84 Mr. C. V. Boys on Apparatus for the 



laterally displaced from the radial position to an extent equal 

 to the radius of the cylindrical part of D D. This causes the 

 true radius, which is parallel to the axis of the tangent-wheel, 

 to intersect the axis of the cylinder. Now, if the pointer P 

 is moved radially in the slot prepared for it, it is clear that 

 the tangent of the inclination of the tangent-wheel t will 

 be proportional to the square of the distance of P from the 

 vertical axis about which the machine can turn, also that, 

 during any turning of the machine about this axis, the sphere 

 will turn about its vertical axis at a proportionate speed. Now 

 it has been shown that, when the sphere is made to turn about 

 its vertical axis, the rate of rotation of the Amsler wheel is 

 proportional to such rotation multiplied by the tangent of the 

 inclination of the tangent-wheel — that is, in this case to r 2 dd. 

 Therefore the whole rotation of the Amsler wheel is a measure 

 of \ r 2 dd ; and so, if the pointer P is taken round any closed 

 curve, the area of that curve may be read off from the Amsler 

 wheel. The wheel W is three times the radius of w ; so that 

 the pointer may, if necessary, be taken completely round the 

 pole, and yet the tangent-wheel will only move 120° on the 

 sphere in latitude. The diameter of the Amsler wheel is one 

 third of that of the sphere, so as to restore the diminished 

 speed. Unlike Amsler's planimeter, this one shows the incre- 

 ment of area for each part of a closed curve, the reason being 

 that it is an exact mechanical equivalent of the polar formula 

 for integration. Though the machine works very well, it can- 

 not be compared to Amsler's as a practically convenient in- 

 strument. 



An exact mechanical equivalent of the formula jj§rdrdO 

 would be produced by retaining all the last machine, except 

 the short limb of the L-shaped piece D D, and mounting on 

 the long limb a tangent-wheel to traverse a cylinder, the rate 

 of rotation of which for a given radial movement of the pointer 

 would be proportional to the distance of the pointer from the 

 pole — that is, to rdr — and the whole rotation would he§rdr. 

 Now, if the cylinder were by its rotation caused to change the 

 inclination of the lever I so that the tangent of the inclination 

 of I was proportional to the whole rotation of the cylinder, 

 then, when the pointer was taken round a curve, the rotation 

 of the Amsler wheel would be^rdrdd. In either case, 

 instead of an Amsler wheel, a cylinder mounted as described 

 on the last page would give the integral. 



