Clerk Maxwell in 1855 designed a planimeter in which pure rolling was substituted for the undesirable partial sliding of the register wheel on the cone or disc which occurred in previous types, but the instrument was never constructed. James Thom son in 1876 investigated the same problem, and in attempting to simplify Maxwell's mechanism, evolved his disc, sphere, and cylinder combination which could be applied to the construction of a planimeter.
I., fig. 4, which represents the original model of Kelvin's harmonic analyser, made in 1876. The plane of the circular disc is inclined at 45°, and the sphere (shown displaced from its proper position) rests against the disc and the cylinder. The points of contact of the sphere are on a generating line of the cylinder and the horizontal diameter of the disc, and the distance of the sphere from the centre of the disc is controlled by the movement of a rod carrying two forks, between which the sphere fits when the instrument is in use. The sphere acts as an intermediate variable gear, which communicates the rotation of the disc to the cylinder at a rate directly proportional to the distance between the axis of the disc and the centre of the sphere. This rod carries near its other end a pointer. When used as a planimeter, the curve whose area is i required is wrapped round the larger cylinder, and as this is made to rotate the pointer is guided so as to follow the point of the curve which is on the topmost generating line. If the instru ment is adjusted so that the sphere is at the centre of the disc when the pointer is on the axis of x, then the distance of the sphere from its central position will be always equal to the ordinate y of the curve, and the measuring cylinder will give, by the amount of its rotation, the value of f y.da—i.e., the area of the curve. Owing to the difficulties and cost of construction, this type of planimeter was never made commercially, but Kelvin adopted the mechanism in his harmonic analyser.
A planimeter of the polar type, in which the recording wheel, kept in the required position by means of a guiding curve, rolled on the paper, was designed about 1856 by Gierer of Fiirth. Bouniakovsky of St. Petersburg and Decher of Augsburg in 1856 each proposed an instrument of this type, in which the guiding curve was replaced by linkwork. C. V. Boys in 1883 invented a polar planimeter in which there was no slipping, but it was never constructed commercially.
Jacob Amsler, about 1854, invented his polar planimeter, which from its simple construction and low price very soon came into extensive use ; up to 1884 Amsler had made over 12,000 examples of this instrument. An example made by Stanley about 1875 is shown in P1. I., fig. 3. In using the instrument, the weighted point F is fixed and the tracing pointer T is guided exactly once round the outline of the figure whose area is to be measured. The difference of the readings on the graduated roller R before and after this operation gives the area of the figure in units depend ing on the radius TP of the tracing arm. Accurate setting of the tracing arm is facilitated by means of a fine screw adjust ment. The example shown is of the "proportional" type—i.e., the unit can be changed by altering the radius of the tracing arm.
About 1893, Coradi introduced his "compensation" polar plani meter, adopting a suggestion made by O. Lang, a Neuweid sur veyor. In this modification of Amsler's polar planimeter the tracing frame and the pole arm are made in separate parts so as to allow the pole arm to be placed on either side of the tracer arm. Any error due to non-parallelism of the axis of the roller and the tracer arm can be eliminated by taking the mean of two readings, one obtained with the pole to the left of the tracer arm, and the other with the pole symmetrically to the right.
Amsler in 1856 invented his "integrator," which will measure larger areas than can be measured by the polar planimeter. It will also measure the moment and the moment of inertia of an area about any axis lying in the plane of the area. The instrument (Pl. I., fig. 5) is carried by a pair of wheels moving in a straight groove in a long steel bar, and a counter balance weight is provided. If the tracing point T be guided so as to describe the outline of a plane figure, the graduated roller A attached to the swinging arm will register the area of the figure. The outer roller M will register the moment of the areas about the straight line described by the point where the axis of rotation of the instrument intersects the paper. The moment of inertia about the same straight line is deduced from the readings on the first roller and the inner one. Two gauges are provided for indicating the exact position of the straight line to which the moments are referred.