54 



OF PREPONDERANCE, AND THE MAXIMUM OF EFFECT. 



559. Theorem. The centre of oscilla- 

 tion of a triiinj^le, suspended at its vertej^, 

 and vibrating in a direction perpendicular to 

 its plane, is at the distance of ^^ of its height 

 from the vertex. 



Calling, for the sake of simplicity, the base of the triangle 

 unity, the fluxion of the rotatory power is x'i, the fluent 

 Jx', the distance of the centre of gravity |r, the product 

 ^«', and the quotient Jr. 



SECTION Xn. OF PREPONDERANCE, AND 

 THE MAXIMUM OF EFFECT. 



560. Theorem. In order that a smaller 

 weight may raise a greater to a given height 

 on an inclined plane in the shortest time 

 possible, the length of the plane must be 

 to its height as twice the greater weight to 



the smaller. 



Let the descending weight be i , the ascending a, and the 

 length of the plane to its height as x lo l, the weights being 

 simply connected by a thread and pulley ; then the portion 

 of the power employed m maintaining the equilibrium is 



—(255), and the remaining portion 1 ; and the weight 



lo be moved being constantly 0+1, the velocity produced 



a a 

 by the acting power 1 will »ary as 1 , and the 



square of the time of describing i, as x : f 1 l (233), or 



XX 



— — , thefluxion of which vanishes when it is a minimum; 



Suppose the two weights fixed nt opposite ends of a lerer, 

 and let it be required to determine their respective distances 

 from the fulcrum, so that the velocity of the ascending 

 weight may be the greatest possible ; let this weight be 

 called a, and its distance from the fulcrum unity, the de- 

 scending weight being I, and its distance x. Then, if the 

 weights were in equilibrium, a wouldbe zzx; and the dif- 

 ference o(x and a, or x—a, is the force tending to raise a j 

 but the mass to be raised is equivalent to a+arx, for tjie 

 mass of the weight 1 acts in the duplicate ratio of its dis- 

 tance from the fulcrum (349), and the velocity of a wiH 



fx — a)ixi 

 =0, hence 



no; and multiplyrngby ■ 



[x—a)' " ^ " ' XX 



therefore — 



X 



a(x — o) — x=:o, X — 2«:=0, and xzz^a, 



361. Theorem. If a given weight, or 

 any equivalent force, be employed to raise 

 another given weight by means of levers, 

 wheels, pullies, or any similar powers, the 

 greatest effect will be produced, if the acting 

 weight be able to sustain in equilibrium a 

 weight about twice as great as the weight to 

 be raised, when thi* weight is very large ; or 

 about twice and a half as great, when the 

 weights are nearly equal. 



X — a ... . ' 



be — , and its fluxion — — 



a+xx o+xx (a+xx)' 



a+xx— 2xx+2axz:o, orrxx — 2ax, and adding aa, xx— 



2nr+(7n=a+oa, {x—ay'=Za+aa, x—azZy/{a+aa),x::: 



a+ </(«+«<')• Hence if am, xz:i + v'2; if az: 00 



x::^ia. And the same reasoning is applicable to any other 



mechanical power. If the mass of the machine be also 



considered, let the weight of each of its parts be reduced to 



the place of a (349), and let b be equivalent to their sum, 



then the velocity will become. , and x:z.a+.J 



^a+i+xx 



(a+i+oa). 



362. Theorem. If the heights of descent 

 and ascent, and the descending weight be 

 given, the operation being supposed to be 

 continually repeated, the effect will be great- 

 est in a given time when the aseending^^ 

 weight is to the descending weight as 1 to- 

 1.61 8, in the case of equal heights; and ia 

 other cases, when it is to the exact counter- 

 poise in a ratio which i& always between 

 1 : 1.5 and 1:2. 



Let the height of descent be i, that of ascent 0, the de- 



1 



seending weight 1, and the ascending - ; then the equili- 



X 



brium would require x^ia (313), and 1 — i» the force acfc- 



X 



ing on I V but the mass, reduced, as before, is l-{- — , and 



X — a 



the relative fbrce , and the space being given, the- 



x-f-aa 



(x+aa \ 

 h233) ; and the whole effect in a 



given time being dtiectly as the weight raised, and inversely 



as the time of ascent, will be as -. ^ I ) • but wheik 



X \ x+aa I 



tbia.. is a maximum, its square is a maximum, aod. 



