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.

2. If P moves along a vertical line, P' does not change, but Q R turns round it, remaining parallel to O B.

FIG. 1, 2, 3.

Without taking the trouble, as I ought to have done, to inquire what previous investigations had achieved in this matter, I thought, three years ago, I could get an apparatus to save me the trouble of drawing sum curves, made somewhat after the following fashion.

P (Fig. 2) is the guide or point to be taken round the primitive. It is attached to a block, D, which works along the bar, B C, which in its turn moves on the four wheels, e e f f, upon the frame R S U T fixed upon the drawing board. O A is fixed perpendicular to R U, and is such that O may be fixed at various points to determine the polar distance. O B D is a light bar passing freely through B and forming one side of a parallel ruler of two or more points, g g, h h, i i. Along i i is a slot and in this works a loaded block containing a wheel P', whose plane is always parallel to i i. This block also passes through a slot in D E, an arm at right angles to B C. A little consideration will show that P', if worked at all, would trace out the sum curve of P.

It was only when I showed the rough idea of this to Professor Kennedy, with the view of ascertaining what would be the amount of back-lash and friction, that I learned that Mr. Boys had already invented a very similar integrator. In his model the double parallel ruler is replaced by two endless strings and pulleys, and the bar, B C, by a T square.

Although this integrator was afterward made in a less crude form, I do not think it has ever been a practical instrument for the draughtsman. Shortly afterward I came across a work by Abdank-Abakanowicz, entitled "Les Integraphes," being a study of a "new kind of mechanical integrator."

The new kind of integrator was really only an independent version of Boys' instrument, but in many respects a great improvement. The real merit will ultimately belong to the scientific instrument maker who constructs an instrument reasonably cheap and capable of efficient practical service. Abdank-Abakanowicz's integrator however certainly went further in the practical direction than any previously constructed. The drawing board machines, it is true, of rather a complex nature, were actually exhibited to the Paris Academy, but no more have been made. The instrument before me was made by Coradi, of Zurich, on conditions laid down by me, namely, that the cost should not exceed £14, and that polar distances should range between one and ten half-inches. The first machine made by Coradi on these lines was, by a misunderstanding, sold in Germany, but the one I exhibit is the first, I believe, that has reached England, and to this extent I may, perhaps, be permitted to call it new. I look upon it rather as a suggestion upon which a still more practical instrument can be made in this country than as a perfect model. I believe there would be a wide sale for such an instrument were it once generally known to exist, and, what is more to work efficiently. It remains for me to point out in what the Abdank-Abakanowicz, or, rather, Coradi, integraph differs from Boys' instrument.

Two points deserve special attention. In the first place, the fixed frame is abolished, and the horizontal motion of P (Fig. 3), the guide point, is produced by putting the whole frame on friction rollers; in the second place, as a necessary result of the first change, the guide point carries about with it its own polar system, which renders the changes in length of "rays" much more manageable. f f, f' f' is a frame moving on four roughed wheels, e e e e, so that it can only move in the direction, f', which we may term horizontal. f f and f' f' are rails guiding the chariots, A and B, from f to f and from f' to f'. Of these chariots, A contains the guide point, P, to trace out the primitive with, and B the pencil, P', to draw the sum curve, i.e., the tracer. The chariot, B, like Boys' tracer, is heavily loaded. g g is a horizontal bar rigidly attached to the crossbars, q q and q' q', of the frame. On g g is a movable pivot, to which h, which determines the pole, k₀ h being the polar distance. k₀ is the position of a second point, k, on the chariot, A, when the guide point, P, is on the initial line, g g. l l is a bar with a long slot in it, in which work the pivots, h and k; this bar represents the "ray." A projecting arm k k' has been introduced to enable me to shorten the polar distance down to 2 in. and under by removing the pivot, k to k'. m m is a bar attached to the block, n, which runs on l l, so that m m is always perpendicular to l l. On the chariot, B, is another bar, m' m', capable of turning round the pivot, d, and always maintained parallel to m m by the rods, m m', m m'. Attached to m' m' is a wheel, w, whose axis is parallel to m' m'. This wheel, therefore, always moves perpendicular to m' m', and therefore to m m; hence it moves parallel to the ray, h k. A pencil, P', attached traces out the sum curve. If we wish to use the machine as an integrator, we have merely to measure the vertical distance traversed by P', or the distance B has run along f' f'. This is done by means of a scale on f f'. If k be brought down to k₀, w runs parallel to g g, or P' traces out a horizontal straight line, which is thus the base line. If k be fixed as near as possible to