MECHANICS, 



17 



The first member of this equation may 

 be taken as the measure of the sensi- 

 bility. 



(41 .) From the result of this investi- 

 gation, the student will find no diffi- 

 culty in drawing the following conclu- 

 sions : 



1. That all other things being the 

 same, the sensibility of a balance is in- 

 creased by increasing the lengths of its 

 arms. 



2. That all other things being the 

 same, the sensibility is increased, dimi- 

 nishing the weight of the beam. 



3. That the sensibility is increased 

 by diminishing the distance between the 

 centres of gravity and motion. 



4. That the sensibility is increased 

 by diminishing the distance of the line, 

 joining the points of suspension from 

 the centre of motion. 



5 . That the sensibility is greater when 

 the load is smaller. 



We cannot here pursue this subject 

 further, although numerous other in- 

 teresting consequences might be de- 

 duced. We have supposed that the line 

 joining the points of suspension is below 

 the centre of motion. This is not al- 

 ways the case, and when it is above it, 

 the formula which we have obtained 

 for the sensibility becomes 

 tan. D a 



E ~Gd (2 \V + E) b. 



"VVe leave the mathematical student 

 to pursue the effects of this modification 

 on the sensibility. 



(42.) In the practical construction 

 of a balance of a high degree of sensi- 

 bility for philosophical purposes, there 

 are many circumstances to be attended 

 to, which are properly enough neglected 

 in balances used for commercial pur- 

 poses. 



Fig. 26. is a representation of a very 

 sensible philosophical balance, by which 

 very minute differences of weight may 

 be determined. The beam S S' has arms 

 of equal length, and of perfectly equal 

 and similar figures. It is very accu- 

 rately placed upon knife-edges at 

 m, which rest upon highly-polished 

 plates of hardened steel. The beam 

 is only allowed to rest upon the plates 

 when in use. Two forks are placed 

 under its arms at G G', supported 

 by vertical pillars, which when raised 

 by screws which are represented at the 

 foot of each pillar, will lift the beam 

 from the plates on which the knife-edges 

 rest. By this, the wear arising from 

 the continued pressure of the edges on 



Fig. 26, 

 m 



the plates is avoided. A needle or index 

 is attached to the centre m and plays 

 upon a graduated arch below, and 

 points to zero on the arch when there 

 is exact equilibrium. A balance such 

 as this is generally inclosed in a glass 

 case, and only opened sufficiently to in- 

 troduce into the dishes the weights and 

 substances to be weighed. 



(43.) Commercial balances are fre- 

 quently misconstructed for fraudulent 

 purposes, by making the arm, from which 

 the substance to be weighed is suspended, 

 longer than that from which the counter- 

 poise is hung, thereby giving the counter- 

 poise, a greater leverage, and enabling it 

 to support a weight proportionally greater 

 than itself. The end to be attained by 

 the use of such a balance may be de- 

 feated in several ways. If the object be 

 merely to detect the fraud, it will be suf- 

 ficient, after equilibrium has been esta- 

 blished between the substance to be 

 weighed and the weights, to transpose 

 them, and put the substance to be 

 weighed in the dish in which the weights 

 were, and vice versa. If the balance be 

 honestly constructed, the equilibrium 

 will be undisturbed ; but if it be fraudu- 

 lent, the substance to be weighed will 

 preponderate, since, after the transposi- 

 tion, it will have the greater leverage. 

 But if the object be not alone to detect 

 the fraud, but to ascertain the true 

 weight of the substance, let the counter- 

 poise which will produce equilibrium 

 after transposition be found, and let 

 this and the former counterpoise be 

 reduced to the same denomination of 

 weight, and let the two counterpoises 

 thus expressed be multiplied together, 

 and the square root of the product ex- 



