FD, F D', being perfectly equal 'and 

 similar, the effect of the weight distri- 

 buted amongst them is the ; same as if 

 the whole weight rested upon any one. 

 Hence the condition of equilibrium is 



P xpp'p"=Vi x w w' ic". 

 Thus, if ic and w be each one foot, and 

 that p and p' be each 1 feet, and if 

 ic" be one inch, and p" be one foot ; we 

 have 



PxlOx 1 Oxl^Vxlxlx^ 



or 100P= A T v ; 

 that is 1200 P = W. 



In which case the weight would be 1200 

 times the counterpoise, and thus a 

 weight of one pound would balance 

 1200lbs. 



By a proper adjustment of the levers, 

 the indicating weight might be made to 

 have any convenient relation to the 

 weight to be ascertained. Thus one 

 ounce might correspond to one hundred 

 weight. This would be effected if w 

 and ir' were each one foot, and p were 

 8 feet, p" 16 feet, and w" one inch, and 

 p" } 4 inches : for then we should have 



Px8x 16 x i| = Wx 1 x 1 x T V 



or 1792 P = W ; 



that is, the weight is 1792 times the 

 counterpoise; and since there are 1792 

 ounces in one hundred weight, it fol- 

 lows, that one ounce will balance an 

 hundred weight. 



(30.) The mechanical efficacy of a 

 machine depends on the ratio of the 

 weight to the power, and is said to be 

 greater or less according as this ratio 

 is greater or less. It might be well at 

 once to define the mechanical efficacy to 

 be the numerical quota arising from the 

 division of the weight by the power. 

 Thus, if the weight be ten times the 

 power, the mechanical efficacy is ten. 

 If three times the weight is equal to 

 twenty times the power, the mechanical 

 efficacy is - or 6. It follows, there- 

 fore, that when this quantity is a proper 

 fraction, what in ordinary cases is me- 

 chanical efficacy, becomes a mechani- 

 cal disadvantage, as when the weight is 

 half the power. We have already seen 

 when the power on the lever is between 

 the weight and prop, the machine acts 

 in this way under a mechanical dis- 

 advantage. 



(3 1 .) The same lever admits of having 

 its mechanical efficacy varied at plea- 

 sure, by changing the positions of the 

 power and weight with respect to the 

 prop, so that it may be made to act with 

 any given mechanical efficacy, or even 

 with any mechanical disadvantage. This 



MECHANICS. 13 



is a property in which it is distinguished 

 from most other simple machines, and 

 one which renders it a convenient stand- 

 ard or modulus for representing all 

 other machines. Whatever be the pro- 

 portion of the power and weight in any 

 machine, a lever may be assigned in 

 which the power and weight will have 

 the same proportion, and which, there- 

 fore, we call an equivalent lever. 



As all simple machines may be repre- 

 sented by equivalent simple levers, so 

 all complex machines may be repre- 

 sented by equivalent systems of com- 

 pound levers. To effect this, it is only 

 necessaiy to determine the proportion 

 of the power to the weight in all the 

 simple component machines, w r hich, in 

 the complex machine under considera- 

 tion, are interposed between the weight 

 and the power, and take, a series of 

 levers whose alternate arms, beginning 

 from the power, bear the same propor- 

 tions to the remaining arms respec- 

 tively. Such a system of levers will be 

 mechanically equivalent to the complex 

 machine. 



Hence, and from the result of (26) 

 it follows, that in any complex machine, 

 the proportion of the power to the 

 weight may be easily assigned, pro- 

 vided the proportions of the power to 

 the weight in the simple component 

 machines are known. For let these 

 proportions be w : p, w : p 1 and w" : 

 p" ; then the proportion of the power 

 and weight in the complex machine 

 will be determined by 



P. p p'p' 1 = W, ic w' w". 



When we say that a lever or a sys- 

 tem of levers can always be found 

 which are mechanically equivalent to 

 any given machine, we would be un- 

 derstood to mean only, that the same 

 power will sustain the same weight in 

 each, but by no means implying that all 

 the mechanical functions of each species 

 of machine can be discharged by a 

 system of levers. 



CHAPTER V. The Balance common 

 Balance with equal Arms its requi- 

 sites sensibility Method of detect- 

 ing a fraudulen t Balance Steel-yara 

 Danish Balance bent leverBalance 

 Brady's Balance. 



(32.) THE BALANCE is an instrument 

 used for determining the weights of 

 bodies, by showing their relation to the 

 weights of some other bodies which are 

 known, or which are assumed as general 



