﻿268 Mr. F. W. Hill on 



triangle, and integrating round the curve there results, for 

 the angle between the initial and final positions of the rod, 

 the expression 



+J(c + jfe + --) 0M{ '-«"' 



where A is the area of the curve, and the integrations extend 

 from 0=0 to 0=27r. Thus if the greatest breadth of the 

 curve is less than the length of the planimeter, the most im- 

 portant term in the expression is A, and it remains to estimate 

 the effect of the other terms. 



First, considering those terms which depend only on the 

 position of 0, 



1 C 4JQ A* 2 



where Ak C2 is the moment of inertia of the area about the 

 polar axis through 0. This term is therefore least when is 

 the mass-centre of the area. 



— r— . I r 6 dd is less than —-, ( - ) , 

 144c 4 J 144 \c/' 



where a is the greatest radius-vector from 0, and, if - is as 



great as ~ , is less than one per cent, of the area. 

 Of the terms depending on cj> the most important is 



^ fr 5 cos (0- </>)</#. 



3c 



In these small terms the value of c 2 <£ at any point may be 

 taken to be the area of that part of the curve already de- 

 scribed by the tracing-point, and the terms can be evaluated 

 when the equation of the curve is known. If, however, the 

 area of the curve is less than that of a square of side c, so 

 that the greatest value of <f) is less than 1, it is easy to 

 approximate. 



■^ fr 3 cos (0 -(/>)#= i L 8 cos 6d0 + ~ fr 3 </> sin 6d0 



3c, 



-~ |r 3 </> 2 cos 6d6+. . . 

 Now, the axis of x being the initial position of the rod, 





