A treatise of mechanics, theoretical, practical, and descriptive
Gregory, Olinthus, 1774-1841
. 61*8 18-08" /■ 60-5 63-00" 112-5 57-51 110-5 46-45 180° ■■ 162-5 47-44 ggo^ t62-5 38-05 212-5 41-49 ^^ / 212-5 32-66 202-6 37-32 V. 262-5 29-27 P 62-5 77-50 / 60-5 60-55 n2-5 56-95 ' 110-5 44-56 163'- 1 162-5 47-22 St" ■< 162-5 35-78 1 212-5 41-26 l' 212-5 31-25 262-5 37-12 '^ 262-5 27-51 > 62-5 75-09 r 62-5 57-50 1125 56-15 \ 112-5 42-75 156° ■« ' 162-5 46-44 12«< 162-5 34-85 1 212-5 41-03 / 212-5 89-65 262-5 36-52 ^^ 262-5 25-86 ^ 62-5 13 38 ( 60-5 55-45 , 112-5 54-75 \ 112-5 40-04 144" ■ 1 162-5 45-35 60' < 162-5 33-05 1 212-5 39-58 / ai2-5 28-25 262-5 37-57 V 262-5 24-17 r 60-5 72'03 { €2-5 .>2-5 1 in-5 53-25 ,) 112-5 38-05 132'' ' ]G2-5 43-75 < 162-5 31-61 1 212-5 3S-26 / 2IL-5 27-56 ^ 2(!2-5 34S0 ^ 262-5 24-30 '' 63-0 6H-32 36- f *2-5 51-15 112-5 00-84 J U2-5 36-96 120" ■■ 16-2-0 41-84 ( 162-5 30-53 212-5 36-62 24" ( 6i-5 4y-4a ^ 262-5 32-77 ] 112-5 3576 61-5 65-85 ( 162-5 30-23 112-5 48-75 12' C ^-^ 49-38 1118° < 162-5 39-50 ] 112-5 35-33 212-5 34-46 ( 162-5 30-01 262'5 31-05 1 The direct resistance on the end of the vessel without an isosceles prow may bs represented by any arbitrary number, as 10000 for insUnce Cha^. VI.] Dr. Youngs Remarks an Resiiiance. 561 And if the angle of the prow, or that formed at the fore-part of the vessel or summit of the triangular sectioni be x, the reaistanee will be expressed by the formula 10000 cos'j:. To correct this expression by means of the foregoing table, we may observe that, when the angle x undergoes a variation of 12% each of the angles at the base of the isosceles prow will vary 6^: calling this latter variation q, M. Bossut deduced from the table — y ' jr and q being expressed in decimal parts of the radius unity. Tlius the whole resistance against the prow, estimated in the opposite direction to that of the motion of the boat, is equal to 10(XK) cos^jr+S*153 ^iV**\ And this formula will be sufficiently exact when the extreme angle of the prow is greater than 12^ : but when that angle is less than 12^ the theorem wil err in excess. 560. M. Lacroix has inserted in the Bulletin de la Soci6t£ Philomadiique, a comparative view of the experiments of Bossut and Vince, on the resistance of fluids striking on oblique surfaces. He does not appear to have assigned a sufficient reason fortfie difference of the results of those experiments; but the subject is oif so much importance, as to deserve such an examination in its present state as may serve to assist in the prosecution of farther expeiiments. The first approximation, says Dr.T. Young, to a determina* tion of the effect of the resistance to a body of a given section, terminated by oblique planes, is to suppose each particle of the fluid to impinge once on the surface, and then to retire for ever : on .this supposition, the resistance ought to vary as the square of the cosine of the angle of incidence. Another part of the resistance is occasioned by the adhesion of the particles of the fluid ; this may be supposed to vary, as the product of the secant and the sine of the angle of incidence; that isi as its tangent A third part depends on the form of the posterior surface of the body, and upon the unknown irregularities produced in the motions of the particles of the fluid, by the difference of the forms 4>f its anterior part. It may be expected, that this nega- tive pressure will be nearly uniform, when the shape of the posterior part of the body remains unaltered, as in Bossut's experiments; but that, when a thin surface is employed, as in Mr* Vince's apparatus, it will be somewhat diminished by the obUquity of that surface, evep supposing the transverse pro- jection of the surface to remain unaltered. Thb portion, how- ever, may naturally be expected to be liable to great irregularities; VOL. I. GO 562 PNEUMATICS. [BookV. and it appears to be somewhat increased when the thin sarfoce is inclined in a small angle only. Mr. Romme has remarked, that the facility wirti which the particles of the fluid can escape before the moving body, is proportional to the angular space of the fluid which remains open to admit them, and that therefore the resistance must vary in proportion to this angle. Without allowing die troth of the observation in its whole extent, we may with proprietj inquire whether or no^the portion of the pressure doifed from impulse may not in part depend on some simple fimcticMi of the angle of incidence ; and whether the whole resistance to an oblique surface may not be considered as composed of a constant portion, k portion varying with the tangent of the angle of incidence, and a third portion proportionate to the square 'of the cosine, diminished in the ratio of a power, or other function, of the angle of incidence. Andit will appetr upon inquiry, that if we take one fifth of the radius, increased by one-tweuty-fifth of the tangent, and add to it fonr-fifths of the square of the cosine, diminished in the ratto'of the circom- fereitce of a circle increased by the angle of incidence, to die rim pie circumference, we may approach always within about one-fiftieth to the number expressing the oblique reaistance, until the angle of incidence becomes greater than 80^. Thus, the direct resistance being unity, and a the angle of incidence, the oblique resistance will •2 + -04 tan a +288 cos* a: (360+a**). A formula, somewhat more accurate than this, deduced from experiment only, is rzz cos* a + •(XXX)0042 1 7a'*'*: the quantity added to the square of the cosine being a little less than the millionth of the cube of the angle of incidence, expressed in degrees. The result of these two theorems are compared with the experiments of Bossut, and the theorem of £ytelweiD (viz. cos* a + 4 versin a) in the following table. Aoglc. Cos* a. TaDo. Form. 1. FOHD. 8. Bossut. Eyielw. 0° 1 -0000 •000 10000 10000 roooo roooo 6 •9890 •105 •9824 •9891 •9893 •9910 12 •9568 •212 •9492 •9580 •9578 '9656 18 •9045 •325 •9022 •9086 •9084 •9241 24 •8346 •445 •8438 •8449 •8446 •8690 SO •7500 •577 •7769 •7710 •7710 •8036 36 -6544 •726 •7049 •6919 •6925 •7308 42 •5523 •900 •6317 •6135 •6148 •6551 48 •4478 J^lll •5606 •5414 •5433 •5802 54 •3455 1-376 •4985 •4816 •4800 •5103 60 •2500 1-732 •4407 •4403 •4404 •4500 Chap. VI.] Dr, Founds Remarkson Resistance. 56$ Angle. Cos*a Tana. Form. 1. Form. 8. Bossut £yt€lw. 66 •1654 2S46 •3924 •4231 •4240 •4026 72 •0955 3078 •3869 •4344 •4142 •8719 78 *0432 4-705 •4166 -4816 •4063 •3600 84 •0109 9*514 •5875 •5658 •3999 •3693 Mr. Romnie found, by numerous experimentSy that when the magnitude of the greatest section of a floating body, and its distlmce from the angular points, were constant, the form of the outline of any section of the body, whether composed of right lines or of curves of any kind, was either wholly, or very, nearly, indifferent to the magnitude of the resistance: hence he infers, that in the construction of ships, the curve of the sides ought to be determined from considerations independent of the resist- ance. In experiments like those of Mr. Vince, the circumstances are materially different: but the accuracy of Mr. Vince's ex- periments on water is, in some measure, confirmed by a com- parison with those of Schober, which were made in a similar manner on air. The results of both these investigations are here exhibited in a table, and compared with a coarse approxi- mation from this formula r=4*-i~6cos a, and with the results of Dr. Hutton. Angle. Coso. Form.S* Schober. Vince. Hutton. 0^ 1-0000 1-0000 1-0000 1-0000 1000 10 •9848 •9909 •9854 •9787 1-010 20 •9397 •9638 •9808 •9743 1*068 30 -8660 •9196 •9200 •9463 I'OOS 40 •7660 •8596 •8308 •8729 •963 £0 •6428 •7857 •7563 -7869 •830 60 •50(X) •7000 •6738 •6610 •662 70 -3420 •6052 •6359 -4913 •453 80 -1736 •5042 •5833 •2779 •304 (Vide Journals of the Royal Institution.) Dr. Hutton's experiments, when reduced to surfaces of given transverse pro- jections, indicate at first an increase of resistance as the surface becomes more oblique: a circumstance analogous to what was remarked by Bossut in reference to float-boards, and mentioned already at the end of book iv. ch. 4. 561. Such are the conclusions and rides which have been deduced from some of the most judicious experiments on the resbtance of fluids to the motion of bodies passing through 56i PNEUMATICS. [BookV. them. Considering the number and importance of these in- farences with respect to the magnitude «nd extent of the expe- riments from which they were drawn, we cannot hut earnestly wish that those who hare IdoBure and opportunity to pursue such enquiries^ would still further extend and vary their re- searches in this department of e^iperimeptal philosophy. Much valuable knowledge may be expected to result from labours of this kind : for it is^ probably, only by reiterated experiments and observations that suiBcient data can be obtained to perfect tlie theory of two momentous branches of mechanical sciencci viz. those relating to the motion of military projectiles, and to tbe manoeuvres c^ vessels at sea. INDEX. {The figures refer to the articles or paragraphs.) A Abctmbnts, 199 Air, mechanical properties of, 484-490 ——, density and elasticity of, 489 — , motioD of, 510—520 Air-pump, theory of,. 521 — 53S Alembert, De. See D'AIembert. Altitudes, measured by barometer aud thermometer, 509 — 510 Amplitude, 250 ■ I of impression, 330 Angle of position, 950 Angular accelerating force, 802 - pendulum, 311 Animal ^rce, 376 — 378 Arohes and piers, 199—208 Archivolt, 199 Areometer, 401—409 Artand nature, difference between,173 Ascent of bodies, 240—245 Atmosphere homogeneous, 495—498 Atmospherical logarithmic, 498 Attraction, centre of, 279 ,capilUry, 426—435 — i— — — ,of sphews, dec......... 293 ,nniversal, 294 Axioms, 21 — 27 Mam'm peritrocbio, 148—147 ^ in motion^... 36% 321, 367,372 B Bmlanoe. See Lever» nd toL U. Ballistic pendnlmiii 913 Aru Banks, on comparative strength of materials^ 191 , on velocity of air, 515 Barometer, 485 Barometer, altitudes meas. by ,502, &o» Bettancourt on steam, vol U. BUKaids, 361 Block of a pulley, 14ff Body, a — , b4rd, soft, and elastic, 329 Boaut's eiperimantson dischac^e of fluids,. 458—461 , OD resistance of water,... 559 BrachystochronoD,..., 277 Breast*wheels, 467, &c. C Capillary tubes, 426—435 Catenarian curve, 19tt Celerity, ^^ 15 Central forces,. 279—299 Centre of attraction,..^ ^ 279 gravity, 102—11*4 gyration, 309— 3t9f osciUation,... 305—311 H ■ percmsion, 317— -dto ■ prflMure, .391—393 ■ I ' spontaneoui rotation* 322—326 ■ ■ '» veliqae,«..«..M noU^ 425 Centtifhgal force, 279 puvp, •••537—338 560 INDEX. Art. Centripetal force, S79 Centrobaryc method, 195 Circalar motioD, 389—888 Clock, problem coDoernhig the jaoction of its hands, 883 CoUision, S29-*<^8 Composautff 89 Composition offerees,. 88—101 ofmotioos,^ 817 Condorcet^n resistance of water, 559 ■ on weight and gravity, 103 Conservatio Tirium vivarom, 341 Cords, 193—198 Coulomb on friction, yol. IL Crown of an arch, 198 Cnpola, 809 Carves, descents along, 863, fcc. , of swiftest descent, 877, 878 Cycloid, ^ 873 D D*Alembert's principle of dyna- mics, 867 — on varied motion,... 834 Deflecting force, • 879 De Lnc on altitudes by barom. ... 506 DensHy, 10, 811 of the planets, 894 Depth of impression, 330 Desaguliers's hydrometer, 403 Descents occasioned by gravity, 240—847 Diminution of gravity, 887 Direction 16 , line of, 106 Discharge of fluids through orifices, 438-457 1 expr. relative to, 457 — 466 Divisibility, 4 Domes, 809 Dynamics,, SIO— 380 £ Bfifects of machrnes, .368^380 Efliux of water, » 438, A&c. Elasticity, 389 Elevation in projectiles, 850 Elliptical motion, 890,891 Emerson on relative strength pf materials 190 English, on stability of canal boats, 485 Equator, centrifugal force at, 885 Equilibrium, 19^ gg — ' , of elasUc fluids, 491—508 ift Equivalent, S9 Expansions of mercory and air, 5B8 Extradoi, m Eytelwein^enbydinQtiCi,., m 40 F Flanks of anarch,..^ 199 Flottatioa, plane of, - 41! Float-boards, 4IQ Fluid, ^ 3, no — — » compressible and incoB- preisible, 381 Fly. SeevoLIL Force, 17,818 — , accelerating, 8M -, retarding, 984 Forcing pump, U7 Friction. See vol. IL Fulcrum, ; 131 — -, pressure upon, 133, 140, 391 Funicular polygon/. 193*197 Galileo, on flilling bodies, 946 ^— on the lever,. 138 on the resifL of solids, 167, &c. Oirard on the distinction between the operations of nature and ait, 173 Gra v i meter. See vol .II. Gravity,.. 108, 106,837, 840 , its proper measure, M42 , centre of,.*. 108 — 184 , diminution of, 886 -, specific, 383,396-^10 , tiibleof, 410 Gunpowder, force of, vol 11. Gyration, centre of, 309, &c. H Halley, on altitudes by barom^.. 499 Hammer, stroke of, 354 Hands of a clock, 883 Height due to a velocity, ^.. 344 Homogeneous bo^ea, 106 ■ atmosphere, .« 495 Horse, strength of, 378 Hunter, Mr. his screw, :. 161 Hutton, on altitudes by barom. 508 ,on practical gunnery ,255'-857 ■ ^on resistance of fluidis,555>558 Hydraulics, 20 INDEX. 567 Art. Hydrometer, 401—409 H3rdrodyiiamics, 4S6— 483 Hydrostotical paradox, 387 HydrOflUticty 380—435 Immateriality of the Sapreme Being, 18 Impact oblique, 360— 36S Impdied point, 365 Impetofl, 250 Impoit of an arch, 199 loeitia, 3, 18 *—, centre of, 108 Inclined plane, 158—157 — »— motion along, 958, S60, 967, 315,369,371 Intradof, 199 Joints of fracture, 199 Joan, Dun, on percutsion... 330, &c. Keygtone of an arelit 199 Laws of motioD, 91—87 Lever, 131—148,366 ■ I-, angular, 140, cor. 7. — — , heavy, #....135, 136 Lifting-pump, 596 Line of direction^ 106 — of support, 411 Logarithmic, atmospherical, 498 M. Macbinei, m,....- 196 ■, maximum, efiects of 363—380 Machinery, simplification of, vol. II, lAan, strength oif,M 378 Mass, 9, 911 Materials, strength of, 167—19^ Matter, 3 Measure of the force of gravity, 5t49 Mechanics, 1 Mechanical powers, 196—166 Jiftm Metacentre, ^ 411 Mobility, 8 Moment, or momentum, 31, 812 Momentum of impulse, i... 365 of inertia, 302 Motion, 11 ■ , absolute and relative, 19,919 , uniform, fcc 915 bylaws of, 91,97 Muffle, 148 N Natnreand art, difference between, 173 Newton on the lever, 140 — on (distance of fluids, 541 Kewtonian axioms, 91—97 Nicholson's hydrometer, 409 O Oblique impact, 360—369 Orbit, 979 Orifice, discharge of water through, 438—457 Oscillation, centre of, 305—308 , of pendulums, 968 , of vesselsn 493,494 Overshot-wheels, •. 467, 8ic. P Papin, on motion of air into a vacuum, 51 1 Pkradox^ hydrostatic, 387 Parallelogram of forces, 49 Pkrcienz's areometer,. 405 Pendulum, angular, 311 -, ballistic, 313 , compound, 308 , conical, 988 ,cycloidal, 973—976 , simpjft. 968—979 Percuasioo, centre of, 317—320 , theory of, 399—369 Periodic Ume, 980 ■ of planet depends on its mean distance solely, 991 Perpetual screw, , 160 Piers and arches...... .199—908 Pinion and tooth, .14G Piston, 524 Pivots, pressure upon, 145 Place, absolute and relative....... ^ PUne, inclined, 159—137 5(» INDEX. Art, Atne, motion along... '.258, S60, 967 » 315, 369 Planets, double motion of, 33*7 Pneumatics. 484^560 Polyspscton, 148 Position, centre of, 109 Power, 17 Pressure, centre of, 391 — 393 — ->— of non-elastic fluids, 384—^95 Projectile force, 279 Projectiles, 348— -357 Pulley, 148—151 ', motion of, 361, 391, 314 Pomps, theory of, 524— 58ft Quantity of matter. See Mass. -of motion, 213,313 Rftdit of small cylinders, to esti- mate, fMe, 408 Radius vector,; ..-....■ 279 Random, or rau;^ 250 Rarity, or rareness, ..-..« 2tt Reduction of forces 55 lU-sistance of solids, 167 .- of fluids, 539—561 Resolution of forces, 28 — 101 — --of motions, 217 Resultant, 29 RetsgrdiUion, 540 Ricochet firing, 254 Robin:»*s exp^H'iments on the re- sistance of flurds, 353 — 554 Robison, on altitudes by barom. 505 on domes, 209 on relative strength, 1 90 Rotation, spontaneous, 322 , cM»iitre of^; 823—326 Rotatory motion, 300—328 Roy, on expansion of mercury, &c. 503 Sailing of ships, 549 Screw, 158—161 Ships, stability of, 411 , oscillations of, 425 Shuckburgh, on alt. by barom. ... 507 Smeaton, on water-wheels, 477—483 — — , on windmills, ?ol. H, Solid, 3 Solids, equal in equal timet....... 991 Space, absolute, ^ $ — -, initial, 39D , relative, $ Span of anarch,.... 199 Spandrils,. 199 Specific gravity, 383, 396—410 Spouting fluids, 45^-456 Spring of an arch,..., 199 Stability of vessels, 411—43$ , indiffereoty') -» negative, > .412,418 — 1 positive, J Statics, 31-309 Steam, its efiRects, vol.11. Steelyard, 137,138 Strain,; 168 Strength and stress, 167^—191 Sucking-pump, 524 Symmetrical bodies^ ..<.. 107 T Taotochrones, 276 Teeth of wheels, the best forms of, 1 47 Tension of a cord, 193 Time, absolute and relative, J 4 Time in which the planets would fall to the sun,. 292 Tooth and pinion-work, 146 Trajectory,...- « 279 Tubes, capillary, 42t>— 43,'» -^ to measure their diamet. note, 426 U Undershot-wheels, 467, kc. Uniformly varied motion,... 224—230 Unitof time,. 14 Variable motion, 231 — 339 Veli que centre, nofe, 425 Veloftity, 15 , accelerated, or retarded, 15 — ^ , virtual,.... noCe, 130 , angular, 279 >' due to a height, 244 Vena contracta, 440, 437 Venturi on motion of fluids,. 462 Vertical projectiles, 245 Vessels, stability of, .....* 411 ', oscillations of, 425 INDEX. 569 Art. Vibrating cone, 311 Vibration of pendolums, 268 Vince, on motion offluids, 465 — - on friction. Sec vol. II. Virtual Telocities, mU, 130 VoanoirSy 199 W Water.wbeels, 467—476 — , Smeaton on, 477 — 483 Wcdjje, 162—166 Weight, 104 Wheel and axle,... 143— 147,367, 370 , pressure upon, 145, 321 Whirling motion of fluid», 444 Windmili-tailii, 547 , Smeaton on. i?ccvol.II. Working point, , ouj Yonnjf, Dr. M. on tlie motion of fluids ^'^ END OF VOL 1. .» f • .1. • i ERRATA IN VOL I. Pagt 39, line 9,ybf c : c, rtwi €f : c. — 36, line 23, for ap.c*, rtadAr^^. ^— 33, line 8, ybr (</ x fft), read + (C x so). — 94, line 15, for Cr.4., read Cor.4. — — 190, line 84, Uke away the sign + before *"*'^. w — 155, line 16, for a -^g read a 4-. •• • .. • — 168, line 85,yar — — + — , read ■» ^ + —. t y i —- . 168, Hue 4 of note, read P£quilibre des — 165, line S^farf « BT, read r « by. — -* 165, line 1 1,/br — oc B, read — oe B« » T — 187, line 6 from bottom, /or /-i-V ^^^{—\ »*— S86, line 5 from bottom, for force, read forcei. r— — 888, line 4 firom bottom, far ay be, read may be. — — 879, line Iffbr x zi oo, tmi? ar=s oo • — ~- 883, line 81 , for ai at its, read at itti — — ^884, line 86, ^orrv^-f, readr^\». •— i- 316, line 86, put i for i in the denominator. -— * 513, line H,for iafertigation, read In? ettigation. ——647, line 3,/or ^ ', rttd *^ j. te- to T. Davimm, Lombard-street, Whitefriars, London. 