ON TLANIMETERS. 



503 



position QoTo andQT,, and « be the angle through which the rod is turned. 



The area generated by the iirst step, the translation, is then = Ip, the area 



of the sector is i I'^d. The whole area swept over by the rod is the sum of 



the areas due to the successive elementary 



steps. It consists, therefore, also of two 



parts, the first being due to the elementary 



translations and equals l'S:p, where 2^ is the 



sum of all the lateral displacements of the 



rod. The second part consists of the sum of 



all the sectors of circles described by the 



turning of the rod, and equals ^ Z'^SW, where 



Sfi denotes the sum of all the elementary 



turnings, and this is the same as the total 



turning of the rod. But the rod returns 



to its original position. The angle Sa is 



therefore either = or it is a multiple of 



Iw, which will happen if the rod has turned 



round once or several times. This gives — 



Theorem 2. — If a rod of finite length per- 

 forms a cyclical motion ivithout itself making 

 a complete rotation, then the area siuept over 

 equals that generated hy the successive mo- 

 tions of translation only ; hut if it completes 

 n rotations, then to this quantity has to be 

 added nTrP, i.e., n times the area of a circle ofradi^is 1. 



To construct a planimeter on these principles, let the rod QT be con- 

 nected by an articulated joint at Q to another rod OQ, which shall pi 

 future be called the 'arm' (fig. 10). Let this arm have at O a needle 



FiG.lO. 



point which can be pressed into the drawing-board. This being done, the 

 point Q will be restrained to move on a circle, whilst T can be guided by 

 hand along any curve. The point O is called the pole of the instrument 

 and the latter is therefore generally called a Polar Planimeter. 



