Let us now obtain the average of the three amplitudes 81, S3, S5 on the left and the two amplitudes 82 and 84 on the right. We have for the first average

expanding the terms in parentheses and rejecting all terms higher than the first degree, since they are negligible with respect to unity, we get as the value of our expression

Proceeding the same way for the value of the right-hand average, we have

We have consequently proved that the two averages are equal (to a first order of approximation); hence, the validity of our procedure for obtaining the zero point by taking the average of an odd number of readings on one side and an even number on the other, and obtaining the mean of these two averages for our zero point.

After one

66. Counterbalancing the Object with Weights. has always made sure that the beam arrests and pan supports are up, the object is placed on one pan (for direct weighings this means always the left-hand pan). Beginning with the largest integral weight of the set, we surmise which is the first weight, as we go downward in the scale, that will establish an inequality of weight in favor of the object, i.e., which is the largest single weight that will not overbalance the object. This weight is placed on the pan, and the beam arrests partially lowered to show whether the weight is too heavy or not. Upon the assumption that it is not too heavy, the beam arrests are raised, and the next weight of the set in order of descension is added, the beam arrests partially

lowered again to show the inequality and then raised. If this weight is too heavy it is removed and the next lower weight substituted for it; if it is not too heavy it is let stay on the pan and the next lower weight tried in addition, and so on down the set in regular manner to the one gram weight, it being always remembered to raise the beam arrests before the addition or removal of any integral weight. During the process of adjusting the integral weights the pan supports are always kept up. When the fractional weights are reached, the beam arrests are kept completely lowered but the pan supports are kept up during the addition or removal of a fractional weight and only released to show the inequality. The fractional weights are likewise added systematically in order of descension down to and including the five-milligram weight.

The adjustment to the nearest milligram is now made by means of the "rider." As already mentioned, § 64, this accessory is a small piece of platinum wire which is placed upon the beam of the balance, usually the right beam. The actual weight of a rider for giving milligrams should be the same number of milligrams as the number of whole divisions into which the beam is actually or constructively divided between the central prismedge and the terminal prism-edge. Thus if the beam is divided into six whole divisions, a six-milligram rider should be used, because (from the principle of moments) such a rider, if placed at division one on the beam, will produce an effect equal to one milligram in the pan; if placed at division two it will produce an effect equal to two milligrams in the pan, etc. Owing to the different calibration of balance beams and the different weights of riders every rider should first be proved before it is accepted for use (cf. § 83).

67. Obtaining the Fraction of a Milligram. It is not practicable to do this to the nearest tenth of a milligram by placing the rider at successive fractional divisions on the beam and noting the displacements of the needle upon releasing the pan supports, because even with the most careful final adjustments of the rider and with the utmost care in releasing the pan supports, the needle will persist in swinging through a small are which corresponds to several tenths of a milligram. Recourse is consequently had

to the procedure known as the "method of swings" which is not only very simple but also time-saving, i.e., the rider having been adjusted to the nearest milligram, the pan supports are lowered and the zero point for loaded pans is determined. The difference between this zero point and the zero point for empty pans, divided by the sensibility for the given load, gives the fraction of a milligram to be added or subtracted from the weights on the pan inclusive of the rider. An illustration of the weighing of a platinum crucible by the balance mentioned in § 63 will make this clear.

Zero point with empty pans

10.2

Zero point when crucible was counterbalanced by 22.487 g. 11.7 Difference in zero points (displacement being to the right) 1.5 Sensibility for this balance for load of 22 g. is 2.8 (§ 63) . ́. correction is 1.5 ÷ 2.8 = 0.5 mg. which must be added because weights were too light.

As soon as the zero point for loaded pans has been determined, the balance is stopped swinging by bringing up the pan supports when the needle is passing by the middle point of the scale. The beam arrests are then immediately raised so as to take the load off the prism-edges.

68. Directions for the Use of the Balance.

1. The balance should be set up on a solid platform fixed to a wall or to a pier which is as free from mechanical vibration as possible; to further reduce vibration it is advisable to interpose glass cups mounted on rubber stoppers between the legs of the balance and the platform. A place should be chosen where neither direct sunlight, nor hot nor cold draughts, will come in contact with the balance, as resulting temperature changes will produce errors in the weighings.

2. When not in use the beam arrests and the pan supports must be up so that the prism-edges are free of contact with their opposing planes, otherwise injury to the edges will result from jarring.

3. In releasing the balance, first bring the prism-edges into contact with their opposing planes very carefully and gently; then lower the pan supports and start the balance swinging by wafting down a gentle current of air on the right-hand pan by a few quick impulses of the hand. Never touch the pan with the forceps nor attempt to start the beam swinging by suddenly lowering it upon its prism-edges.

4. As the first step of a weighing, always ascertain the zero point of the balance. If it is more than one division away from the center point of the scale, proper adjustment should be made by turning the small nut Y on

the right-hand beam. The student, however, should not make this adjustment himself, nor any other adjustment that might be needed, but should bring the matter to the immediate attention of the instructor.

5. During a weighing or upon any occasion when objects are being added to or removed from the pans, both the beam arrests and pan supports must be raised so as to protect the prism-edges.

6. As the last step in a weighing, first bring the pans to rest by raising the pan supports just when the needle passes by the center of the scale. Then raise the beam very carefully and gently, remove the weights and the object, and finally close the balance case.

7. No chemicals, minerals, salts or like substances which can possibly injure the balance pans should ever be placed upon them, but upon a watchglass or in some suitable vessel like a weighing bottle or crucible. The practice of keeping an open dish of concentrated sulphuric acid or anhydrous calcium chloride to maintain a dry atmosphere inside the balance case has been shown by O. Kuhn to be of no value and should be avoided as it endangers the balance through the possibility of the dehydrating agent overflowing or upsetting.

8. Objects should never be weighed while hot, as the air currents produced inside the balance case will introduce very serious errors in the weighing.

9. The weights should be handled only with the pincers provided for that purpose. The large weights should be placed near the center of the pan and the fractional weights should be arranged on the pan in some systematic order with respect to their denominations.

10. The weight of the object or substance should be recorded, first, by adding up the weights missing from the box (in which count every weight should have its own place) and, second, by checking them off as they are returned to the box. This procedure gives a check which is most desirable as it happens more often than would be supposed that the first count was incorrectly made.

11. Care should be taken not to overload a balance by weighing bodies heavier than those for which the balance is designed to carry (cf. § 63).

69. Errors of Weighing. The chief errors that may occur in weighing are those due to the following causes: the inequality of the length of the balance-arms, the inaccuracy of the weights, the buoyant effect of the air, a change in the composition of the substance during the time of weighing, a change in the condition of the container.

70. Inequality of Length of Balance-arms. Since the length of the two arms of a balance are never equal, it is evident that an object will apparently have different weights according to whether it is placed in the left-hand or in the right-hand balance-pan. As

O. Kuhn, Chem. Ztg. 34, 1097 (1910).

it can be shown that the true weight lies between these two apparent values, it follows that every "direct weighing" is attended with a certain error. Whether this error is allowable depends upon its magnitude and the accuracy which is being sought. In a good analytical balance the relative lengths of the arms seldom differ by more than one part in 40,000, but in many of the cheaper balances the difference is much greater than this.

It might be thought that whatever the ratio of the relative lengths might be, it could be determined at the start, and then used as a factor to correct subsequent weighings. This procedure, however, is not valid because the ratio is not a constant quantity, but varies from time to time on account of temperature changes inside the balance case, or, indeed, it may acquire an entirely different value due to a displacement of a prism-edge through accident or jarring. For the former reason a balance should be protected from direct sunlight and from all hot and cold draughts of air. Moreover, any object to be placed on the balance-pans should be at, or very near to, the same temperature as obtains inside the balance case. For the latter reason a balance should be protected from jarring or vibration, and the prism-edges always brought down upon the agate planes very gently.

With reference to the accuracy being sought, it must be emphasized that whenever the element of absolute value enters into the determination in any way, either directly or indirectly, it is necessary to eliminate the error introduced by the inequality of the balance-arms. This elimination can be accomplished by either of the two methods to be described presently: (1) Gauss's method of double weighing; (2) Borda's method of substitution. As an illustration to show where the element of absolute value enters into a determination, let us consider the calibration of a piece of volumetric apparatus. Its capacity is determined by dividing the weight "in vacuo" of the water (W) which is contained at a given temperature, by the weight "in vacuo" of one cubic centimeter of water (w) at the same temperature. The values for (w) have been determined once and for all in terms of the absolute unit, and are so given in the tables. It therefore becomes obvious that if a "direct weighing" is made, the recorded weights when reduced to "in vacuo" (W') will contain the error