(2) There is no relation, mathematical or other, between constant and mean errors, and they not only may be but must be isolated and studied separately, if an investigation is to be conducted in the interests of scientific exactness.
It will be necessary to reject the method of Dr. Woodworth if the most reliable results are desired, in which case the planimeter is a necessity.
The theory of the planimeter cannot be developed at this place; every physicist and engineer is acquainted with it. The writer believes he was the first to apply the planimeter to the calculation of results from psycho-physical data for averaging both mean and variable errors. More than 340,000 lines were involved, each demanding two measurements. The best type of planimeter for general use and the one used here is the Amsler adjustable-arm form.
In Fig. D is shown a record taken at twenty beats per minute that will both explain the method of computation and show how the planimeter has been used to find the constant and mean errors.
Fig. D
The record, as made and ready for computation, is not provided with the line cd or with the dotted lines that connect the ends of the ruled lines. The line ab is drawn by turning the drum of the apparatus with a pencil held at the end of the normal or left-hand line af, which was here 100 mm. long.
The tracing-point of the planimeter being placed at a, a reading is taken, which was in this case 1486; after following with the tracing-point the dotted path to g and returning, via gb and ba, a second reading is taken, which was 1248; subtracting gives 238, which should be read 2380 square mm. for the area of the space agba; dividing by the distance ab, in this case 119 mm., gives the average height, which is + 20.0 mm., the plus sign suggesting that the distance thus found, which is the constant error for the series, be laid off in addition to or beyond Fa.
This being done, a line cd is drawn parallel to and 20.0 mm. from ab, as the mean line of constant errors.