three, determine the exact amount to be cut off. As, however, it will not do to make it too light, we cut off a somewhat less quantity and weigh again; by a few repetitions of this process we may reduce the error to a very small amount.

This method is sometimes the only one available, but it should not be too generally used, as it encourages guessing at results, and tends to destroy habits of accuracy.

GRAPHICAL METHOD.

Suppose that we have any two quantities, x and y, so connected that a change in one alters the other. Then we may construct a curve, in which abscissas represent various values of x, and ordinates the corresponding values of y. Thus suppose we know that y is always equal to twice x. Take a piece of paper divided into squares by equidistant vertical and horizontal lines. Select one of each of these lines to start from. The vertical one is called the axis of Y, the other the axis of X, and their intersection, the origin. Make x= 1, y will equal 2, since it is double x; now construct a point distant 1 space from the origin horizontally, and 2 vertically. Make x = 2, y = 4, and we have a second point; x=1, gives y=-2, &c., and x = 0, gives y = 0. Connecting these points we get a straight line passing through the origin, as is evident by analytical geometry from its equation, y = 2x. Again, let y always equal the square of x, and we have the corresponding values x 0, y 0; x 1, y 1; x = - −1, y 1; x = 2, Y 4; connecting all the points thus found we obtain a parabola with its apex at the origin, and tangent to the axis of X. As another example, suppose we have made a series of experiments on the volumes of a given amount of air corresponding to different pressures. Construct points making horizontal distances volumes, and vertical distances pressures. It will be found that a smooth curve drawn through these points approaches closely to an equilateral hyperbola with the two axes as asymptotes. Now this curve has the equation xy a, or y: =ax, that is, the volume is inversely proportional to the pressure, which is Mariotte's law. Owing to the accidental errors the points will not all lie on the curve, but some will be above it and others below, and this will be true however many points may be observed.

In general, then, after observing any two quantities, A and B, construct points such that their ordinates and abscissas shall be these quantities respectively. Draw a smooth curve as nearly as possible through them, and then see if it coincides with any common curve, or if its form can be defined in any simple way. To acquire practice in using the Graphical Method it is well to construct a number of curves representing familiar phenomena, as the variation in the U. S. debt during the late war, the strength of horses moving at different rates, and the alterations of the thermometer during the day or year. It is by no means necessary that the same scale should be used for vertical, as for horizontal distances, but this should depend on the size of paper, making the curve as large as possible. The greatest accuracy is attained when the latter is about equally inclined to both axes.

It is sometimes better when one of the variables is an angle to use polar coördinates. In this case paper must be used with a graduated circle printed on it. The points are constructed by drawing lines from the centre in the direction represented by one variable, and measuring off on them distances equal to the other. For ordinary purposes circles may easily be drawn, and divided with sufficient accuracy by hand. Laying off the radius on the circumference divides it to 60°; bisecting these spaces gives 30°, and a second bisection 15°. By trial these angles may be divided into three equal parts, which is generally small enough, as the observations are usually taken at intervals of 5°.

Interpolation. All kinds of interpolation are very readily performed by the Graphical Method. After constructing one curve to find the value of y, for any given value of x as x', we have only to draw a line parallel to the axis of Y, at a distance x', and note the ordinate of the point where it meets the curve. Inverse interpolation is performed in the same manner, and this method is equally applicable, whether the observations are at equal intervals or not. As by drawing a smooth curve the accidental errors are in a great measure corrected, this method of interpolation is often more accurate than that by differences.

Residual Curves. The principal objection to the Graphical Method, as ordinarily used, is its inaccuracy, as by it we can rarely obtain more than three significant figures, although Regnault, by

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 x, 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 a', y' be the observed coördinates, and y = f(x), the assumed curve. Construct a new point, whose coördinates are x' and a [y' — f(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 change 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 ∞. 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.

B

C

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 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.

A

Fig. 2.

E

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.