236 OX A NEW FORM OF THE PLATOMETER. 



The tracing point has now traversed the boundary of a rectangle, whose 

 length is the circumference of the wheel A' and its breadth is equal to CD ; 

 and during this operation, the index-sphere has made a complete revolution. 

 360* on the sphere, therefore, correspond to an area equal to the rectangle con- 

 tained by the circumference of the wheel and the distance CD. The size of 

 the wheel K being known, different values may be given to CD, so as to make 

 the instrument measure according to any required scale. This may be done, 

 either by shifting the position of the beam 755, or by having several sockets 

 in the carriage E for the pin which directs the arm to work in. 



15. If I have been too prolix in describing the action of an instrument 

 which has never been constructed, it is because I have myself derived great 

 satisfaction from following out the mechanical consequences of the mathematical 

 theorem on which the truth of this method depends. Among the other forms 

 of apparatus by which the action of the two spheres may be rendered available, 

 is one which might be found practicable in cases to which that here given 

 would not apply. In this instrument (Fig. 4) the areas are swept out by a 

 radius-vector of variable length, turning round a fixed point in the plane. The 

 area is thus swept out with a velocity varying as the angular velocity of the 

 radius-vector and the square of its length conjointly, and the construction of the 

 machine is adapted to the case as follows : 



The hemisphere is fixed on the top of a vertical pillar, about which the rest 

 of the instrument turns. The index-sphere is supported as before by a ring and 

 framework. This framework turns about the vertical pillar along with the tra- 

 cing point, but has also a motion in a vertical plane, which is communicated to 

 it by a curved slide connected with the tracing point, and which, by means of a 

 prolonged arm, moves the framework as the tracing point is moved to and from 

 the pillar. 



The form of the curved slide is such, that the tangent of the angle of 

 inclination of the line joining the centres of the spheres with the vertical is 

 proportional to the square of the distance of the tracing point from the vertical 

 axis of the instrument. The curve which fulfils this condition is an hyperbola, 

 one of whose asymptotes is vertical, and passes through the tracing point, and 

 the other horizontal through the centre of the hemisphere. 



The other parts of this instrument are identical with those belonging to 

 that already described. 



