THEOKY OF THE BALANCE.] MECHANICAL PH I LOSOPHY. STATICS. 



723 



The scale-pans and the cords, or apparatus by which 

 they are suspended, must also be made as light as can 

 be, compatible with the uses of the balance, as they in- 

 crease the values of P and Q, whose sum is an element 

 which diminishes the sensibility. 



When equal weights act on A and B, the extremities 

 of the beam, its stability is measured by its tendency to 

 resume a state of rest, after its equilibrium has been 

 disturbed. This tendency will depend upon the mag- 

 nitude of the sum of the moments of the forces, about 

 the point C, for any given position of the beam ; i.e., for 

 any particular value of 6. 



The sum of the moments of the three forces P, Q, and 

 W about C, when A B is inclined at an angle to the 

 horizontal, or C D at the same angle to the vertical, 

 will be 



W-Cw+Q-Cg P -Op; 



orWbsin. + Q(csin. + a cos. 0)-P(acos. 0-csm.0). 

 When P and Q are equal, this expression becomes 

 W 6 sin. -f- P c sin. -f P c sin. ; 



or(W-6+2P-c)sin.0. 



When Is zero, sin. 6 is also zero, and this expression 

 becomes the same, or the sum of the moments is zero, 

 which it must be, in order that there 'should be equili- 

 brium. But for any other finite value of 0, the magni- 

 tude of the liability will be greater, as W ' & -f- 2 P c is 

 greater ; that is, as W, P, 6 and c are greater. 



These conditions are contrary to those required for 

 sensibility, which for any given value of o, demand that 

 W, P, b and c shall be as small as possible. 



Hence, while the arm of the balance remains of the 

 same length, any increase of sensibility is made at the 

 expense of its liability ; but the quantities b, c, W and 

 P remaining the same, we may increase the sensMIity 

 without injury to the stability by increasing a, or the 

 length of the arm of the balance. 



Win-re minute differences of weight are not matters of 

 importance, and quickness of determination essential, as 

 in weighing substances rapidly, which are not of great 

 value in proportion to their weight, stability is of more 

 importance than sensibility. The reverse, however, is 

 the case where the substance is of great value in pro- 

 portion to its weight, or where extreme accuracy is 

 rcquirrd. 



In all that has been said, it must be remembered that 

 the weight P includes that of the scale-pan, together 

 with the apparatus by which it is suspended. 



The scale-pan must be so suspended, that its centre of 

 gravity, as well as that of the weight or substance placed 

 in it, may be exactly in the vertical line passing through 

 the extremity of the beam, from which it is suspended. 

 From the mathematical expressions given above for sen- 

 tibility and itabiliiy, both depending upon P, it follows 

 that the same balance will have different degrees of these 

 qualities according as it is used for determining greater 

 or less weights. The quantities of substances which are 

 used in philosophical or chemical balances being generally 

 very small, a balance which is very sensible when no 

 weight is placed in the scales, will be of almost equal 

 sensibility for every weight with which it is intended to 

 be used. 



A needle is usually fixed to the beam in the direction 

 of the line CG, which points vertically upwards or 

 downwards when the beam is in a horizontal position. 

 A graduated arc or scale attached to the support of the 

 balance to indicate the arcs described by the extremity 

 of this needle, as the beam oscillates, is a very convenient 

 addition. When G is very near C, the oscillations will 

 be very slow ; in this case, the equality of the weights 

 in the scale-pans may be ascertained by means of the 

 icdex-needle, before the balance comes to a state of rest. 

 When the weights are equal, the extremity of the needle 

 will describe equal arcs on both sides of the vertical line; 

 when they are unequal, the scale which preponderates 

 will be indicated by a greater arc being described on that 



side than the other. This method renders stability of 

 much less importance than sensibility. 



Where great accuracy is required, the disuse of this 

 long index-needle, almost in contact with the graduated 

 arc, and the substitution in its place of a graduated arc 

 attached to one end of the beam has been recommended. 

 This arc is viewed through a fixed compound microscope, 

 having a horizontal wire in the focus of the eye-piece ; 

 or by a mirror attached to the beam, iu which the re- 

 flected image of a scale is viewed through a telescope. 



In very accurate instruments the scale-pans are sus- 

 pended from steel knife-edges, resting on agate planes 

 fixed to the extremities of the beam. The balance should 

 be constructed as much as possible of brass, as steel and 

 iron are apt to acquire magnetic properties. Palladium 

 has been used for the construction of the beam, and 

 platinum is to be preferred for the scale-pans. When 

 glass scale-pans are used, care must be taken not to 

 excite their electrical properties ; a difference of half a 

 grain may be produced by merely cleaning one of the 

 glass scales with a dry silk handkerchief. In construct- 

 ing the beam, care must be taken by making it hollow, 

 or if solid, by cutting out portions of it, to secure the 

 greatest degree of lightness with the greatest length of 

 arr' which is compatible with its rigidity, or the re- 

 ten- -n of its form, under all the weights to which it is 

 intei. >)d to be subjected. 



All \ cry sensitive balances have a contrivance by which 

 the knife-edges, which support the beam and scale-pans 

 are lifted from the planes on which they play, whenever 

 the instrument is not in use. The knife-edges are thus 

 preserved from becoming blunt, or wearing the agate 

 planes, and the beam is freed as much as possible from 

 every strain, which would tend to alter its shape. 



In addition to all these precautions, the balance should 

 be enclosed in a glass case, for its preservation from dust 

 and injury : to prevent error from the action of currents 

 of air, a window at the side affords the means of intro- 

 ducing weights and the substance to be weighed, without 

 removing the case. A cup containing quick-lime, or 

 some other powerful absorbent of .moisture, should also 

 be kept within the case. 



Ramsden constructed a balance for the Royal Society 

 of such extreme sensibility, that when weighted with 10 

 pounds, it turned with about the thousandth part of a 

 grain. 



From the above description, it is manifest, that the 

 construction of a perfect balance may be regarded as 

 impossible, though one may be nearer than another to 

 perfection. Borda invented a very simple method, by 

 means of which very accurate results may be obtained 

 by a balance sufficiently sensitive and well-constructed 

 on its knife-edges, though the points of suspension of 

 the scale-pans may not be equidistant from its fulcrum. 

 Fig. 103. 



a 



His ingenious device is to weigh the article whose 

 weight is to be determined by weights or any other sub- 

 stance, such as sand, placed in the other scale : when 

 equilibrium has been thus obtained, the article is re- 



