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.1. PHILOSOPHY. 8TATIO& [THBOUTOF THB BAI 



a dish is gincndil, in one of which the substance to be 

 - placed, and in the other the weight* by which 

 its value is determined. 



The lever is to constructed as to be capable of moving 

 on it* axis in a vertical plane ; and when a given weight 

 is placed in one dish, ami a substance equal in weight to 

 it in tin- other, alter vibrating some little time, it as- 

 sumes a horizontal p->iti..ii \\heii a slight excess of 

 weight is iiiMe.l t<> cither dish, the lever again vibrates 

 and auMiinc* such a position of rest, that the extremity 

 above the dish containing an excess of weight over the 

 .* tin- horizontal line. The lever is called 

 .ml t'h,- two dishes, scale-pan*, 

 are three requisites in a balance : 

 When equal weights are placed in the scale- 

 pans, the two extremities of the beam should rest in a 

 perfectly horizontal line. This is called its hori&mtality. 

 Second. On the slightest addition of weight to either 

 scale, the beam should lose its horizontal position. This 

 is called its ttnsibility, and is measured by the smallest 

 weight which causes the beam to depart from its liori- 

 tontalitii. 



Third. After any disturbance the beam should as- 

 sume a state of rest as speedily as possible : this is called 

 its liability. 



iiall now proceed to consider the mathematical 

 conditions which must bo satisfied, in order to obtain 

 these requisites ; for this purpose we must first confine 

 our attention to the construction of the beam. 



Let A B (Fig. 101) represent the beam, G its centre 

 of gravity, and C the point on which its axis or fulcrum 

 is supported. 



Fig. 101. 



Join A B, and produce C G to meet A B in the point 

 D. 



To secure the horizontality of the beam, the points C 

 and G must not coincide, for if they did, from the pro- 

 perties of the centre of gravity, we should have in- 

 different equilibrium, and the beam, either by itself or 

 when loaded at its extremities with equal weights, 

 would rest in every position in which the line A B might 

 be placed. 



\\ IH-II C does not coincide with G, by Prop. XIX., 

 and the beam is at rest, C G will assume a vertical 

 position; hence in order that A B may be horizontal, 

 C D must be perpendicular to A B. In order that this 

 horizontal position may be that of equilibrum when 

 equal weights are suspended from A and B, we must 

 also have A D = B D. 



The axis is generally a prismatic piece of metal, 

 which pierces and also is firmly fixed at right angles to 

 tin- IK-IUII. This prism rests on one of its edges, tech- 

 nically called the Knife-edge, on a plane or curved surface, 

 laced as a support on each side of the beam, that 

 Igo or line of support about which the beam oscil- 

 lates is horizontal and perpendicular to the plane in 

 w hi. -li it oscillates. The axis might be a cylinder or 

 cone working in a socket, but the hiife-edye is generally 

 I i. f.-rred, in order to avoid friction as much as pos- 

 sible. 



"ii'litions of horizontality given above require, 

 that tin- piano passing through the line of 

 support and the centre of gravity of the beam shall be 

 at right angle* to, and also bisect, the line passing 

 through its extremities, or the points from which the 

 scale- pans are suspended. 



To determine the other two conditions, retaining the 



me letters as before, let two unequal weights P and Q 

 wnted in magnitude and direction by 

 A P and B Q, be suspended from A and B. 



Let be the angle A B makes with the horizontal line 

 ;.' '/' drawn through the point D, and let p g cut A P 

 in p aud B Q in q. 



Through C draw p g parallel to pf g', and cutting A P 

 Fif. 103. 



f 



A 



in p, and B Q in q. Let W, represented in magnitude 

 and direction by G W, be the weight of the beam and its 

 axis, and let the line G W cut p q in w, p' q" in /. 



Also let A D = p B = a, C G = 6, and C D e. 



Since C L) is at right angles to A B, and is the angle 

 A B makes with the horizontal line p' y', it will also 

 make the angle with the vertical line if G w'. 



Also because p 9 is parallel to p' q" and P p', w w' and 

 q q' are parallel to each other, p w = p' w , and q tr 

 = (f vf. 



Taking the moments of the three forces P, Q, and W, 

 about the point C, we shall have, as a condition of equi- 

 librium, 



W-Ctc-f Q-Co = P-Cp. 

 ButC 3 = ' 



- CGsin. 0+ GDsin. 6 + DBcos. 



6 sin. + (c b) sin. + a cos. 



e sin. -j- a cos. 0. 

 Again Cp = p to C to = p' w 1 Cto 



= p'D w'l) Cw 

 = a cos. (c t) sin. 6 sin. 

 = a cos. csin.0. 

 Also C 10 = b sin. 0. 



Substituting these values of Cq, Cp, and Cw in the 

 equation W - C w + Q- C q = P- Cp, we have 

 \\ -//sin.e-f-Q (csin. -fa cos. 0) P (a cos. 6 csin.0). 

 Dividing both sides by cos. 6 

 W-itan. + Q(cfcin. G + o) = P (a ctan. 0) ; 

 Or (W b + Q 

 tan. 

 P Q 



Now the sensibility of the balance for a given difference 

 of P and Q will be greater, the greater the angle also 

 for a given value of 0, the sensibility will be greater the 

 smaller the difference between P and Q. Since tan. is 



tnn. 

 greater, the greater the angle 0, it follows that p -- Q is 



a measure of the sensibility of the balance. 



Hence the greatest sensibility will he attained when 

 a WA + (I> + Q)c 



great> r " ~~ " " 



as possil.li-. 



In making this calculation wo have neglected the 

 friction of the edge of the fulcrum on its supports, 

 which will have a tendency to diminish the sensibility. 



In instruments where great accuracy is required, such 

 as chemical, philosophical, and assay balances, this 

 friction is diminished as much as possible by making the 

 knife-edge of the fulcrum of hard polished steel, and the 

 support on which it rests a plane of polished agate. 

 Having thus obviated the diminution of sensibility duo 

 to the friction, it appears from the above expression for 

 the sensibility of the balance, tliat it will be greater, 

 the greater a is, and the less W, 6, and c are. 



We must, therefore, male thu beam as light and as 

 long as we possibly can, and the distances of G and D from 

 C as small as may bo consistent with other conditions- 



