From this it follows that the error of the planimeter is less than 0.1 per cent. and that of the integraph about 0.5 per cent.  Obviously we could make this error much less if we excluded small areas measured with large polar distances, or such polar distances that the cross bar must be shifted.  Excluding such cases, we see that the accuracy of the integraph scarcely falls behind that of the planimeter and is quite efficient for practical purposes.  It must be borne in mind that the above measurements were made with the “control lineal,” an arrangement which carries the guide round a circle of the exact test area.  In most cases the curve has to be followed by hand, and the error will be greater—­greater probably for the integraph than for the planimeter, as the former is distinctly hard to guide well.

I think, then, we should be safe in saying that the error of the integraph is not likely to be greater and is probably less than 2 per cent., so that in this respect the instrument may be considered a practical one.

5.  A further condition for a good integraph is that it should have a wide range of polar distances, and that it should be easily set at those distances.

One of the conditions I gave to the maker of the instrument was that it should be able to take all polar distances from one to ten half-inches.  This condition he can scarcely be said to have fulfilled.  With polar distances of 1/2 inch and 1 inch, the machine works unsatisfactorily, which indeed might have been foreseen from the construction of its sliding bars.  It works best from 2.5 inches to 5 inches, and this is the range to which I think we ought to confine the present type of instrument.  As the last conditions I may note that: 

6.  A practical integraph ought to be easy to read.

7.  Draw a good clear curve.

The scale on the present instrument is very inconvenient, as it is often almost out of sight; the curve it draws, on the other hand, I consider very satisfactory, when the pencil is loaded, say, with a planimeter weight.  On the whole, I think you will agree with me that this integraph goes a good way, if not the whole way, toward fulfilling the conditions of a practical instrument.

I next turn to its construction and the claim it has to be considered in any way new.  Let me briefly remind our members of the process by which an element Q R of the sum curve (Fig. 1) corresponding to the point P on the primitive is drawn; P M being the mid-ordinate of L N, a horizontal element, P B is drawn perpendicular to any vertical line A B; and O A being a constant distance termed the base or “polar distance,” Q R is drawn between the ordinates of L and W, parallel to O B. If P’ be the point where P M meets Q R, we note the following relationship of P’ to P.

1.  If P moves along a horizontal line, O B remains unchanged, and, therefore, Q R or P’ must move in the straight line Q R parallel to O B.