Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/891

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thefe parts are equal, that is, when G H is equal f.6 1? C, or B D equal to 2 B C. It appears, therefore» that when the power F and weight W are given, and W is to be raifed by an inclined plane, from the level of a given point A to the given point B in the lead time poflible ; we are firft to find the plane B C, upon which W would be Curtained by P, and to take the plane B D double in length of the plane B C ; or we are to make ufe of the plane B D, upon which a weight that is double of W could be fuftained by the power P.

For another example, fuppofe a fluid, moving with the velo- city and direaion A C, (Fig. 3.) ftrike the plane G E ; and fuppofe that this plane moves parallel to itfelf in the di-

Fig. 3.

■ tectiort C B, perpendicular to C A, or that it cannot move in any other direction. Then let it be required to find the moft advantageous pofition of the plane C E, that it may receive the greateft impulfe from the action of the fluid. Let A P be perpendicular to C E in P, draw A K parallel to G B, and let P K be perpendicular upon it in K, and A K will meafure the force with which any particle of the fluid impells the plane E C, in thediieSion C B. For the force of any fuch particle being reprefented by A C, let this force be refolved into A Q_parallel to E C, and A P perpen- dicular to it ; and it is manifeft, that the latter A P only has any eftl-fl upon the plane C E. Let this force A P be re- folved into the force A L perpendicular to C B, and the force A K parallel to it ; then it is manifeft, that the for- mer, A L, has no effect in promoting the motion of the plane in the direaion C B : fo that the latter A K, only, meafures the effort by which the particle promotes the mo- tion of the plane C E in the direaion C B. Let E M and E N be perpendicular to G A and C B, in M and N ; and the number of particles, moving with dirc&ions parallel to A C, incident upon the plane C E, will be as E M. There- fore the effort of the fluid upon C E being as the force of each particle, and the number of particles together,, it will lie as A K xE M ; or, becaufe A K is to A P (= E M,) as E N to C E, as EM^x EN ; fo that CE beinggiven,

the problem is reduced to this, to find when E M * X E N is the greateft poffible, or a maximum. But becaufe the fum ofEM-mdofEN 1 (= C M *1 is given, being al- ways equal to C E % it follows, that EN » X £ M *, is greateft, when EN'r jCE'; in the fame manner as it was demonftrated above, that when the fum of A C and CB (Fig. 1.) was given, ACxCE' was greateft, when A C = | A B. But when EN X E M 4 is greateft, its fquareroot ENxEM» is of ncceffity at the fame time greateft. Therefore the aBion of the fluid upon the plane CE in the direaion C B is greateft when EN' Si CB % and confequently E M ■ = § G E * ; that is, when EM, the fine of the angle ACE, in which the ftream ftrikes the plane, is to the radius, as the J 2 to J 3 ; in which cafe it eafily appears, from the trigonometrical tables, that this an- gle is of 54°. 44'.

Several ufeful problems in Mechanics may be refolved by what was (hewn in the preceding paragraph. If wereprefen't the velocity of the wind by A C, a feftlon of the fail of a wind-mill perpendicular to its length by C E, as it follows from the nature of the engine, that its axis ought to be turned direaiy towards the wind, and the fail can only move in a direaion perpendicular to the axis, it appears, that, when the motion begins, the wind will have the greateft effeft to produce this motion, when the angle AGE, in which the wind ftrikes the fail, is of 54". 44'. I„ the fame manner, if G U reprefent the direaion of the motion of a (hip, or the pofition ot her keel, abftraaing from her lee-way, and A C be the direaion of the wind, perpendicular to her way, then the moft advantageous pofition of the fail C E, to pro- mote her motion in the direaion C B, is when the an.de AG E, in which the wind ftrikes the fail, is of 5 i°. 44/ The beft pofition of the rudder, where it may have the <rreat- ett efrea in turning round the (hip, ij determined in like manner. And the fame angle enters likewife into the deter-

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ihination of the figure of the rhorhbules that form the bales ot the cells in which the bees depofite their honev, in the (noil frugal manner. See the article Bee, Suppl. But it is to be carefully obferved, that when the fine of the angle A C E is to the radius as ,/ 2 to J 3 ; on which is the fame thing, when its tangent is to the radius, as the dia- gonal of a fquare to its fide ; this is the moft advantageous angle only at the beginning of the motion of the engine ; fo that the fails of a common wind-mill ought to be Co (mated, that the wind may indeed ftrike them in a~greatcr angle than that ot 54°. 44'. For it is dcrnonftrabki that when any part of the engine has acquired the velocity c, the effort of the wind upon that part will be greateft, when the tangent of the angle m which the wind ftrikes it is to the radius, not

as the J 2 to i, but ^ a X 111. X IL to 1, the velocity

of the wind being feprefehted by a. If, for example, c = i a ; then the tangent of the angle ACE ought to be double of the radius ; that is, the angle AGE ought to be of 63° 26'. If c = a ; then A C E ought to be of 74° 10'. This obfervation is of the more importance, becaufe, in this engine, the velocity of the parts of the fail remote from the axis, bears a confiderable proportion to the velocity of the wind, and perhaps fometimes is equal to it ; and becaufe a learned author, Mr. Daniel Bernoulli, has drawn an oppo- site conclufion from his computations in his Hydrodynamics, by mifiakmg a minimum for a maximum ; where he inlets, that the angle in which the wind ftrikes the fail, ought to decreafe as the diftance from the axis of motion increafes ; that if c = a, the wind ought to ftrike in an angle of 45- ; and that if the fail be in one plane, it ought to be inclined to the wind, at a medium, in an angle of 50°. How he fell into thefe miftakes, is (hewn by Mr. Mac Laurin, in his rluxions, §. 914.

In like manner, though the angle A C E of 54°. 44'. be the moft advantageous at the beginning of the motion, when a (hip fails with a fide wind ; yet it ought to be enlarged aftei- wards as the motion increafes. In general, let A "a, paral- lel to C B, be to A C, as the velocity which the engine has already acquired in the direaion CB, to that of the ltream ; upon A C produced take A D to A C as 4 .to 3, draw 1 D G parallel to G B, and let a circle defcribed from the Center C with the radius C a, meet D G in g ; and the plane C E (hall be in the moft advantageous fituation for promoting the motion of the engine, when it biffcas the angle a Cg. It is generally fuppofed, that a direa wind always promotes the motion of a (hip, the fail being perpendicular to the wind, more than any fide-wind ; and this has been affirmed in feveral late ingenious treatifes ; but, to prevent miftakes, we are obliged to obferve, that Mr. Mac Laurin has demon- ftrated the contrary, in his Treatife of Fluxions, §. qiq ; where other inftances of this fecond general problem in Me- chanics are given, to which we refer. See Mac LaurhA Account of Sir Ifaac Newton's Philofophical Difcoveries, B, II. c. 3. p. 173. feq.

MECHOAGAN, (Cycl.) the name by which the great American Convolvulus, or bindweed, is known in the (hops. See the article Convolvulus, Suppl.

M&QlC-vetcbing; a name by which fume call the Onobnchis, a dittina genus of plants. See the article Onoerychis, Suppl.

MEDICA (Suppl)— &7/W-Medica, the Engliffi name of a genus of plants, calied by botanical writers Medicago. See the article Medicago, Suppl.

MEDLAR (Suppl,) — Parjlcy-leavcd Me d l as, the name By which fome call a fpecies of Crataegus. See the article Cra- taegus, Suppl.

MEDUSA, in zoology, the name of a genus of infeas, of the order of the Gymnarthria. See the article Gymnarthria, Append.

The body of the Mcdufa is of an orbiculated convex figure of a gelatinous fubftance, and not hairy. The tentacula, or the plica;, which ferve inftead of them, are fituated in the cen- ter of the under part of the animal. Authors have defcribed them under the names Urtica marina, and Puhuones marini. See the article Urtica Marina, Suppl.

MELANCHOLY tbijlk, Cirjium, in botany, the name of a fpecies of Tbifile. See the article Thistle, Suppl.

MELANTHIUM, in botany, a name ufed by fome authors for a fpecies of Nigella. Sue the article Nigella, Suppl.

MELOE, in zoology, the name of a genus of four-winged flies, whofe antenna; are (lender and filiform ; and the' exte- rior wings dimidiated. It is called in Englifh the oil-beetle, as being foft and mucuous to the touch. It is black, but not at all glofiy, and has been defcribed by authors under the names Scarabaus mollis, Cant bants uncluofus, and Projcara- baus. See the article Scarabjeus, Suppl.

MELON (Suppl.)— (Pater-MnLOrt, the name of a genus of plants, called by botanifts Anguria. See the ankle An- guria, Suppl.

TTu/rVt'-MELoN, CacJus, in botany, the name of a genus of plants. See the article Cactus, Suppl. and Append.

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