A NEW PLANIMETER. V 



sliding the parabola. Now while the pointer is moving from A to 

 B the wheel revolves through the arc CD. 



Measure of sector BOA = |OA^a. 

 " arc CD = ODa. 

 But fiom the properties of a parabola 

 OB^ = bOB. 

 Therefore 



Measui^e of sector BOA = ^bODa, 

 while 



Measure of arc CD - ODa. 



Hence the distance through which the wheel I'evolves while the 

 pointer is tracing the arc AB varies as the area of the sector BOA, 

 and by suitably graduating the wheel the area of the sector BOA is 

 registered on the wheel. 



The boundary of the figure X may be considei'ed as made up of 

 an infinite number of infinitely small arcs having the centre O and 

 different radii. As each arc is traced the wheel registers the area of 

 the corresponding sector, while a change of radius causes the wheel 

 to slide without revolving. Hence, while the tracing pointer is 

 describing the boundary of the figure from P to Q in the direction 

 of the hands of a watch, the wheel registers the area of the figure 

 contained by OP, OQ, and the part of the boundary of X passed 

 over by the tracing pointer. As the tracing pointer goes on from 

 Q to P, the wheel revolves in the opposite direction, registex'ing. the 

 area of the figure contained by OP, OQ, and the part of the boundary 

 of X meanwhile ti"aced. It is therefoi'e easy to see that when the 

 tracing pointer has described the whole boundary of X just once, 

 the wheel will indicate the area of X. 



Mr. G. B. Abrey exhiibted one of Amsler's Planimeters 

 and said that the introduction of the curve of the parabola 

 was new so far as he was aware. Other curves could be in- 

 troduced, and the principle extended indefinitely. He hoped 

 to have an instrument made that would be a practical success. 

 He explained the theory of another form of the instrument, 

 simpler in practice, though more complex in theory. 



Mr. Macdougall called attention to a Planimeter invented 

 by Mr. John Say, of Kirkaldy, the leading feature of which 

 was the employment of a cone, acting on an index-wheel 



