None
36. If x = c cosh ^ cos tj, y = c sinh ^ sin ij, prove that 37. If ^, 7j be current coordinates, the equations of the tangent and normal at any point (x, y) on the curve are (^ - a:) <^„ + (r, - 2/) <j!.^ = 0, and (^ - a;)/^„ = (77 - y)/.^„, respectively. 38. If 3/ be a function of x determined by the equation <^ (x, y) = 0, prove that ^ = - '^yVo^ - 2fe<^,.^a.i/ + -^^Vw. 39. Prove that the curvature at any point of the curve <j) {x, y) = 0 40. Prove that if at a point on the curve <^ (re, y) = Q we have c^a; = 0, ^y= 0, simultaneously, then two (real or imaginary) branches of the curve pass through that point, whose directions are given by the quadratic <t>^^2ci>^£+^^(£) =0. Hence shew that the point is a node, a cusp, or an isolated point, according as {^xyTz'I'xx'I'mi- 616 APPENDIX. NUMERICAL TABLES. A. Table of Squares of Numbers from 10 to 100. 0 1 2 3 4 5 6 7 8 9 1 100 121 144 169 196 225 256 289 324 361 2 400 441 484 529 576 625 676 729 784 841 3 900 961 1024 1089 1156 1225 1296 1369 1444 1521 4 1600 1681 1764 1849 1936 2025 2116 2209 2304 2401 5 2500 2601 2704 2809 2916 3025 3136 3249 3364 3481 6 3600 3V21 3844 3969 4096 4225 4356 4489 4624 4761 7 4900 5041 5184 5329 5476 5625 5776 5929 6084 6241 8 6400 6561 6724 6889 7056 7225 7396 7569 7744 7921 9 8100 8281 8464 6649 8836 9025 9216 9409 9604 9801 B. 1. Table of Square-Roots of Numbers from 0 to 10, at Intervals of -1. ■0 -1 -2 -3 •4 -5 6 -7 •8 -9 0 0 •316 -447 -548 -632 -707 •775 -837 -894 -949 1 1-000 1-049 1-095 1-140 1-183 1-225 1-265 1-304 1-342 1-378 2 1-414 1-449 1-483 1-517 1-549 1-581 1-612 1-643 1-673 1-703 3 1-732 1-761 1-789 1-817 1-844 1-871 1-897 1-924 1^949 1-975 4 2-000 2-025 2-049 2-074 2-098 2-121 2-145 2-168 2-191 2-214 5 2-236 2-258 2-280 2-302 2-324 2-345 2-366 2-387 2-408 2-429 6 2-449 2-470 2-490 2-510 2-530 2-550 2-569 2-588 2-608 2-627 7 2-646 2-665 2-683 2-702 2-720 2-739 2-757 2-775 2-793 2-811 8 2-828 2-846 2-864 2-881 2-898 2-915 2-933 2-950 2-966 2-983 9 3-000 3-017 3-033 3-050 3066 3-082 3-098 3-114 3-130 3-146 10 3162 APPENDIX, 617 B. 2. Table of Square-Roots of Numbers from 10 to 100, at Intervals of 1. 0 1 2 3 4 5 6 7 8 9 1 3-162 3-317 3-464 3-G06 3-742 3-873 4-000 4-123 4-243 4-359 2 4-472 4-583 4-690 4-796 4-899 5-000 5 099 5-196 5-292 5-385 3 5-477 5-568 5-657 5-745 5-831 5-916 6-000 6083 6-164 6-245 4 6-325 6-403 6-481 6-557 6-633 6-708 6-782 6-856 6-928 7-000 5 7-071 7-141 7-211 7-280 7-348 7-416 7-483 7-550 7-616 7-681 6 7-746 7-810 7-874 7-937 8-000 8-062 8-124 8-185 8-246 8-307 7 8-367 8-426 8-485 8-544 8-602 8-660 8-718 8-775 8-832 8-888 8 8-944 9-000 9-055 9-110 9-165 9-220 9-274 9-327 9-381 9-434 9 9-487 9-539 9-592 9-644 9-695 9-747 9-798 9-849 9-899 9-950 C. Table of Reciprocals of Numbers from 1 to 10, at Intervals of -1. -0 -1 -2 -3 -4 -5 -6 •7 •8 -9 1 1-000 -909 -833 ■769 •714 -667 -625 ■588 ■556 ■526 2 •500 -476 •455 •435 -417 •400 -385 ■370 ■357 -345 3 -333 -323 •313 ■303 -294 -286 ■278 ■270 ■263 •256 4 •250 -244 -238 ■233 -227 -222 •217 ■213 ■208 -204 5 •200 •196 •192 -189 -185 -182 ■179 •175 •172 •169 6 -167 •164 •161 ■159 -156 ■154 ■152 ■149 •147 ■145 7 •143 •141 -139 •137 -135 ■133 ■132 ■130 •128 ■127 8 •125 •123 ■122 -120 •119 •118 •116 ■115 ■114 ■112 9 •111 -110 -109 •108 •106 •105 ■104 •103 ■102 ■101 D. Table of the Circular Functions at Intervals of One-Twentieth of the Quadrant. eiiT sin 0 coseo 6 tan $ cot e seo 0 cos 0 0 0 00 0 00 1-000 1^000 100 ■05 •078 12^745 ■079 12^706 1-003 •997 -95 •10 •156 6-392 ■158 6^314 1-012 ■988 -90 •15 •233 4-284 ■240 4^165 1028 •972 -85 •20 •309 3-236 ■325 3-078 1^051 •951 ■80 •25 •383 2613 ■414 2-414 1-082 •924 -75 •30 ■454 2^203 ■510 1-963 1-122 ■891 -70 •35 •522 1^914 •613 1-632 1-173 ■853 -65 •40 ■588 1^701 •727 1^376 1^23e -809 •60 •45 ■649 1^540 ■854 1171 1-315 •760 -55 ■50 •707 1^414 1^000 1-000 1-414 •707 •50 COB e seo $ cot 8 tane cosec 0 sine eiiT 618 APPENDIX. E. Table of the Exponential and Hyperbolic Func- tions of Numbers from O to 2-5, at Intervals of -1. X ex e-x cosh X ginh X tanh X 0 1-000 1-000 1-000 0 0 •1 1-105 -905 1-005 •100 •100 •2 1-221 •819 1020 ■201 •197 •3 1-350 •741 1-045 -305 •291 ■4 1-492 -670 1-081 •411 -380 •5 1-649 -607 1-128 -521 -462 •6 1-822 -549 1-185 -637 -537 •7 2-014 •497 1-255 •759 -604 •8 2-226 •449 1-337 ■888 -664 •9 2-460 •407 1-433 1^027 -716 10 2-718 •368 1-543 1^175 •762 11 3-004 •333 1-669 1-336 •801 1-2 3-320 ■301 1-811 1-509 •834 1-3 3-669 •273 1-971 1-698 -862 1-4 4-055 •247 2-151 1-904 -885 1-5 4-482 •223 2-352 2-129 -905 1-6 4-953 •202 2-577 2-376 -922 1-7 5-474 •183 2-828 2-646 -935 1-8 6-050 •165 3-107 2-942 -947 1-9 6-686 •150 3-418 3-268 -956 20 7-389 •135 3-762 3-627 ■964 21 8-166 •122 4-144 4-022 •970 2-2 9-025 •111 4-568 4-457 ■976 2-3 9-974 •100 5-037 4-937 ■980 2-4 11-023 -091 5-557 5-466 ■984 2-5 12-182 -082 6-132 6-050 ■987 F. Table of Iiogarithms to Base e • -0 ■1 -2 -3 -4 -5 -6 ■7 8 •9 1 0 -095 •182 •262 -336 •405 ■470 ■531 •588 -642 2 ■693 -742 ■788 •833 -875 ■916 ■956 ■993 1^030 1-065 3 1-099 1-131 1^163 1^194 1-224 1-253 1^281 1^308 1^335 1-361 4 1-386 1-411 1^435 1^459 1-482 1-504 1-526 1^548 1-569 1-589 5 1-609 1-629 1-649 1-668 1-686 1-705 1-723 1-740 1^758 1-775 6 1-792 1-808 1-825 1-841 1-856 1-872 1-887 1-902 1-917 1-932 7 1-946 1-960 1-974 1-988 2-001 2^015 2028 2-041 2-054 2-067 8 2-079 2-092 2-104 2-116 2-128 2^140 2-152 2-163 2-175 2-186 9 2-197 2-208 2-219 2-230 2-241 2-251 2-262 2-272 2-282 2-293 log 10=2-303, log 102=4-605, log 103=6908. INDEX. [The numerals refer to the pagei."] Acceleration, 68 angular, 69 tangential and normal, 397 Accidental convergence, 10 Algebraic functions, continuity of, 26 Amsler'a planimetre, 250 Anchor-ring, surface of, 272 volume of, 259 Approximate integration, 275 Arbitrary constants, 166 Arc of a curve, formula for, 264, 266, 268 Archimedes, spiral of, 354, 367 Area, definition of, 241 sign of, 245 swept over by a moving line, 249 Areas of plane curves, formulffl for, 242, 247 mechanical measurement of, 250 of surfaces of revolution, 270 Astroid, 359, 438 Bernoulli, lemniscate of, 370, 389 Binomial theorem, 564, 574, 584 Bipolar coordinates, 886 Calculation of tt, 273, 554 Cardioid, 356, 358, 869, 370, 383 Cartesian ovals, 388 Caasini, ovals of, 389 Catenary, 342 arc of, 265 curvature of, 397, 400 parabolic, 465 Centre, instantaneous, 434, 439 of curvature, 395, 400 of mass, see 'Mass-Centre' of pressure, 296 of rotation, 433 Centrodes, 445 Change of variable in integration, 179 Chord of curvature, 395, 403 Circle, perimeter of, 5 area of, 241 involute of, 354, 428 of curvature, 395 osculating, 406 Circular arc, mass-centre of, 293 disk, radii of gyration of, 315, 320 Circular Functions, continuity of, 33 differentiation of, 71, 72, 78 graphs of, 34 Circular motions, superposition of, 359 Cissoid, 337 Clairaut's differential equation, 485 Complementary function, 474, 502, 509 Concavity and convexity, 157 Cone, right circular, mass-centre of, 304, 305 surface of, 270 volume of, 257 Continuity of functions defined, 15 Continuous functions, properties of, 17, 18, 23, 52, 54 Continuous variation, 1 620 INDEX. Conjugate point, 333 Contour lines, 98, 606 Conveigence of infinite series, 4 essential and accidental, 10 uniform, S43 Convergence of a definite integral, 211 of power-series, 643 Corrections, calculation of small, 133, 138 Cotes' method of approximate in- tegration, 276 Crossed parallelogram, 447 Cubic curves, 333 Curvature, 394, 402, 406 centre of, 395 chord of, 395, 403 circle of, 396 radius of, 395, 397, 400, 402, 406 Cusp, 336 Cusp-locus, 420 Cusps, circle of, 444 Cycloid, 347 arc of, 348 area of, 350 curvature of, 398, 442, 444 evolute of, 423 Definite integral, see ' Integral ' Degree of a differential equation, 463 Density, mean, 288 Derived Function, definition of, 64 geometrical meaning of, 66 properties of, 100, 103, 104, 106 DifEereutial coefficients, 64, 96, 144 Differential Equations, 456 exact, 466 homogeneous, 469 integration of, by series, 659 linear, 473, 476 of first order and first degree, 462 of first order and higher degree, 484 of second order, 490 simultaneous, 529 Differentials, 132, 138 Differentiation, 69, 71 of a sum, product, quotient, etc., 73, 74, 76 of a function of a function, 80, 139 of a definite integral, 219 of implicit functions, 98, 139, 607 Differentiation of inverse functions, 85 of power-series, 557 partial, see ' Partial differentia- tion' successive, 144 Discontinuity, 21 Displacement of a plane figure, 433 Distributed stresses, 322 Elasticity of volume, 69 Elimination of arbitrary constants, 456 Ellipse, area of, 243, 248 perimeter of, 267, 548 curvature of, 398, 402, 404 evolute of, 422 Ellipsoid, volume of, 260 radii of gyration of, 317, 821. 328 of revolution, surface of, 273 Elliptic disk, radii of gyration of, 316, 327 Elliptic segment, mass-centre of, 326 ElUptic-harmonic motion, 346 Elliptic integrals, 268 Envelopes, 413, 415 contact-property of, 418 Epioyolies, 359 as roulettes, 448 Epicycloid, 350 arc of, 353 curvature of, 398, 442, 444 evolute of, 425 double generation of, 450 Epitrochoid, 354 Equations, theory of, 104, 161 Equiangular spiral, 366 curvature of, 399 Even and odd functions, 41 Evolute, 421 arc of, 425 Exact differential, condition for, 698 Exact differential equations, 466 Expansion, coefficient of, 69 Expansions by means of differential equations, 562 by Maclaurin's theorem, 573, 583 Exponential function, 35, 660 graph of, 38 Flexure of beams, 299, 323 Function, definition of, 12 graphical representation of, 13 INDEX. 621 Functions, algebraic and transcen- dental, 26 implicit, 97 inverse, 44, 85 Qeometrical representation of mag- nitudes, 2 Goniometric functions, 45 differentiations of, 87 graphs of, 86, 88 Gradient of a curve, 67 Graph of a function, 18 Gregory's series, 652 Gnldin, see 'Pappus' Gyration, Badius of, 313 Hart's linkage, 381 Hemisphere, mass-centre of solid, 804 mass-centre of surface of, 301 Homogeneous differential equations, 469 Homogeneous functions, Euler's theorem, 608 Homogeneous strain, 324, 327 Hyperbola, area of sector of, 243 Hyperbolic functions, 41 differentiation of, 84 graphs of, 42, 43 inverse, 48 differentiation of, 93 Hypocycloid, see ' Epicycloid ' Hypotrochoid, see ' Bpitrochoid ' Implicit functions, 97 differentiation of, 98, 140, 608 Indicator diagram, 211, 247 Inertia, moment of, 318 Infinite series, 4, 9 addition and multiplication of, 11, 35 differentiation and integration of, 549, 656 Infinitesimals, 61 Inflexion, points of, 157 Instantaneous centre, 434, 439 Integrals, definite, 208, 214 convergence of, 211 properties of, 217, 232 rule for finding, 221 approximate calculation of, 275 Integrals, multiple, 282 Integration, 166 by parts, 187 Integration by substitution, 179, 184 of irrational functions, 176, 203 of power-series, 549 of rational fractions, 171, 193, 195, 197 of trigonometrical functions, 182, 229 Interpolation by proportional parts, 156 Intrinsic equation of a curve, 397 Inverse functions, 44 differentiation of, 85 Inversion, 378 mechanical, 380 Involutes, 427 Involute teeth, 452 Leibnitz' theorem, 146 Lemniscate of Bernoulli, 370, 383 Lima(fon, 357, 368, 383 Limit, upper and lower, of an as- semblage, 3, 61 Limiting values, 8, 54, 55, 68 Limits, upper and lower, of a defi- nite integral, 209 Linear differential equations of first order, 473, 476 of second order, 501 with constant coefficients, 509, 513 Line-density, 289 Line-roulette, 443 Lissajous' curves, 344 Logarithmic differentiation, 91 Logarithmic Function, 46 graph of, 47 differentiation of, 89 Logarithmic series, 652, 676, 585 Maclaurin's Theorem, 573, 576, 682 Magnetic curves, 389 Mass-centre, 291 of an are, 293 of an area, 298, 295 of a surface of revolution, 301 of a soHd, 303, 306 Maxima and minima, 106, 160, 688 by algebraic methods, 111, 136 of functions of several variables, 135, 603 Mean density, 288 pressure, 296 Mean values, 279 Mean-value theorems, 129, 217 622 INDEX. Modulus (in logarithms) 48 Moment of inertia, 313 Multiple integrals, 282 Multiple roots of equations, 161 Newton's treatment of curvature, 402 Node-locus, 420 Node, 338 Older of a differential equation, 456 Orthogonal projection, 324 Orthogonal trajectories, 478 Pappus, theorems of, 307 Parabola, arc of, 265 curvature of, 398, 402, 403 evolute of, 421 Parabolic segment, area of, 243 mass-centre of, 294, 295 Paraboloid, volume of, 259, 260 mass-centre of, 304 Parallel curves, 428 Parallel projection, 324 Partial derivatives, 95 of higher orders, 695 Partial differentiation, commuta- tive property of, 596, 600 Partial Fractions, 172, 194, 195, 197 Particular integral of a linear dif- ferential equation, 474, 602 PeauceUier's linkage, 380 Pedal curves, 382 Pedals, negative, 384 Pericycloid, 350 Perimeter of a circle, 5 Planimeter, 250 Point-roulette, 433, 440 Polars, reciprocal, 384 Positions, curve of, 68 Power-series, continuity of, 547 diiferentiation of, 557 integration of, 549 Pressure, centre of, 296 Primitive of a differential equation, 457, 459 Prismoid, volume of, 261 Proportional parts, 156 Quadrature, approximate, 275 Eadius of curvature, lee ' Curva- ture' Badius of gyration, 813 Batioual fractions, graphs o^ 28 integration of, 171, 193, 195, 197 national integral functions, 26 Beciprocal polars, 384 Beciprocal spiral, 868 Bectification of curves, 264, 266, 268 of evolutes, 425 Beduotion, formulee of, 189, 190, 229 Bemainder in Taylor's and Mac- laurin's Theorems, 581, 582 Bing, radius of gyration of a, 317 surface of a, 272, 308 volume of a, 259, 308 Bolle's theorem, 105 Boots of algebraic equations, sepa- ration of, 105 Boulettes, 433 curvature of, 440, 443 Second derivative, 144 geometrical meaning of, 151, 154 Sector of a circle, see ' Circle ' Semi-cubical parabola, 336 Separation of roots of an algebraic equation, 105 of variables in a differential equation, 464 Series, see ' Convergence ' and ' In- finite Series ' Sign of an area, 245 Similar curves, 376, 470 Simpson's rules, 260, 277 Simultaneous differential equations, 529 Sin X, expansion of, 562, 675, 605 Sin~> x, expansion of, 556 Singular solutions, 487 Sphere, radius of gyration of, 817, 320 surface of, 272 volume of, 259 Spherical sector, mass-centre of a, 305 Spherical segment, volume of, 258 Spherical shell, radius of gyration, 316 - Spherical surface, area of a, 272 mass-centre of a 301, 302 Spherical wedge, mass-centre of a, 306 INDEX. 623 Spiral, equiangular, 366 of Archimedes, 354, 367 reciprocal, 368 Stationary point, 396 Stationary tangent, 167, 396 Stationary values of functions, 110, 604 Subnormal, 122 Subtangent, 122 Successive differentiation, 144, 589 of a product, 146 Surface-density, 289 Surface of revolution, area of, 270 mass-centre of, 301 Tangent to a curve, 66 Tangential polar equation, 371 Taylor's Theorem, 154, 572, 573, 682 expansions by, 574, 583 extension of, 599 Teeth of wheels, 450 Tetrahedron, volume of a, 257 JTheory of equations, 104, 161 Tirtie-integral, 210 Total variation of a function, 136, 601 Tractrix, 344 Trajectories, orthogonal, 478 Transcendental functions, 26, 33 Triangle, mass-centre of a, 293, 294 Trigonometrical Functions, , see 'Circular Functions' Trochoid, 350 Uniform convergence of series, 543 Variable, dependent and indepen- dent, 12 Variable, change of, in integration, 179, 184 Variation, continuous, 1, 2 Velocities, component, 267, 270 Velocity, 68 angular, 69 Volumes of solids, 255, 328 of revolution, 258 'Witch' of Aguesi, 337 Work, 210 CAMBEIDOE : PRINTED BT J. & 0. F, CLAY, AT THE UNIVEBBITT PBESS. mm. 