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.

[Illustration: 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 formin