MACHINE. 



Tor it is tVie diiTerence between ttie velocities W an'l V 

 which is efficient, and theaftion, being conllant, will vary as 

 tlie fqiiare of the efficient velocity. Hence we (hall have 

 this analogy, 



: F :: (W-o)'-; (W - V)', 



and, confeqaently, 



¥ = <p 



velocity of the force, in order that the work done may be S 



maximum. 



Akhotigh the prolTure of an animal is not a£lually uniform 

 during the whole time of its aftion, yet it is nearly fo, and 

 therefore in geweral we may adopt this hypotbelis, in or- 

 der to approximate to the true nature of animal adion. On 

 which fuppofition the preceding propofition, as well as the 

 following one, will apply to animal exertion. By retaining 

 the fame notation^ we have alfo 



Or the 'oiorh done hy an animal Is the greatejl when the 'vg' 

 locity with which it moites, is ene-third of the grentejl •vetocitj 

 ivith which it is capable of moving ivhen not impeded. 



Agaui, lince we have 



T = <p 



W = 



V a/(P 



,/ - ^/ F 



Avhich formula, applied to the motion of animals, gives the 

 following theorem. 



The ulnic/l velocity with which an animal, unimpeded, can move, 

 is to the velocity ivith tvhich it m.ves when impeded with a given 

 rejijlance ; as the fquare root of its abfolute force to the dif- 

 Jercnce of the fquare roots of its abfolute and efficient forces. 



Again, to inveftigate expreffions by means of which the 

 maximum effeft.. in machines whofe motion is uniform, may 

 be determined. 



1. it follows from the obfervations made in the preceding 

 -part of this article, that when a machine, whether limple or 

 compound, is put into motion, the veloc;t es of the impelled 

 and working points are inverfely as the fwrces which are in 

 equilibrio when applied to thofe points in the direftion of 

 their motion. Confequently, if /" denotes the refillance 

 when reduced to tlie working point, and v its velocity ; 

 while F denotes the force acting at the impelled point, and V 

 its velocity, we (hall have F V :=; _/"■!), or introducing t, the 

 time, F V ^ zi: y "u / Hence 



In all working machines which have acquired an uniform mo- 

 tion/ the performance of the machine is equal to the momentum of 

 .the impulfe. 



2. Let F be the effort of a force upon the impelled point 

 of a macTiine, when it moves with a velocity V, the velocity 

 being W, when F = o, and let the relative velocity W — 

 V ^ u. 



Then, fince F = 



/W - 

 W 





^)-, b, 



the foregoing pro- 



pofition, the momentum of impulfe F V becomes 



becaufe, fince W — V = «, we have V = W — //. 



Now making this expreffion for F V a maximum, or fup- 

 preffing the conftant quantities, and making 



,a' (W — a) ^ a maximum, 

 we have, by throwing it into fluxions, 



a a « W — 3 a' a = O, or 2 W = 3 M, or a = I W ; 



whence, again, V = W- a =: W - JW = 4 W. 



•Confequently, when the ratio of V to ii is given by the 

 condruftion of the machine ; and the refillance is fiifceptible 

 of variation, we ought to load the machine more or lefs, till 

 the velocity of the impelled point is one-third of the greateH 



u 



W' " '' W 

 in the cafe of the maximum, we have alfo ' 



• FV=|?)V=$^.i W=/^<pAV, 



for the momentum of impulfe, or for the work done when 

 the machine is in the bed (late. 



Confequently, when the refiftance is a given quantity, we 

 mud make 



V : •u : : 9/ : 4. ?:, 

 •which ftnidlure of the machine will give the maximum effedl 

 = ,V ^ W. . 



If we enquire the greateft effect on the fuppofition 

 that 1^ only is variable, we mud make it infinite in the 

 above expreffion for the work done, which would then be- 

 come 



WF, orW -/, or W-//, 



V V 



including the time in the formula. 



Whence we come to this important conclufion, viz. 



Thai the fum of the agents employed to movea machine may le 

 infinite, <whHe the -effed is finite. 



For the variations of I, Avhich are proportional to this 

 fum, do not influence the above expreffion for the effeft. 

 The lad theorem may be applied to the aftion of men and 

 of horfes, with more accuracy than might at fird be fuppofed. 

 Obfervations have been made on men and horfes drawing a 

 lighter along a canal, and working feveral days together. 

 The force exerted was mealiired by tlie curvature and weight 

 of the track rope, and afterwards by a fpring ileelyard. 

 The produil of the force thus afcertained into the velocity 

 perhowt, was confidered as the momentum ; and in this way 

 the adlion of the men was found to be very nearly as 

 (VV — V)'. The aftion of the horfes, loaded fo as not to 

 be able to trot, was nearly as (W — V) •", or as (W — V^|^ 

 Hence the hypothefis above adopted may, in many cafes, be 

 fafely alTumed. According to the bed obfervations, the 

 force of a man at red is on an average about feventy pounds, 

 and the utmod velocity with which he can walk is about fix 

 feet per fecond, taken at a medium. Hence in the above 

 theorems ^ = 70, and W = 6 ; confequently F =: ^ 9 

 = ji^lbs., the greated force a man can exert when in mo- 

 tion, and he will then move at the rate of i W, or two feet 

 per fecond, or rather lefs than 1 4 mile per hour. 



The drength of a horfe is generally reckoned about fix 

 times that of a man, that is, about 42olbs. at a dead pull. 

 His utmod walking velocity is about ten feet per fecond ; 

 and therefore his maximum aflion will be i x 420=3 iS6fibs. 

 and he will then move at the rate of \ of 10, or 3^ ieet per 

 fecond, or nearly 2^ mi es per hour. In both thefe initances 

 we fiippofe the force to be exerted in drawing a weight, 

 by a cord running over a puUey, which makes its direction 

 horizontal. 



The theorem above given may ferve to fhew under what 



points of view machines ought to be coiifid-ered by thofe 



who would labour beneficially for tlieir improvement. The 



ikft objeCil of utihty is in furiiilhing the means of giving 



4 to 



