G6‘

LECTURE VII.N

a force equivalent to two ounces, drawing the lever upwards; and instead ofone of the weights, we may place the end of the lever under a linn obstacle,and the equilibrium will still remain, the lever being now of the second kind.Here therefore, the weight remaining at the other end of the lever counter-balances a force of two ounces, acting at halt the distance from the new ful-crum; and we may substitute for this torce a weight of two ounces, acting atan equal distance on the other side ot that tulcrum, supposing the lever to besufficiently lengthened, and there will still be an equilibrium. In this case thefulcrum will sustain a weight of three ounces; and we may substitute for it aforce of three ounces acting upwards, and proceed as before. In a similarmanner the demonstration may be extended to any commensurable proportionof the arms, that is, any proportion that can be expressed by numbers; andit is easy to show that the same law must be true of all ratios whatever, evenif they happen to be incommensurable, such as the side of a square, comparedto its diagonal, which cannot be accurately expressed by any numbers what-ever; the forces remaining always in equilibrium, when they are to each otherinversely as the distances at which they are applied.

It is sometimes more convenient to have a series of levers acting on eachother, with a moderate increase of power in each, than to have a single leverequivalent in its effect. Wc may also bend either arm of a lever in any man-ner that we please, without altering its power, provided^that the direction ofthe force be perpendicular to the line drawn to the fulcrum; or if the force beapplied obliquely, it may always be imagined to act at the end of a lever equalin length to the perpendicular let fall from the fulcrum on the direction of theforce. Thus, if two levers are connected by a rope or bar, when the directionof one of them nearly coincides with that of the rope, a force applied trans-versely to the lever acts with a great mechanical advantage against the rope;but as the inclination increases, the advantage gradually diminishes, andchanges, at last, to an equal advantage on the side ot the rope and the otherlever to which it is attached. When therefore a great force is required in thebeginning of the motion, and afterwards a much smaller force with a greatervelocity, this apparatus may be extremely convenient: thus, in opening asteam valve, the pressure of the steam is at first to be overcome, and after this,little or no additional force is required ; and Mr. Watt has very ingeniouslyapplied this arrangement ollevers to the purpose in his steam engines. In the