## Problems Involving Averaging.

### Method of Measurement.

The use of the Polar Planimeter in measuring these diagrams and obtaining their mean height is exactly the same as for the plain diagram used in our description, and as follows:

Let abce in Fig. 3, Plate III, be the indicator diagram from which we desire to ascertain the mean effective pressure of steam in the cylinder of the Engine from which the diagram is taken. The paper or cardboard on which the diagram is drawn should be pressed down smoothly on a very level board or table, and if, as is generally the case, the paper or cardboard is too small to contain the entire path of the Planimeter Wheel during the tracing of the diagram, a supplementary sheet of paper having a surface favorable for the purpose must be smoothly lain down in such a position relative to the diagram that the entire path of the wheel shall be within its area during the entire tracing of the diagram.

The base of the diagram having been decided on, the length of the Tracer Arm FT is made equal to the length of the base of the diagram. This adjustment may be made in the manner already described by measuring the length of the base by scale, or, if the Planimeter has special attachment for making this adjustment, it can be used in the manner shown in Fig. 3 of Plate III referred to.

The adjustment for length being completed, the Planimeter is placed in the most favorable position, the Tracer brought to the place of beginning already determined, and the instrument brought to a Zero Reading.

The periphery of the diagram is then traced in the direction of the hands of a watch, in exactly the same manner as for the measurement of the area of a plane figure, and the tracer having completed its circuit of the diagram, the Reading of the Planimeter is taken. Unlike the Reading for the measurement of an area however, which, as we have seen, we consider as expressed in Vernier Units, the Reading in the present case is expressed in complete revolutions and tenths, hundredths and thousandths of a revolution, the only difference between the two being in the position of the decimal point. This Reading then multiplied by the circumference of the wheel gives the actual distance rolled by the Wheel during the tracing of the diagram and is, as we have just proved, the average or mean height of the given diagram.