MACHINE. 



furcd by an uniform motion, in confcqiicnce of the period- 

 ical and regular repetition which may exill in the accele- 

 ration and retardation. Thus, the motion of a fecond pen- 

 dulum, confidered in relation to a fingle ofcillation, is ac- 

 celerated during the firll half fecond, and retarded during 

 the fecond ; but the fame motion taken for many ofcilla- 

 tions may be confidered as uniform. Suppofc, for example, 

 that the extent of each ofcillation is five inches, and that 

 the pendulum has made ten ofciUalions ; its total cfl'eft will 

 be to have run over 50 inclies in 10 feconds, and as the 

 fpace defcribed in ejch fecond is the fame, we may compare 

 the effects to a moveable, which moves for 1 o feconds at 

 the rate of five inches per fecond. We fee, therefore, that 

 the theory of machines, whofe motions are uniform, con- 

 duces naturally to the eftimation of the efFedts of ihofe 

 whofe motion is alternately accelerated and retarded, fo 

 that what follows wi'l be directed to thofe machines only, 

 whofe motion falls under the fecond head, fuch problems 

 bcinij of far the greatell utility in practice. 



We have had already frequent occafion to make ufe of 

 the terms mover, or moving force and rcjijlancc ; and in wliat 

 follows, they will be uled in the fame general fcnfe. By 

 the lirft is always to be underllood any caufe of motion 

 whatever, and by the latter, any thing that is oppofed to 

 the adion of the former. The impelled point of a machine, 

 is that to which the aAion of the moving power may be con- 

 fidered as immediately applied ; and the •working point is that 

 where the reliilance arifing from the work to be performed 

 immediately afts, or to which it ought all to be reduced. 

 Thus in the wheel and axle, Plate ^-JJg- 6. Mechanics, where 

 the moving power P is to overcome the weight or refillance 

 W, by the application of the cord to the wheel and to the . 

 axle, A is the impelled point, and E the working point. 

 The velocity of the moving power is the fame as the velocity 

 of the impelled point ; and the veLcity of the refijlance, the 

 fame as that of the working point. The perjormunce or 

 rJjeLl of a machine, or the '■^vork done, is mealured by the 

 product of the refiftance into the velocity of the working 

 point; and the momentum of impulfe h mcafured by the pro- 

 duct of the moving force into the velocity of the impelled 

 point. 



Thefe definitions being cdabliflied, we may exhibit a 

 few of the moft ufeful problems relative to the effect of 

 machines, and with which we mull conclude this article. 



Let A B {P late 'KX.y^ll. Mechanics, fg. 1.) reprefent the 

 velocity of a ftream, A C the velocity of the part of the 

 engine which it llrikes, when the motion of the machine 

 becomes uniform, and C B will reprefent their relative 

 velocity, upon which the effeft of the engine depemis. It 

 is known that the aftion of a fluid upon a given plane,' 

 is as the fquare of this relative velocity ; confequently the 

 •weight raifed by the engine, wheii its motion becomes uni- 

 form, being equal to this aftion, it is lik'jwife as the fquare 

 of C B. Let this be multiplied by A C, the velocity of 

 the part of the engine, impelled by the fluid ; arid the ef- 

 ft'fi of the engine in a given time will be proportional to 

 AC X C B- = (fuppofing C B to be bifi-fted in D) 

 ACx2CDx2DR-4ACxCDxDB; coniV- 

 <juently, the cffeft of the engine is greateil when the produdt 

 of A C, C D, and D B is greatelL But it is cafy to fee, 

 that this product is greatell when the parts AC, C U, ai;d ' 

 D B, are equal ; for if you defcribc a femicircle upon 

 A D, and the perpendicular C E meet the circle in E, 

 then ACxCb=CE, and is greatell when C is the 

 centre of the. circle ; fo that in order that AD x C D 

 X D B may be the greatell poffibie, A D mult be bifotled 

 in C ; and C B having beea bifcdcd in P, it follows 

 1 



that A C, C D, D B, mud be equal ; or that A C. t!tt 

 velocity of the part of the engine impelled by the ftream, 

 ought to be but one-third of A B, the velocity of the 

 dream. Iji this cafe, when, (abftrafting from friction) 

 the engine aCls with the utmoll advantage ; the weight 

 raifed by it is to the weight that would jiilt fullain the 

 force of the dream, as the fquare of C B, t!ie relative 

 velocity of the engine and llream, to the fquare of A B, 

 which would be the relative velocity, if the engine wa& 

 quiefcent ; that is, as 2 x 2 to 3 x 3, or 4 to 9. There- 

 fore, that the engine may have the greatell cfiVct pofiible, 

 it ought to be loaded with no more than ^ths of the weight, 

 which is juft able to fullain the efforts of the llream. See 

 Maclaurin's Account of fir Ifaac Newton's Difcoveries, 

 p. iji, and Fluxions, art. 908. 



Again, fuppo'ethat a given weight P. (Jig. 2.) defcend- 

 ing by its gravity in the vertical line, raifts a greater weight 

 W, likewife given, by the rope P M W, (that pafles over 

 the fixed pul'ey M) along the inclined plane B D, the height 

 ot which B A is given ; and let it be required to find 

 the pofition of thi.s plane, along which W will be raifed 

 in the lead time, from the horizontal line A D to B. 

 Let B C be the- plane upon which, if W was placed, it 

 would be exactly fudained by P ; in which cafe, P is to 

 W as A B to B C. But W is to the force with which 

 it tends to defcend along the plane B D, as B D to A B ; 

 confequently the weight P is to that force, as B D to B C. 

 Therefore the excels of P above that force (which excefs 

 is the power that accelerates the motion of P and W) is to 

 P, as B D - B C to B D ; or taking B H upon B C equal 

 to B D, as C H to B D. But it is known that the fpaces 

 defcribed by motions uniformly accelerated, are in the com- 

 pound ratio of the forces which produce tliem and the 

 fquaros of the times ; or, that the fquare of the time is di- 

 rectly as the fpace defcribed in that time, and invcrfely 

 as the force ; confequently, the fquare of the time in which 

 B D is defcribed by W, will be direflly as B D, and 



inverfely as 



that 



.ind will be lead when 



BD^ 

 CH' 



when 



BC^ 

 C H 



-f C H + 2 B C, or (becaufe 



B C- 

 2 B C is invariable) when ~— rj + C H, is a minimum. 



Now as when the fum of two quantities is given, their pro- 

 duct is a maximum when they are equal to each oiIkt ; fo 

 it is manifed, that, when their product is given, their fum 

 mull be a minimum when they are equal. Thus it is evi- 

 dent, that as in Jig. i, the redtangle or produdt of the 

 equal parts A C and C D was equal to C E' ; fo the reiftan- 

 gle of any two unequal parts, into which A D may be di- 

 vided, is lefs than C E , and A D is the leall fum of any 

 two quantities, the produdt of which is equal to C E'. But 



B C" 



the produdt of and C H is B C, and confequently. 



given : therefore the fum of and C H is lead when 



thefe parts are equal, that is, wJien C H Ls equal to B C, 

 or B D equal to 2 B C. It appears, therefore, that wlien 

 the power P and weight W are given, and W is. to be riiifed 

 by an inclined plai.e, from the level of a given point A to 

 the given point B in the lead time poflible ; we are fird 

 to find the plane B C, upon which W would be fudained by 

 P, and to take the plane B D double in length of the plane, 

 B C ; or wc arc to make ufe of the plane B D, upon which 



a weight 



