502 



REPORT 1894. 



sense as in the other. The point Q may be either within or without the 



area. 



If the curve cuts itself, the area swept over Avill be the same as in the 

 former case (fig. 5), and all the remarks made there will be applicable. 



This is the mode in which areas are generated when using polar co- 

 ordinates. Planimeters based on this principle might therefore be called 

 polar planimeters. But as this name has been appropriated for an 

 altogether different type, I call these simply planimeters of Type II., or 

 Polar-co-ordinate Planimeters. 



Third Mode of Generating an Area. 



Whilst in the last two cases an area was generated by a line of 

 variable length which had either a motion of translation or a motion of 



Fig. 8. 



turning alone, we shall now suppose that a line has both motions but a 

 constant length. 



Let QT again denote the generating line now of constant length l. 

 Let it be moved from the position QqTo (fig. 8) to a near position, QT. 

 This motion can be decomposed into a small translation to QTj and a 

 turning about Q which bi-ings it to its final position, QT. For both these 

 motions our rule of sign holds ; it will therefore also hold for the 

 actual motion of the line where both motions go on simultaneously. On 

 applying the rule the following theorem will be seen to be true in which a 

 motion of a line is called cyclical if it ultimately returns to its initial 

 position (compare fig. 9). 



Theorem 1. — If a line QT of constant length performs in a -plane a 

 cyclical motion, then the area generated is equal to the area enclosed by the 

 path of the end point T diminished by that enclosed by the path of Q, both 

 areas being talcen in their proper sense as determined by the sense in which 

 each boundary is described. 



Corollary. — If the 2}oint Q is moved to and fro along a curve so that its 

 path does not enclose an area, then the area generated ivill be equal to the 

 area enclosed in the piath of T. 



The measurement of this area is much facilitated by the following 

 considerations. Let ^j denote the perpendicular distance between the 



