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Problems Involving Averaging.

Application To or Discharge of River and Similar Diagrams.

As a further example of the use of the instrument in this particular form of problem, let us take a case of very frequent occurrence in engineering work— that of obtaining the average discharge of a stream or river from a set of observations extending over a given time.

In Fig. 2 of Plate IV is given a diagram taken from a Report of the United States Geological Survey, and is the graphical representation of a set of observations taken twice a day, and covering a period of one month or thirty days of the observed volumes of discharge of a river.

In all these diagrams it makes no difference in their reduction by means of the Planimeter whether the diagram be a plotted one, or whether it be the record of a self-recording mechanism, providing only that their size be such as to come within the capacity of the instrument.

In the case under discussion a line AB is taken as a base, the height of which is entirely arbitrary, provided only it does not exceed the maximum length of the tracer arm of the planimeter to be used in its reduction. This base AB is divided into thirty equal parts to correspond to the thirty days covering the period during which the observations were taken, and through each point of this division of the base perpendiculars of indefinite length are drawn.

As two observations were made each day, each one of the thirty divisions is divided into two equal parts, and through these intermediate parts other perpendiculars are drawn as shown on the given figure.

A vertical scale is now assumed in which a unit of length is made to represent graphically a certain number of thousand or million gallons discharge. In the particular diagram under description a linear inch was made to represent say one million gallons, and the rate of discharge observed each half day is graphically represented by making the length of the perpendicular which represents the day or half day on which the observation was taken equal in length to the observed rate of discharge, thus making the perpendicular above the base AB as many inches and decimals of an inch long as there was million gallons and decimals of a million gallons discharge at the time of the observation.

It is evident from this description of the construction of the diagram that it is simply plotting the “curve” of the variable to rectangular coordinates, the times of observations being plotted as abscissas and the corresponding discharge ratio as ordinates.

If now we connect the top of each perpendicular with the top of its adjacent perpendiculars by straight lines we shall have the broken line which forms the upper side of the figure we are discussing. This broken line is then the graphical representation of the fluctuation of the discharge of the river for the thirty days during which the observations on the discharge were made. By increasing the number of observations it is evident that the number of small lines which unite the tops of the perpendiculars will be proportionally increased while their length is diminished, and if the number of observations be very large the lengths of these connecting lines will have become so small as to be practically elements of a curve. This is what happens when the diagram is automatically drawn by a self-recording mechanism, in which case the observations being continuous are infinite in number and the connecting lines are infinitely small.

It is evident then that both in the case of a diagram constructed as above described with a comparatively few numbers of observations, and in the case of the self-recorded diagram in which the observations are infinite in number, the average or mean discharge will be average or mean of all the observations and will be represented by a perpendicular whose length is the mean height of all the perpendiculars or the mean distance between the base AB and the broken line or curve EF of the given diagram.

If the Planimeter we are using is of the simple form and has a Tracer Arm of fixed length it is obvious that to obtain the length of this mean perpendicular or the mean distance between the base AB and the line EF we have simply to measure the Area ABCD of the given diagram, and since we know the length AB of the base the average or mean distance will be equal to the measured area divided by the given length of base. This being done we have simply to multiply this distance by the number of gallons per linear unit to which the observations have been plotted, and the result will be the average or mean discharge for the entire period through which the observations were made.

If on the other hand we have a Planimeter with an adjustable length of Arm, graduated into half millimeters and having the attachment already described, we adjust the Tracer Arm so that it is equal in length to the base AB of the given diagram and trace the diagram in the usual manner. The Reading for the given diagram multiplied by the circumference of the Wheel will then give the required mean height of the diagram in millimeters and this height divided by 25.4001, the number of millimeters in a linear inch and multiplied by one million, the vertical scale to which the observations are plotted gives the average or mean rate of discharge for the month during which the observations were made.

The application of this operation to the finding of the position of the center of gravity of any plane area is obvious, and although of very great importance will on that account not be given.

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