## Problems Involving Averaging.

To illustrate the application of this Equation in this connection let us take a rectangle having an area equal to the area of any given plane figure.

The area of this rectangle will evidently be

A' = Base × Altitude ... (2)

Since the area of the given figure and the rectangle are equal, or A = A', we have

t × c × r = Base and altitude ... (3)

If now we make the length of the Tracer Arm t equal to the base of the rectangle, we have

c × r = Altitude ... (4)

and since c × r is the distance rolled by the Wheel during the tracing of the given figure, Eq. 4 shows that the rolling of the Wheel measures the altitude of the given area whose base is equal to the length of the Tracer Arm.

If instead of a rectangle the figure is in the form of the figure given in Fig. 1 of Plate V, the formula for the area still holds good with the exception that the altitude in this case is the mean distance between the base AB and the curved line CD, which is evidently the mean height of the figure ABCD.

Formula 4 then becomes

c × r = Mean Ht. ... (5)

and hence to obtain the Mean Ht. of any figure of a form similar to ABCD, we have simply to make the length of the Tracer Arm equal to the length of the Base AB and trace the figure ABCD in the usual way. The number of complete revolutions and fractions of a revolution made by the Integrating Wheel during the tracing of the given figure multiplied by the circumference of the Wheel is then the quantity c × r of Eq. 5, and is the mean or average height of the given figure, since it is the distance rolled by the Wheel. The same result is obtained without making the length of t equal to the base of the figure by tracing the figure with any length base and dividing the Area thus obtained by the length of the given base obtained by direct measurement by scale, and this is the method necessarily adopted when the Planimeter used has a fixed length of Tracer Arm, as is the case in the cheaper grades of the instruments.