using a large plate of copper and a dividing engine to construct his points, attained a higher degree of precision.

It will be found, however, that in many of the most carefully conducted researches the fourth figure is doubtful, as for example, in Regnault's measurements of the pressure of steam, and even in Angström's and Van der Willingen's determinations of wavelengths.

By the following device the accuracy of the Graphical Method may be increased almost indefinitely. After constructing our points, assume some simple curve passing nearly through them. From its equation compute the value of y for each observed value of, and construct points whose ordinates shall equal the difference between the point and curve on an enlarged scale, while the abscissas are unchanged. Thus let x', y' be the observed coördinates, and y = f(x), the assumed curve. Construct a new point, whose coördinates are x' and a [y' —ƒ(x')], in which a equals 5, 10, or 100, according to the enlargement desired.

Do the same for all the other points, and a curve drawn through them is called a residual curve. In this way the accidental errors are greatly enlarged, and any peculiarities in the form of the curve rendered much more marked. If the points still fall pretty regularly, we may construct a second residual curve, and thus keep on until the accidental errors have attained such a size that they may be easily observed. To find the value of y corresponding to any given value of x, as x, we add f(x) to the ordinate of the corresponding point of the residual curve, first reducing them to the same scale. Most of the singular points of a curve are very readily found by the aid of a residual curve. See an article by the author, Journal of the Franklin Institute, LXI, 272.

Maxima and Minima. To find the highest point of a curve, use, as an approximation, a straight line parallel to the axis of X, and nearly tangent to the curve. Construct a residual curve, which will show in a marked manner the position of the required point. The same plan is applicable to any other maximum or minimum.

Points of Inflexion. Draw a line approximately tangent to the curve at the required point. In the residual curve the changa of curvature becomes very marked.

Asymptotes. Asymptotes present especial difficulties to the Graphical Method, as ordinarily used. Suppose our curve asymptotic to the axis of X; construct a new curve with ordinates unchanged, and abscissas the reciprocals of those previously used, that is equal to 1÷x. It will contain between 0 and 1 all the points in the original curve between 1 and co. It will always pass through the origin, and unless tangent to the axis of X at this point the area included between the curve and its asymptote will be infinite. When this space is finite, it may be measured by constructing another curve with abscissas as before equal to 1x, and ordinates equal to the area included between the curve and axis, as far as the point under consideration. Find where this curve meets the axis of Y, and its ordinate gives the required area. A problem in Diffraction is solved by this device in the Journal of the Franklin Institute, LIX, 264.

Curves of Error. This very fruitful application of the Graphical Method is best explained by an example. Suppose we wish to draw a tangent to the curve B'A, Fig. 2, at the point A. Describe

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a circle with A as a centre, through which pass a series of lines, as AB, AD, AE. Now construct C by laying off BC equal to AB', the part of the curve cut off by the line. We thus get a curve CD, called the curve of error, intersecting the circle at D, and the line AD is the required tangent. This is evident, since if we made our construction at this point we should have no distance intercepted, or the line AD touching, but not cutting, the curve. A similar method may be applied to a great variety of problems, such as drawing a tangent parallel to a given line, or through a point outside the curve.

Fig. 2.

Three Variables. The Graphical Method may also be applied where we have three connected variables. If we construct points whose coördinates in space equal these three variables, a surface is generated whose properties show the laws by which they are connected. To represent this surface the device known as contour lines may be used, as in showing the irregularities of the ground in a map. First, generate a surface by constructing points in which

ordinates and abscissas shall correspond to two of the variables, and mark near each in small letters the magnitude of the third variable, which represents its distance from the plane of the paper. If now we pass a series of equidistant planes parallel to the paper, their intersections with the surface will give the required contour lines. To find these intersections, connect each pair of adjacent points by a straight line, and mark on it its intersections with the intervening parallel planes. Thus if two adjacent points have elevations of 28 and 32, we may regard the point of the surface midway between them, as at the height 30, or as lying on the 30 contour line. Construct in this way a number of points at the same height, and draw a smooth curve approximately through them; do the same for other heights, and we thus obtain as many contours as we please.

They give an excellent idea of the general form of the surface, and by descriptive geometry it is easy to construct sections passing through the surface in any direction. An easy way to understand the contours on a map is to imagine the country flooded with water, when the contours will represent the shore lines when the water stands at different heights. This method is constantly used in Meteorology to show what points have equal temperature, pressure, magnetic variation, &c. Contour lines follow certain general laws which are best explained by regarding them as shore lines, as described above. Thus contour lines have no terminating points; they must either be ovals, or extend to infinity. Two contours never touch unless the surface becomes vertical, nor cross, unless it overhangs. A single contour line cannot lie between two others, both greater or both smaller, unless we have a ridge or gulley perfectly horizontal, and at precisely the height of the contour. In general, such lines should be drawn either as a series of long ovals, or as double throughout. There will be no angles in the contour lines unless there are sharp edges in the original surface. A contour line cannot cross itself, forming a loop, unless the highest point between two valleys, or the lowest point between two hills, is exactly at the height of the contour.

The value of contour lines in showing the relation between any three connected variables, is well illustrated in a paper by Prof. J.

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

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To acquire facility in using the Graphical Method, it is well to apply it to some numerical examples. Thus take the equation = 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 2ayx +a2y=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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30"

32"

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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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