ON A NEW FORM OF THE PLATOMETER. 



231 



line descends one inch, it sweeps out a number of square inches equal to the 

 number of linear inches in its length. 



Therefore, if we have a machine with an index of any kind, which, while 

 the generating line moves one inch downwards, moves forward as many degrees 

 as the generating line is inches long, and if the generating line be alternately 

 moved an inch and altered in length, the index will mark 

 the number of square inches swept over during the whole 

 operation. By the ordinary method of limits, it may be 

 shown that, if these changes be made continuous instead 

 of sudden, the index will still measure the area of the 

 curve traced by the extremity of the generating line. 



3. When the area is bounded by a closed curve, as 

 ABDC, then to determine the area we must carry the tra- 

 cing point from some point A of the curve, completely round 

 the circumference to A again. Then, while the tracing point 

 moves from A to C, the index will go forward and mea- 

 sure the number of square inches in ACRP, and, while it 

 moves from C to D, the index will measure backwards the 

 square inches in CRPD, so that it will now indicate the 



square inches in ACD. Similarly, during the other part of the motion from 

 D to B, and from B to D, the part DBA will be measured ; so that when 

 the tracing point returns to D, the instrument will have measured the area 

 ACDB. It is evident that the whole area will appear positive or negative 

 according as the tracing point is carried round in the direction ACDB or ABDC. 



4. We have next to consider the various methods of communicating the 

 required motion to the index. The first is by means of two discs, the first 

 having a flat horizontal rough surface, turning on a vertical 



axis, OQ, and the second vertical, with its circumference rest- 

 ing on the flat surface of the first at P, so as to be driven 

 round by the motion of the first disc. The velocity of the 

 second disc will depend on OP, the distance of the point of 

 contact from the centre of the first disc; so that if OP be 

 made always equal to the generating line, the conditions of the instrument will 

 be fulfilled. 



This is accomplished by causing the index-disc to slip along the radius of 



