Measurement of Mechanical and Electrical Forces. 83 



After the cylinder and disk, the most simple form for an 

 integrating surface is that of a sphere. Let a sphere be sup- 

 ported, with its axis hori2ontal, on a frame which can be 

 made to reciprocate about a vertical axis which would, if con- 

 tinued, pass through the centre of the sphere ; then, if a tan- 

 gent-wheel is fixed so as to lie, when its inclination is nothing, 

 in the horizontal plane which passes through the axis of the 

 sphere, angular reciprocation, which must be less than 180°, 

 will cause the tangent-wheel to describe on the sphere a me- 

 ridian when it is in its neutral position, or a rhumb line if 

 inclined at a constant angle. As the speed of rotation of the 

 sphere is inversely proportional to the radius of contact — that 

 is, to the cosine of the latitude of the point of contact — some 

 means must be adopted whereby the rotation recorded is less 

 than the rotation of the sphere in the same ratio. The most 

 simple plan is to use Amsler's principle, and mount a small 

 sliding and rolling wheel so as to be in contact with the sphere 

 at the highest point on the equator (*. e. 90° from the tan- 

 gent-wheel), but with its plane passing through the centre of 

 the tangent-wheel : then the rotation of the Amsler wheel is 

 always less than the rotation of the sphere, in the same ratio 

 that the rotation of the sphere is too great. Instead of an 

 Amsler wheel, a cylinder capable of moving longitudinally on 

 its horizontal axis, and in contact with the sphere at a point 

 exactly opposite to the tangent-wheel, would, by pure rolling 

 and without any sliding, take off the correct proportion of 

 motion, since it and the tangent-wheel always touch the 

 sphere at points having the same radius. 



Fig. 4 is a perspective view of a polar planimeter in which 

 the integration is effected by a disk sphere and Amsler wheel, 

 as described. All the parts marked a belong to a rigid frame, 

 which balances on and can turn about a vertical spindle, 

 the top of which is just visible below the tangent-wheel t. 

 The vertical spindle is fastened to the stationary wheel ic, 

 which rests on three feet. The segmental wheel W in gear 

 with to is secured to a vertical spindle, the upper end of which 

 carries the crutch C. Screws in the crutch form the hori- 

 zontal axis about which the sphere S may rotate. The tan- 

 gent-wheel t is mounted in a frame which can be turned 

 about a horizontal axis e by means of a lever I. The Amsler 

 wheel rests by its weight on the highest point of the equator 

 of the sphere, which is shown dotted. DD is an L-shaped 

 piece, which carries at the angle the pointer P. At the end 

 of the long limb is a slot embracing a pin, as shown. A part 

 of the short limb is made cylindrical : against this part rests 

 the edge of the lever I. This edge is not truly radial, but is 



