518 



REPORT — 1894. 



If we decompose a very small displacement of QT during this motion 

 into a sliding along QT and a turning about Q, then the latter alone will 

 generate an area, and if the rod turns through a small angle (i the area 

 generated equals ^Pd^h ^« where a=:ld is the small arc described by T 

 during this turning. 



Hence, if a line OP=QT be drawn from any fixed point O and kept 

 always parallel to QT it will generate an area equal to that generated by 

 QT itself. Thus in fig. 15 the area OATQO equals the area of the circular 

 sector OAPO if OP is parallel to QT. 



In particular, if T is moved to an infinite distance, QT will fall into 

 AX, and will be parallel to OB ; from which follows at once the well-known 

 result, that the area between the tractrix and its asymptote from the cusp 

 to infinity equals that of a quadrant of a circle with radius I. 



Let now a closed curve be given, A a point on it, and let the planimeter 

 QT be placed in the position OA. Next let T be moved from A rovtnd the 



Fig. 16. 



curve in the sense of the arrow ; then Q will describe a curve somewhat 

 like OSB with a cusp at S (fig. 16). 



If the given curve is less simple this Q-curve may be more complicated, 

 but the following reasoning will always hold. 



The area described by QT is found to be equal to that of the section 

 OAC, where 00 is parallel to BA. In order to make the Q-curve closed, 

 suppose the line QT tui-ned about A from AB back to AO. Hereby the 

 sector BAO is described, which equals OAC both in magnitude and sense, 

 for the sense of QT is from Q to T ; hence from B to A. 



We have now a cyclical motion for QT. According to Theorem 1 



