ON I'LANIMETERb. ' 



517 



The knife-edge prevents Q from moving sideways. The only motions 

 possible are therefore a sliding in the direction of QT together with a 

 turning about Q. Hence, if T is moved along any curve, Q will follow, 

 always moving towards T. The Q-curve, as the path of Q may be called, 

 is therefore a curve of pursuit. The line QT is always tangential to it. 

 Its length will be denoted by /. 



To use this instrument as a planimeter the inventor gives the following 

 rules : — 



If the greatest extension of the area exceeds II the area has to be 

 divided and the parts measured separately. 



Take a point R as near as you can guess to the centre of gravity of the 

 area, and join it to a point A on the boundary ; put the tracer T on R, and 

 make with the knife-edge a mark on the paper. Next guide T from R to 

 A, round the boundary and back to R ; make again a mark with Q, and 

 measure the distance Cj between these marks. Next turn the instrument 

 tlirough 180° about R and guide T as before, but in the opposite sense, 

 round the boundary. Let the distance between the marks in this case be 

 €■2. Then is the area given by 



2 



(-©' 



where N^ denotes the mean value of the squares 

 from the boundary. He then adds a table givinj 

 values of N/^. They vary from -002 to "016 



Fig. 15. 



of the distances of R 

 ^ (N/2iyior different 

 and are therefore small. 



He afterwards gives his theory. This leads him to an integral for the 

 area swept over by QT, which cannot be worked out in a finite foi-m, and 

 has to be expanded in an infinite series of which the first terms lead him 

 to the result stated. 



The same problem has also been treated analytically by Mr. F. W. Hill 

 in a more elegant manner, the result being given in a slightly different 

 form. 



These investigations are, however, somewhat unsatisfactory. It seems 

 to me that the following geometrical reasoning affords us a deeper insight 

 into the real nature of the theory of this instrument. 



If T describes a straight line AX (fig. 15) whilst Q starts from O, 

 where AO is perpendicular to OX, then Q will describe a curve known as 

 the tractrix, which will extend to an infinite distance to the right, having 

 AX as asymptote. If T is moved to the left, Q describes the other half of 

 tractrix, which has a cusp at O. Similarly, if T describes any curve, Q 

 will describe a curve which will have a cusp whenever QT becomes normal 

 to the T-curve. 



