A NEW PLANIMETER. 27 



A NEW PLANIMETER. 



Mechanical integrators are constructed on the assumption that if 

 a wheel is moved over a smooth surface in the direction of its axis it 

 will slide without revolving; if moved at right angles to its axis it 

 will revolve without sliding, and if moved in any other direction it 

 will both slide and revolve. For example, if it is moved a distance 

 ^ in a direction making an angle O with its axis it will revolve 

 through a distance I sin O and slide through a distance I cos O. 



If the wheel is perfectly turned and truly mounted this assumption 

 is absolutely accurate; and, from experiments that have been made, 

 it has been shown that it is possible to turn a wheel so perfectly and 

 mount it so truly that this assumption shall be verified to within 

 less than one-tenth of one per cent. However, even slight imperfec- 

 tions in the turning and mounting will lead to results far from 

 accurate, and it is therefore of the utmost importance in the con- 

 struction of instruments for mechanical integration to make the 

 wheel as ])erfect as possible. 



To make an instrument for integrating any quantity it is necessary 

 and sufficient to so arrange its parts that while a tracing pointer is 

 describing a diagram involving the independent dimensions of the 

 quantity the wheel shall revolve through a distance which varies as 

 this quantity. 



In the case of areas and other quantities of two dimensions this 

 has been accomplished in several ways, some exceedingly ingenious, 

 notably so that of Prof. Amsler, whose planimeter is widely known. 



Lately a planimeter has been made from designs furnished by Mr. 

 Butler, C.E., and the writer, on a plan, so far as I know, not hitherto 

 adopted in any mechanical integrator. Moreover the same idea 

 may be applied in constructing instruments for integrating quantities 

 of more than two dimensions, such as volumes of embankments) 

 moments of inertia, total pressure of water on irregular submerged 

 surface not horizontal, and many others. 



This planimeter is constructed as follows : — 



A semi-parabola is ari'anged to slide on a right angled triangle, so 

 that the axis of the parabola shall be always parallel to CD, one 

 side of the triangle. A wheel is placed at B, the vertex of the par- 

 abola, with its axis in a line with that of the parabola. A tracing 



