ON PLANIMETERS. 521 



We also have, if a recording wheel be fixed on the line AB whoso roll 

 for a complete circuit is iv, 



(AB)=AB . tv, (BC)=BC . w, &c. 



Hence, if a line performs a cyclical motion without turning completely 

 round, then the area swept over by any segment on the line is propor- 

 tional to the length of the segment. 



Let now A, B, P be points in a sti-aight line and AB=rt, AP 

 =zp, .•.PB=rt— p; then is 



(AB)=atv={B)-{A), (AP)=ptv=(P) - (A) 



On eliminating lo we get 



p (B)-« (P)=0.-«) (A), or {a-p) (A)=fl (P)-;j (B) 



This is Holditch's Theorem. 



From this it follows that we can always find one point P in the line 

 AIJ which describes a curve of zero-area. For it 



{p-a) (A)=p (B) 

 (A) 



This point shall be called the zero-point in the line. 



If (A)^(B), then is 7^=00 , and the zero-point is at infinity. 



For any other point P (not at infinity) we have now 



a (P)=« (A), .-. (P) = (A) 



Jf two j)oiiUs in a line describe closed cicrves of equal area, all points in 

 the liite do the same. 



The area enclosed by any point in the line encloses always an area 

 tchich is p>'>'oportional to its distance from tlie %ero-poxnt. 



In Amsler's planimeter, or any planimeter of Type I., the point Q is 

 the zero-point of the ' rod ' QT. 



Let on this rod a point T' be taken so that QT=^QT' ; then, whilst T' 

 is guided along the boundary of an area (T'), the point T will descrilje a 

 closed curve of k times the area (T). 



J. Amsler proposed {Jx., p. 29, 1856) to have a tube inserted at T' per- 

 pendicular to the paper. At the bottom this carries near the paper a 

 glass plate with a small circular mark, and at the top a lens. The point 

 T is now moved so that the mark at T' follows the small curve. The 

 point T describes a closed curve whose area is registered by the wheel in 

 square inches, say. The area (T') is therefore registered k times to the same- 

 scale. We have thus a planimeter which registers a magnified area, and 

 is suitable to measure small areas. The advantage is this. In guiding a 

 tracer round a curve the motion of the tracer will be more or less jerky. 

 These jerks at T will be reduced at T'. 



This arrangement, however, has one drawback. The figures desci'ibed 

 by T and T' are not similar, and this makes it difficult to guide T so that 

 T' follows a given curve. But this can be overcome as follows : Let us 

 consider a linkage as in fig. 17. OQT is an ordinary planimeter with 



