strain, 150; electric, 250. Dissipation function, 241, 25, 25. Dissociative multiplication, 11. Distortion of elements, 212, 229; of viscous fluid, 238. Distributive, multiplication of vector by scalar, 4; by vector, 8; property of scalar of product, 6; of product, 9; of differential, 66; of linear function, 88; multiplication for hyper-space, 303. Disturbance in electromagnetic field propagated by waves or by diffusion, 255. Divergence of vector, 212. Division, of vectors reduced to multiplication, 11; homographic, 41, 65, 152, 264. Dodecalron, 104. Double points, on wave surface, 248, 261; apparent, 292; on Jacobian, 298. Duality for point symbol, 265. Dynamics, of a particle, chap. xv., 1st; of system and rigid body, chap. xv., 194; of continuous medium, 235; electro-motive, 249. Eight square points of linear function, 99; umbilical generators, 125; u- Generalization of, 279, 28(5; generators which are also normals, 279. Elastic solid, isotropic; 222, 239; aniso-tropic, 242 et seq.; symmetry, 245. Electro-magnetic theory, 249, 7; of light, 256. Element, rate of change of, 212, 229. FJ> -mi iifx of Qmih nrion* referred to, 1, 3, 7, 29, 31, 34, 45, 53, 55, 56, 59, 82, 85, 114, 118, 120, 121, 132, 156, 157, 197, 211, 264; appendix to, 99, 135, 211. Elimination of a vector, 39, 10. Ellipse, vector equation of, 63, 82; projected into circle, 83; parallactio, 85; abbreviational, 85; differential equation of surface generated by, 149; related to astatics, 163; locus of feet of perpendiculars on generators of cylindroid, 166; in conical refraction, 261. Ellipsoid, Hamilton's construction for, 114; vector equation of Ellipsoidal linear function, 178. Elliptic, logarithmic spiral, 82; coordinates, 124; functions, 198; generalized, coordinates, 2Sl>. Elongation, 181. Emanant, 131, 138. Energy equation, for particle, 184, 187; system of particles, 194; rigid body, 197; for impulses, 200; for impulses, 200; for elastic solid, 243; for dielectric, 252. Envelope, examples, 128, 129: differential equation of, 149, 151; wave surface as, 248, 257. Epicycloid, 83. Equality of vectors, 1: vector-arcs, 17; points, 263. Equilibrium, static, 156; astatic, 160. Equipotential surfaces, 227. Euler's angles, 33; four square identity, 16; exponential formulae, : theorem on curvature, 143; equations of motion of rigid body, 196; of thud, 230, 23s. Evoked wrench, 204. Evolutes on polar developable, 139. Sec Linear vector function, the use of ;ui ellipsoid Being to a great extent superseded. Exact differential, 74, 86, 214. Excentricity of orbit, 187. Excess, spherical, 33. Expansion, of quaternion function, 79, 85 ; of vector of curve in terms of arc, 134 ; in series of spherical harmonics, 223, 224. Exponential of quaternion, 28, 34 ; differential of, 86 ; for hyperspace, 308. Extreme points on line of congruency, 154. Families of curves and surfaces, 148. Family of equipotential surfaces, 227 ; of curves, q= (f+t)ma, 286 ; q=e(-fa, 301. Five vectors, 43, 44, 54 ; quaternions, 43, 269 ; points linearly transformed into live, 272 ; surfaces, 291. Flat space, 303 ; symbol of, 308. Flow of a vector, 231. Fluid, motion, 72, 229, 236 ; viscous, 238, 240 ; motion of solid in, 241. Flux through circuit, 233 ; strength of tube of, 233, 235 ; in electro-magnetic Theory, 249; of radiated energy, Poynting, 252, 257. Focal, property of quadrics, Salmon's, 114; form of equation, 116; for sphere-conic, 120; comes on developable, 126; points on line of concubine, 153; comes related to astatics, 162. Foci of central sections of quadric, 129. Force, moment of, 23; in statics, 156; in dynamics, 184, 194; central, 186; impulsive, 200; electric and magnetic, 251; in electro-magnetic field, mechanical, 255. Forces, reduction to two, 158; conservative, 184, 238; of interaction, 194; system of forces, see Wrench. Formula, A, 11; B, 8; of differentiation, 66. Formula, depending on products of vegetarians, chap, in., 23; of trigonometry, 25, 30. Four numbers involved in quaternion, 9; squares, identity connecting, 16; vegetarians, identities connecting, 24; symmetrical relations for, 42; linear function rendering four vectors parallel to, 92; particles equivalent to rigid body, 199; system of screws, 206; Consecutive points on tangents to surface, surface through, 298. Fourth proportional to three vectors, 31. Fractions, relations reduced by partial, 122. Freedom, degrees of, 204. Fresnel, 163, 262. Frictional constraint, 190. Function, anharmonic, of collinear points, 41, 45, of points in space, 56, on a conic, 267; linear vector, 88; elliptic, 198; dissipation, 240; energy, for elastic solid, 243, for dielectric, 252; combinatorial, 270, 304; linear quaternion, 272. See Linear function. Fundamental formulae of trigonometry, plane, 25; spherical, 30. Gate, self-closing, 207. Gauss, operator, 104; measure of curvature, 144, 147; integration theorem, 215. Generalised, normal, 279; curvature, 286, 295; geodesic, 287. Generation of ruled quadric, 65 : of ellipsoid, 114 ; of ruled surface, 137- Generators of quadric, 103, 116; um- bilical, 125 ; common, and of linear complex, 278 ; generalized umbilical, 279, 286 ; eight, are also normals, 279 ; complex of, of doubly infinite family of quadrics, 301. Geodesic on cylinder, 137 ; differential equation of, 141, 152; curvature, 141, 148 ; Joachimstal's theorem, 152 ; motion of particle along, 190 ; gen- eralized, 287. Geometrical meaning of invariants, 98, 288. Geometry of Three Dimensions, Salmon's, 291, 292, 298. Geometry, projective, chap, xvn., 263, 308. Gilbert's theorem on confocals, 124. Grassmann, 306. Graves, R. P., Life of Hamilton re- ferred to, 16,211. Gravitating body in field of force, 225. Green's theorem adapted to quaternions, 219. Groups, theory of, examples relating to, 80 ; referred to, 104. Half-line, half-cone, 45. Harmonic, mean of two vectors, 41, 50, 56, 109 ; properties of triangle, 45, of polar and quadric, 50, 109 ; spherical, 70, 76, 222 et seq. Hathaway, A. S., 270. Heaviside, Oliver, 11, 249, 250, 253. Helix, vector equation of, 64, 82 ; vector twist of, 133 ; constant curva- ture and torsion, 137 ; osculating, 152 ; particle moving on, 191. Herpolhode, 198. Hessian of surface, 297. Hexagon, Pascal, 121. INDEX, 315 Ht<ih> r Plant Cnm*, Salmon's, referred t-, 101, 105. Holograph, 83, 187, 189. Homographic, ranges, 41, 42, 264 ; locus of line joining corresponding points, 65 ; locus of mean centre of points on, 152 ; screw-systems, 208 ; correspondence of points on twisted cubics, 285. Homography, chiastic, 208. Hooke's law, '243. Hydrodynamics, 72, 228 tt .*»/. Hyperbola. 64; section of quadric, rectangular, 111 ; focal, 114. Hyperboloid, homographic generation, 65, 264; locus of transversals, 103, 27; generators of, 116; line of traction of, 140; equilibrating forces on generators of, 158. Hyperspace, chap. xviii., 303. Hyperspace, S3. Hysteresis, 251. Icosian calculus, 104. Identity, Eider's four square, 16; connecting four vectors, 24; rive quaternions, 269. Ikosahedron, 104. Imaginary, of algebra, 3, 20, 58; with roots of quaternions, 28; roots and axes of linear function, 95, 96, 177; conjugate, vectors, 95, 224, 307; united points of linear transformation, 275, 276. Impact of two bodies, 203. Impulse, 200. Impulsive wrench, 201, 204; generating motion of solid in fluid, 241. Indeterminateness of versor of null quaternion, 19; of tensor of bi-quaternion, 21; of a differential, 87; in solution of equations, 92; of axes of linear function, 95. 96 ; of square roots of function, 99 : in value of function, 216 ; related to conical refraction, 248, 260 : of normal to plane in hyperspace, 3« >3. Index-surface, 248, 261. Induction, magnetic, 250. Inertia function for rigid body, 196 ; Binet's theorem on axes of, 197 ; deduced from observed motion, 199, 21 >2. 207 ; related to Jibw, 22.".. Infinites in field of integration, 216, 219. Infinity, anharmonic equation of plane at, 44, of circle at, 54 ; vector to circular points at, 96, 126, 307 ; vector representing point at, 'J<i3 : equation of plane at, 266. Inflexion on curve, 83. Initial positions in astatics, 160. Inscription of polygon to sphere, 55,.~>ii. Instantaneous twist-velocity, 170, 201 ; orbit, 188. Integrability, condition of. 74. Mi. 21 t. Integrals, line, 73, 215, 219, 231 : surface, 72, 215, 219, 233 ; variation of, 192 Intensity of wrench, 163. Interpretations and formulae, chap in., 23; for protective geometry, 263 et seq. Intersection of, line and plane, 35, 267, 269; planes, 39, 267, 306; two lines, 39, 267; line and sphere, 49; spheres, 50, 54; confocals, 121, 123, 125; quadrics, 285; generalized confocals, 286; curve and complementary, 292; of two surfaces, osculating plane to curve of, 296. Invariants, of linear vector functions, 91, 97; geometrical meaning of, 98; of two functions, 100; derived by operation of V, 102; depending on V, 211; of linear quaternion function 274; of quadrics and linear transformations, 288. Inverse, or reciprocal of vector, 11; of product 12; similitude, 14; transformation, 90; operations of V, 218. Inversion, geometrical, 52, corresponding elements in, 69; of linear functions, 90; of 0-K0, 100. of V, 218 : of linear quaternion function, 273. Involution on ruled surface, 140. Irrotational distribution of vectors, 234. Isotropic solid, 222, 239. Jacobi, differential equations, 86. Jacobian, or functional determinant, 213; of four quadrics, 293; of surfaces, 295, 298. Joachimsta's theorem on geodesies, 152. Joulian waste of energy in electromagnetic field, 252. Kinematics, chap, xn., 168 ; of continuous medium, 228. See also Motion. Kinetic energy, of particle, 184, 187 ; of system of particles, 194; of rigid body, 197 : changed by impulse, 211, 207 : of portion of continuous medium, 239. INDEX. Kinetics of a particle, 184; of rigid body, 194; of continuous medium, 236. Knott, C. G., 11. Lagrange, motion of fluid, 230, 238. Laplace's operator, 75, 227; inversion of, 220. Latent roots of linear function, 93, 96; linear quaternion function, 274, 276. Lectures on Quaternions referred to, 1, 7, 21, 59, 114, 115, 121, 211. Lie, Sophus, 86. Light, electro-magnetic theory, 256. See Optics. Limiting, points of coaxial spheres, 51; ratios, 63. Line, chap, v., 35; six coordinates of 40; inverse of, 53; of striction, 138, 140; of curvature, 144; in point symbols, 266; traced on surface, 287, 295. See Complex, Curve, Generator, Interrogative. Linear, relation connecting four vectors, 5, 24, 25; and distributive function, 66; vector function, chap. VIII., 88; related to quadrics, chap. ix. 106 ; to surfaces, 142 ; to astatics, 159 ; to theory of screws, 164, 205 ; to acceleration of point of body, 172 ; to strain, 177 ; to angular momentum of rigid body, 196 ; to angular momentum of rigid body, 196 ; to operator V, 211 ; to stress, 237, 243 ; to electro-magnetic field, 251 ; to theory of light, 258 ; equation of continuity, 230 ; relations connecting five quaternions, 268 ; equation of continuity, 230 ; relations connecting five quaternions, 268 ; to magnetic force, 249 ; to maximum, 80, 111, 127. Logarithmic spiral, 82, Lorentz, H. A., 229, 249, 251. Lunar theory, example on, 188. M'Aulay, A., 21, 211, 218. MacCullagh, index-surface, 248. Magnetic force, 249; permeability, 251. Maximum and minimum, 80, 111, 127. Maxwell, J. Clerk, sense of rotation, 7 ; curl of vector, 213 ; electro-magnetic theory, 249. Mean, point, 5; harmonic, of two vectors, 41, 50, 56, 109; centre of corresponding points, 152; in point symbols, 264. Measure of curvature, 144, 147. Mechanical force in electromagnetic field, 251. Medium, continuous, 228, 236, 251. Meusnier's theorem, 141. Minchin, 183. Minding's theorem, 162. Mölre's, de, theorem, 27. Momentum, of force, 23; resultant, 156; quaternion, 157, 159; of momentum, 184, 195, 196; of inertia, 196. Momentum, 184; moment of, 195, 196; of portion of medium, 236; of solid and fluid, 241. Monomial equation of circle and sphere, 55. Motion, three-bar, 60, 85; of point on curve, 62; generating roulette, 83, 84; apparent, 84; relative, 171, 174; of body under no forces, 198; of continuous medium, 228, 236. Moving axes, 171; for curve, 134; for surface, 14b; for orbit, 188; for body, 196; for electro-magnetic field, 253. Multiple-valued function, 216; point on Jacobian, 298. Multiplication, by scalars, 3; distributive, 9; associative, 11; of versors, versor-arcs, 16; symbolical, table for S, V, K, T, U, 19; hyperspace, 303; in Aiwdf./tnungxlehre, 306. Mutual potential, 223. Mutually rectangular vectors, system of three, 10; relations connecting two systems, 33; axes of function, 96, 97; vectors transformed from, 98; normal to confocals, 123; related to curve, 134; to surface, 146; examples relating to curve, 173. Negative unity, square of unit vector is, 10, 17; square-root of, 3, 20, 58; see Imaginary. Non-central surfaces, 117. Non-commutative, multiplication, 8; addition, 16; displacements, 168. Nonion, see Linear vector function. Normal, to surface, 65, 139. 144 ; to quadric, 108, 123; to curve, 134; and tangential resolution of force, 185, 189; solutions, 256; generalized, 279; generator as well as, 279. Notation, conventions respecting, 19; for projective properties of surfaces, 296. See Symbol. Nullitier, 21. Number of constants of linear function, 88, 178, 272, 283. Numerous characteristics, order of cone and surface, 101; of curves, 290. O'Brien, Rev. M., 11. Octahedron, regular, 45. KM. Octonions, 21. Ohm's law, 251. Operator, quaternion as, 14: V, 70 et seq., chap. xvi., '211; various expressions for V, 74, '22-") : applications of V to Taylor's theorem, 79; to theory of groups, 80, 80; to generation of invariants, 102: to strain, 181; to calculus of variations, 1. To curvature, 21.5: V. 75, 220, 227: V. 75, 220, 227: Gaussian, 104: 1), analogue of V for projective geometry, 296. Opposite of vector, 1. Optics, examples from, reflection, 19: refraction, 22. Astronomical refraction, 85, 175; bright curves, 87; rotating mirror, 130; characteristic surfaces in, 225; electro-magnetic theory of, 256. Orb, 186; instantaneous, 188. Order of surface, 101; of curve, 290; of multiple points on Jacobian, 298. Origin, variable, 157. Orthogonal spheres, 51, 52, 54; confocals, 121, 123, 125; surfaces, 227. Oscillation of particle, 185; of rigid body, 207. Oscillating plane, 132, 267, 296; circle, 134, 136, 152; sphere, 136; quadric, 144; helix, 152; curve of intersection of two surfaces, 296. Parabola, 64, 267. Parallelepiped, volume of, 22; integra-tion over faces of, 71. Nuclear, vector involving, 62, 64, form of V suitable for, 74, 221;parameter of distribution, 138. Partial differentiation, 67, 229: fractions involving linear functions, 122; differential equations, 86, 148, 151, 153: involving V, 226. Particle, dynamics of, chap. xiv., 184. Particles, "system of, 194; four, dynamically equivalent to rigid body, 199. Pascal hexagon, 121. Pedal of quadric, 109; of three-system of screws, 164. Permanent screws, 209. Permutation, cyclical, of quaternions under S, 16; cyclical, of linear functions in product, 100; of symbols in comminatorial function, 270. Perpendicular, on line, 36; on plane, 36; to two lines, 40; line, to itself, 96; on tangent plane, 109; on generator of hyperboloid, 116; on axis of screw, 156; in astatics, 163; of three-system, 164; of e\vlindr..id, Kit; in hyperspace, 133. Perspective, 4ti, 27s. Perturbed orbit, 188. Pfaff, 86. Philosophical transactions, 1 ol. -j- Pitch, of ruled surface, 138; of screw, 156; in astatics, 161; oftentimes, 164; of two-system, 165; of two-system, 165; of infinite displacement, 169; of impulsive and of instantaneous, 202. Plane of quaternion, 13; straight line and, chap. v., 35; polar, for sphere, 50; for quadric, 108; radical, 50; inverse of, 53; cyclic, 113; oscillating, 132; generating developable, 135; of no virial, 157; central, in astatics, 160; of elastic symmetry, 2-5; polarised wave in elastic solid, 247; in dielectric, 257; projective symbol for, 265; equation of, 266; united, of linear transformation, 274; to quadric, polar, 276. Pliickers coordinates of a line represented by (ff, r), 40. Poinsot, central axis, 156. Point, stationary, 63, 83; of inflexion, 83; circular, 96, 126, 307; double, on wave-surface, 248, 261; on Jacobian, 298; apparent, 292; symbol, 263 et seq., 308; united, of linear transformation, 274, 276. Polar, harmonic, 46; plane of point with respect to sphere, 50; to quadric, 108, 276; line to quadric, 109; developable, 136, 139; general theory of, 296. Polarised waves in elastic solid, 247; in dielectric, 258. Pole, see Polar. Pole, spherical harmonic referred to its, 224. Polhode, 198. Polygon, inscribed to sphere, 55; inscription of, 56; loci related to variable, 300. Potential, operator T~2, 220; expression for, 223; surfaces, equi-, 227; velocity, due to vortices, 235. Power of vector, 28, 69, 173; of quaternion, 29; of point with respect to sphere, 49, 50. Poynting flux of rad Mediation energy, 253; parallel to ray- velocity, 257. Principal, axes of section of quadric, 111; normal to curve, 134; curvature, 143; axes of inertia, 197; screws, 209; circuit, 232. Product, of two vectors defined, 8; associative property of, 11; reciprocal of, 12; of two quaternions, construction for, 14; conjugate of, 15; spherical representation of, 16, 31; differential of, 68; of linear functions, 95, 100; of vectors in hyperspace, 303; Urassmann's, 306. Projection, of point on plane, 37; of ellipse into circle, 83; of vectors, invariants relating to, 98; of curvature, 141. Projective geometry, chap. xvn., 263; in hyperspace, 308. Propagation of disturbance, 255. Proportional to three vectors, fourth, 31. Pure strain, 177; converting general wave-surface into Fresnel's, 262. Pyramid or system of three planes, examples on, 38; invariant relations for, 98. Quadratic equation satisfied by quaternion, 29; by special linear function, 95; quaternion function, 274. Quadric surfaces, chap, ix., 106; harmononic generation of, 65; oscillating surface, 144; pitch, 165; elongation, 181; general, in point symbols, 275; inscribed in developable, 279; tangent-line developable for, 282; invariants of, 288. Quadrilateral, spherical, 34; complete, 46; inscribed to sphere, 56; common to quadric and linear complex, 274. Quartic, Steiner's, 164, 166; symbolic, of linear quaternion function, 274. Quartic, Steiner's, 164, 166; symbolic, of linear quaternion function, 274. 56 ; invariants of linear vector function, 97, 159, 212 ; moment of force, 157 ; as symbol of point, 263 ; of plane, 265 ; function, linear, 272 et seq. Quotient, of parallel vectors, 3 ; of vectors, 13. Radical plane of spheres, 50 ; axis, 51 ; centre, 52, 53. Radius of quadric, 107, see Conjugate, Curvature. Rank of curve, 292. Ratio, of vectors, 13 ; of torsion to curvature, constant, 137. Rayleigh, Lord, 240. Reaction, 194, 200, 236 ; of constraint, 189, 204. Reality of roots of self-conjugate vector function, 96 ; of principal screws, 209 ; of united points, 276. Reciprocal, of vector, 11 ; of product, 12 ; of quadric, 110 ; screws, 204 ; of quadric, 276. Reciprocity, for a surface, relations of, 294. Reciprocity, quadric of, 266, 309. Rectangular vectors, system of three mutually, 10; relations connecting two systems of, 33; axes of function, 96, 97; vectors transformed from, 98; normals to confocals, 123; related to curve, 134; to surface, 146; examples relating to, 173; in hyperspace, 303. Rectifying developable, 136, 139. Reduced wrench, 205. Reflection in plane mirror, 19; in moving mirror, 130; of force for brachistochrone, 193. Refraction, 22; astronomical, 85, 175; conical, 248, 260. Regression, edge of, 136; see Developable. Regular solids, rotations related to, 104. Relative, magnitudes and directions of two vectors, 13; motion, 171. Remainder of a series, 79. Resolution of vector into components, chap, in., 23; of linear function, 96, 99; of strain, 178; of force, tangential and normal, 185; of linear quaternion function, 282. Resultant of statical forces, 156. Revolution, cone of, 45; cylinder of, 45; condition for quadric. of, 114; tangent cylinder of, 115; motion of particle on, 190. Rigid, see Body, Dynamics. Root, of a quaternion, with, 28; differential of square-, 77; of linear vector function, latent, 93; square-, 99; linear function, symbolic, with, of unity, 105; linear quaternion function, latent, of, 272, 276; square of, 282. Rotation, convention respecting sense of, 7; conical q.v., 18; forces, 160; finite displacement, 168 et seq.; strain, 177 et seq.; of elements, 212; in hyperspace, 307. Roulette, 83, 84. Royal Irish Academy, see Academy. Ruled, hyperboloid q.v., 65, 116, 264, 270; surfaces, 128, 137 et seq.; surface, differential equation of, 149, 153. Russell, Robert, 22, 61. S symbol for scalar, 6, 19; differential of 87, 68. Salmon, 114, see Geometry of Three Dimensions, Higher Plane Curves. Sealar, 3, 6; of product, 13: point, 263. Screws, theory of, applied to, motion of emanant, 137: static, 1.15, 159, 163: displacements, 169: dynamics, 3 HI, 204. Segments, theorem of six, 46. Self-conjugate, tetrahedron to sphere, 52. 276: vector function, 80, 96, 97: tetrahedron of two quadrics, 277. Sense of rotation, 7. Series, exponential, 28; Taylor's 79; of spherical harmonics, 223, 224. Sextic curve, Jacobian, 293, 293. Shaw, J. B., -263. Shear, 178. Shortest distance between lines, 40, 138, 134. Similitude, direct and inverse, 14. Six coordinates of line (r, T), 40: segments, 4(5: constants of self-conjugate function, 96: screws, 166; co-reciprocal, 206. Sixteen, constants in linear quaternion function, 272: square roots of linear quaternion function, 282. Solenoidal distribution of vectors. Solid, harmonic, 7. 76, 222 tt *cq. : elastic. 222, 239, 242 tt *q. : moving in fluid, 241. Solution of equations, involving linear function, 92, 117 : involving v, 218. Sphere, chap, vi., 49; inversion of, 32 : through four points, 53, 35, 58 ; touching four planes, 54 ; and polygon, 55, 56; solid, 59; generating ellipsoid, 115; osculating, 136; envelope of, 151 ; surface generated by, 155 ; of reciprocation, unit, 266. Spherical, trigonometry, chap, iv., 29; excess, 33 ; harmonics, 70, 76, 222 : curve, 136 : astronomy, examples, 174. Sphero-conic, 118. Spin-vector, 96, 97 ; of \f/t 97 : in strain, 181, 182 : of element, 212. Spiral, logarithmic, 82. Square-root of quaternion, differential of, 77; of linear function, 99, 112, 124, 177 : of linear quaternion function, 282. Standard form of V, 75; of two linear functions, 100; of screw-system, K>4 : of quadric and linear complex, 27s. Statics, chap. XL, 156. Steiner's quartic, 164, 166. Stokes's theorem, 215. Storage of energy, elastic solid, 243 ; electric and magnetic, 252. Strain, chap. xin., 177, 212, 238; stress in terms of, 243. Strength of tube, 2:;}, 235. M r*.». •_'!, tt •>"/. : in viscous fluid, 23s : in isotropic solid, 239 : in ten strain, 243 : in electro-magnetic field, 955. Striction, line of, 138 : of quadric, 140. Subtraction of vector, 2. Sum of vectors, 2 ; of scalar and vector, 9 : of quaternions, 9 ; of weighted points, 264 ; of area vectors in hyper-space, 306. Supplemental triangles, 29 : related to axes of function and conjugate, 94 ; to propagation of light, 258. Surface, in terms of Permanent, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108,  33, 99, 163, 192, 214, 218. Tangent, to sphere, 49; curve, 63; surface, 65; quadric, 108; confocal, 124; generalized confocals, 285; interindic, 295; meeting surface in four consecutive points, 289. Tangential equation of quadric, 11; and normal components of force, 185, 189; transformation, 273; equation of quadric and linear complex, 270. Taylor's series, 79. Telescope, examples on composition of rotations, 175. Tensor of vector, 4; quaternion, 12; biquaternion, 20; of sum, development of, v. Tetrahedra, in perspective, 46; corresponding vertices of, joined by generators of hyperboloid, 103; reciprocal, of united points of linear transformation and its components, 274. Tetrahedron, formulae relating to Regular, 45, 104; anharmonic relations of point and, 46; self-conjugate to sphere, 52; and sphere, 53; forces on edges of, 158; of reference, 268; self-conjugate to two quadrics, 277; invariants relating to, 288; loci relating to variable, 300. Thomson and Tait, 200. Three-bar motion, 60, 85; system of screws, 164, 205; plane polarised waves in solid, 248. Tore, 59. Torsal generator, 140. Torse, see Developable. Torsion, 132, 134. Total curvature, 148. Transformation, effected by linear vector function, 89; by self-conjugate function, 97; Cremona, 101; of screws, 208; general linear, 272, et seq.; examples of, 285; invariants of, 288. Transversals of lines, 103. Triangle, and point, harmonic properties, 45; and conic, 48. Trigonometry, formulae for plane, 25; de Moivre's theorem, 27; spherical, 21). Trilinear function, 243. Trinomial form for linear function, 89; for pair of functions, 100. Tube, motion of particle in rotating, 191; in fluid motion, 233, 235. Twist of curve, vector, 133; velocity, HO, 171, 201. Twisted, cubic q.i\, 93, 104, 109, 123, 133, 172, 267, 278, 285, 293. Two linear functions, 100; reduction to forces, 158; angular velocities, 172. U, symbol for versor, 4, 13, 19; differential of Uq, 68; development of Uq + q), 85. Umbilical generator, 125; generalized, 279, 286. Unicursal curve, 152. Unit, of length, 4; vector denoted by Ua, 4; vectors, system of mutually (q.v.) rectangular, 10, 96, 98, 134, 146; point of anharmonic coordinates, 44; weight, points of, 263; sphere of reciprocation, 266; vectors in hyper-space, 303. United, screws, 209; points of transformation, 274, 276. V, symbol for vector, 7, 19; differential Variable origin, 157. Variations, calculus of, 192; of interrogals, 231, 233. Vector, as directed right line, 1; as operator, 14; arc, 17; area, 23; of curve, 62; of surface, 64; function, linear, 88; spin-, 96; equation of confocals, 124; emanant, 131; as difference of two points, 263; as point at infinity, 263; area, in hyperspace, 303. Sec. Linear vector function, etc. Vectoral algebra, 11. Velocities, virtual, 157, 254. Velocity, 63; hodograph, 83, 187, 189; of emanant, 138; twist-, 170, 201; relative, 171; of particle, 184 et seq.; arc, 186, 188; of element of medium, angular, 212; potential, due to vortex rings, 235; wave-, 247, 257; ray-, 248, 257. Versor of vector, 4, 14; of quaternion, 13, 16. Versors, coplanar, chap. iv.. 27. Vibration of particle, 185. Virial, 157. Virtual, velocities, 157, 254; co-efficient of two screws, 206. Viscous fluid, 238, 240. Volume of parallelepiped, 23 ; of tetrahedron, 265, 269 ; directed, 304. Vortex motion, 235, 238. Wave-surface, 163, 248, 261 ; velocity, 247, 257. Waves, propagation of disturbance by, 255. Weierstrass, 199. Wrench, in statics, 156, 163 : impulsive, 201, 204; evoked, 204; reduced, 205. Zero, square-root of, 29. GLASCOV: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSK AND CO. LTD. BINDING SECT. JUL 6 1981 PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY 257 KtAS Joly, Charles Jasper A manual of quaternions. 