Thomson, Proc. of the Royal Society, Nov., 1871, also in Nature, V, 106.

To acquire facility in using the Graphical Method, it is well to apply it to some numerical examples. Thus take the equation У ax3 + bx2 cx+d, assume certain values of a, b, c and d, and compute the value of y for various values of x. We thus get a curve with two maxima or minima, and a point of inflexion. Find their position first by residual curves, and then by the calculus, and see if they agree. In the same way the curve yx2 2αyx +a3y=b, has the axis of X for an asymptote. Assume, as before, positive values of a and b, and determine the area between the curve and asymptote, first by construction and then analytically.

PHYSICAL MEASUREMENTS.

The measurement of all physical constants may be divided into the determination of time, of weight and of distance, the apparatus used varying with the magnitude of the quantity to be measured and the degree of accuracy required.

Measurement of Time. A good clock with a second hand, and beating seconds, should be placed in the laboratory, where it can be used in all experiments in which the time is to be recorded. Watches with second-hands do not answer as well, as they generally give five ticks in two seconds, or some other ratio which renders a determination of the exact time difficult. The true time may be measured by a sextant or transit, as described in Experiment 16. This should be done, if possible, every clear day by different students, and a curve constructed, in which abscissas represent days, and ordinates errors of the clock, or its deviations. from true time. Short intervals of time may be roughly measured by a pendulum, made by tying a stone to a string, or better, by a tape-measure drawn out to a fixed mark. We can thus measure such intervals as the time of flight of a rocket or bomb-shell, the distance of a cannon or of lightning, by the time required by sound to traverse the intervening space, or the velocity of waves, by the time they occupy in passing over a known distance. After the experiment we reduce the vibrations to seconds by swinging our pendulum, and counting the number of oscillations per minute.

By graduating the tape properly, we may readily construct a very serviceable metronome.

Where the greatest accuracy is required, as in astronomical observations, a chronograph is used. A cylinder covered with paper is made to revolve with perfect uniformity once in a minute. A pen passes against this, and receives a motion in the direction of the axis of the cylinder, of about a tenth of an inch a minute, causing it to draw a long helical line. An electro-magnet also acts on the pen, so that when the circuit is made and broken, the latter is drawn sideways, making a jog in the line. To use this apparatus a battery is connected with the electro-magnet, and the pendulum of the observatory clock included in the circuit, so that every second, or more commonly every alternate second, the circuit is made for an instant and then broken. Wires are carried to the observer, who may be in any part of the building, or even at a distance of many miles, and whenever he wishes to mark the time of any event, as the transit of a star, he has merely, by a finger key (such as is used in a telegraph office), to close the circuit, when it is instantly recorded on the cylinder. When the observations are completed the paper is unrolled from the cylinder, and is found to be traversed by a series of parallel straight lines, Fig. 3, one corresponding to each minute, with indentations corresponding to every two seconds. The time may be taken directly from it, the fractions of a second being measured by a graduated scale. One great difficulty in making this apparatus was to render the motion

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Fig. 3.

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of the cylinder perfectly uniform, as if driven by clock-work it would go with a jerk each second. This is avoided by a device known as Bond's spring governor, in which a spring alternately retards and accelerates a revolving axle when it moves faster or slower than the desired rate. The seconds marks form a very delicate test for the regularity of this motion, since in consecutive minutes they should lie precisely in line, and the least variation is very marked in the finished sheet. It is a very simple matter by this apparatus to measure the difference in longitude of two points. It is merely necessary that an observer should be placed at each station,

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with a transit and finger key, a telegraph connecting them with the chronograph. They watch the same star as it approaches their meridian, and each taps on his finger key the instant it crosses the vertical line of his transit. Two marks are thus made on the chronograph, and the interval between them gives the difference in longitude. The advantage of this method of taking transits is not so much its accuracy, as the ease and rapidity with which it is used. Observers can work much longer with it without fatigue, and can use many more transit wires, thus greatly increasing the number of their observations. It is called the American or telegraphic method, in distinction from the old, or "eye and ear" method of observing transits, where the fractions of a second were estimated, as described in Experiment 15.

The chronograph is exceedingly convenient in all physical investigations where time is to be measured, and nothing but its expense prevents its more general application.

A simple means of measuring small intervals of time with accuracy, is to allow a fine stream of mercury to flow from a small orifice, and collect and weigh the amount passed during the time to be measured. Comparing this with the flow per minute we obtain the time. A less accurate, but much more convenient, liquid for this purpose is water, using, in fact, a kind of clepsydra. Where very minute intervals of time are to be measured they are commonly compared with the vibrations of a tuning-fork instead of a pendulum. A fine brass point is attached to the fork which is kept vibrating by an electro-magnet. If a plate of glass or piece of paper covered with lampblack, is drawn rapidly past the brass point, a sinuous line is drawn, the sinuosities denoting equal intervals of time, whose magnitude is readily determined when we know the pitch of the fork. A second brass point is placed by the side of the fork and depressed from the beginning to the end of the time to be measured. The length of the line thus drawn, compared with the sinuosities, gives the time with great accuracy. Recently a clock has been constructed, in which the pendulum is replaced by a reed vibrating one thousand times a second. The clock is started and stopped, so that it is going only during the time to be measured, and the hands record the number

of vibrations made. The reed produces a musical note, and any irregularity is at once detected by a change in its pitch.

Measurement of Weight. This is done almost exclusively by the ordinary balance, whose principle is so fully explained in any good text-book of Physics that a detailed description is unnecessary here. We test the equality in length of its arms by double weighing, that is, placing any heavy body first in one pan and then in the other, and seeing if the same weights are required to counterpoise it in each case. The center of gravity should be very slightly below the knife-edges. If too low the sensibility is diminished, if too high the balance will overturn, and if coincident with them the beam, if inclined, will not return to a horizontal position The three knife-edges must be in line, otherwise the centre of gravity will vary with the weight in the scale pans, and of course the friction must be reduced to a minimum. A high degree of accuracy may be obtained with even an ordinary balance by first counterpoising the body to be weighed, then removing it and noting what weights are necessary to bring the beam again to a horizontal position. A spring balance is sometimes convenient for rough work, from the rapidity with which it can be used. It may be rendered quite accurate, though wanting in delicacy, by noting the weight required to bring its index to a certain point, first when the body to be weighed is on the scale pan, and then when it is removed.

Measurement of Length. Distances are most commonly measured by a scale of equal parts, that is, one with divisions at regular intervals, as millimetres, tenths of an inch, &c. This scale is then placed opposite the distance to be measured, and the reading taken directly. To obtain greater accuracy than within a single division, we may divide them into tenths by the eye, as in Experiment 1. The steel scales of Brown & Sharpe are good for common measurements, and may be obtained with either English or French graduation. Instead of dividing into tenths by the eye, a vernier is frequently used. Thus to read a millimetre scale to tenths, nine spaces are divided into ten equal parts, each of which will be a tenth of a millimetre less than the divisions of the scale, as in Ex periment 2.

One of the best devices for measuring very minute quantities is

the micrometer screw. A divided circle is attached to the head of a carefully made screw, so that a large motion of the former corresponds to a very minute motion of the latter. Thus if the pitch of the screw is one millimetre, and the circle is divided into one hundred parts, turning it completely around will move the screw but one millimetre, or turning it through one division only one hundredth of a millimetre. One of the best examples of this instrument is the dividing engine, which consists of a long and very perfect micrometer screw with a movable nut. See Experiment 21, also Jamin's Physics, I, 25. It is much used in engraving scales, but it has certain defects which are unavoidable, and have caused some of our best mechanicians to give it up. For example, it is impossible to make a screw perfectly accurate, and every joint, of which there are several, is a source of constantly varying error. For these reasons, and owing to its expense, the instrument described in Experiment 22 is for many purposes preferable. Two blocks of wood are drawn forward alternately step by step, through distances regulated by the play of a peg between a plate of brass and the end of a screw. As all joints are thus avoided, and the interval is determined by the direct contact of two pieces of metal, great accuracy is attainable by it.

Where several scales are to be made with the utmost accuracy, one should first be divided as correctly as possible, and its errors carefully studied by comparing the different parts with one another, or with a standard. It may then be copied by laying it on the same support with one of the other scales, and moving both so that one shall pass under a reading microscope, the other under a graver. We may thus copy any scale with great accuracy, but the process is very laborious. A good way to construct the first scale is by continual bisection with beam compasses, as is done in graduating circles. The finest scales are ruled with a diamond on glass. M. Nobert has succeeded in making them with divisions of less than a hundred thousandth of an inch. The intervals are so minute that until within a few years no microscope could separate the lines. The method of making them is kept a secret. Mr. Peters, by a combination of levers, has succeeded in reducing writings or drawings to less than one six thousandth their original size. He exhibited some writing done by this machine, which