_:f4 ON A NEW FORM OF THE PLATOMETER. 



figure, then the rest of the framework will be turned through an equal angle, 

 and the index-sphere will roll on the hemisphere till it come into the position 

 represented in the figure. Then, if there be no slipping, the arc AP = BP, and 

 the angle ACP = SOP. 



Next, let the instrument be moved backwards or forwards, so as to turn 

 the wheel Kk and the hemisphere U, then the index-sphere will be turned 

 about its axis Ss by the action of the hemisphere, but the ratio of their veloci- 

 ties will depend on their relative positions. If we draw PQ, PR, perpendiculars 

 from the point of contact on the two axes, then the angular motion of the 

 index-sphere will be to that of the hemisphere, as PQ is to PR ; that is, as 

 PQ is to QC, by the equal triangles POQ, PQC; that is, as ED is to DC, 

 by the similar triangles CQP, CDE. 



Therefore the ratio of the angular velocities is as ED to DC, but since 

 DC is constant, this ratio varies as ED. We have now only to contrive some 

 way of making ED act as the generating line, and the machine is complete 

 (see art. 2). 



11. The arm CF is moved in the following manner: Tt is a rectangular 

 metal beam, fixed to the frame of the instrument, and parallel to the axis AH. 

 cEe is a little carriage which rolls along it, having two rollers on one side and 

 one on the other, which is pressed against the beam by a spring. This carriage 

 carries a vertical pin, E, turning in its socket, and having a collar above, 

 through which the arm CF works smoothly. The tracing point G is attached 

 to the carriage by a jointed frame eGe, which is so arranged that the point 

 may not bear too heavily on the paper. 



12. When the machine is in action, the tracing point is placed on a point 

 in the boundary of the figure, and made to move round it always in one 

 direction till it arrives at the same point again. The up-and-down motion of 

 the tracing point moves the whole instrument over the paper, turns the wheel 

 I\, the hemisphere LI, and the index-sphere Bb ; while the lateral motion of 

 the tracing point moves the carriage E on the beam Tt, and so works the arm 

 CF and the framework CO; and so changes the relative velocities of the two 

 spheres, as has been explained. 



13. In this way the instrument works by a perfect rolling motion, in what- 

 ever direction the tracing point is moved ; but since the accuracy of the result 

 depends on the equality of the arcs AP and BP, and since the smallest error 



