﻿the Hatchet Planimeter. 269 



i fr» cos &/0= * f*rfA= A ", 



where x is the abscissa of the mass-centre of the area. The 



other terms are multiplied by (-) , and being integrals of 



oscillating quantities are very smail. 

 Thus 



, A . Ale 2 Ax P 



where P is very small. 



If is the mass-centre of the curve k 2 is a minimum and 

 i£ = 0. In this case 



very nearly, where A/3 2 is the moment of inertia about the 

 polar axis at the mass-centre. 



If is not near the mass-centre, let the tracing-point be 

 moved again round the perimeter, following the same path 

 as before, but with the initial direction of the instrument 

 exactly reversed ; the effect of this is to change the sign of c 

 and to give 



oW = A + Af-4f_P ; 



2c 2 c c z 

 hence 



c<£> being the distance between two indentations may perhaps 



be measured to within ^J^ inch, so that there will be an 



error in c 2 <I> of at least A sq. inch, if c is 10 inches ; if, 



A& 2 

 therefore, ^- is less than ^ sc l- inch, it may be neglected. 



Of curves with the same greatest breadth, the circle has the 



A& 2 

 greatest area, and ^3- is less than ^ sq. inch for a circle of 



radius 1^ inches. 



Finally, then, if be near the mass centre, and the greatest 



breadth be less than 3 inches, the formula c 2< 3> = A will give a 



result as accurate as the nature of the instrument will allow ; 



A& 2 

 for greater areas the value of c y^- must be estimated and 



taken into account. Also it is easily verified that if <£ is 

 less than ^, or the area less than 23 sq. inches, it is sufficient 

 to measure the chord of the arc c<£> instead of the arc. 



