510 REPORT — 1894. 



Dr. A. Amsler in his valuable paper, ' Ueber den Flacheninhalt und 

 das Volumen durch Bewegung erzeugten Curveii und Flachen, und iiber 

 mechanische Integrationen ' (Inaugural Dissertation, SchafFhausen, 1880), 

 starts with proving that the area passed round by the tracer equals 



I \?,\i\u.ds 



'Y 



where ds is an element of the boundary and o the angle it makes with the 

 rod, the integration being taken over the whole boundary. It is therefore 

 a true line-integral. 



Where tlie only object is to explain Anisler's planimeter, Macfai'lane 

 Cray's theory is perhaps the simplest. It is given in Carr's ' Synoj^sis of 

 Mathematics.' 



In 185.5, hence at very nearly the same time as Amsler, Prof. Miller 

 Hitter von Hauenfels invented a planimeter based on the same principles 

 as Amsler's. It is described in the ' Handbuch der niederen Geodasie ' 

 (2nd to 7th ed.), by Prof. Pr. Hartner, and also in Dyck's ' Catalogue,' 

 p. 190. Gustav Starke, of Vienna, simplified the arrangement of the 

 recording wheel, and manufactured ' in the course of years hundreds of 

 them.' It is known as the Miller-Starke Planimeter. 



As Amsler's polar planimeter is universally used, the following discus- 

 sion of its errors may be of interest. 



First error : The diameter of the wheel and the length of the ' rod ' 

 are not in the proper relation — i.e., if u is the unit division of the recording 

 wheel, I the length of the rod, the product lu does not give the accurate 

 unit of area intended. This error is generally very small. Otherwise all 

 readings have to be multiplied by a factor of reduction. 



Second error : The axis of the wheel is not parallel to the rod, but 

 makes an angle i with it. 



If W is the ' roll ' of the wheel and S the slipping (due to translation 

 of the rod only), the area then is 



A^ ^ cos £ . W -f- ^ sin e . S + nirl- 

 Or, as £ is practically very small, 



A = ZW+?S£ + 7f7rZ2 



This introduces an error, ZSf, which may be appreciable. 



If a rod QT' be fixed to the rod QT, making an angle i with the normal 

 to the latter and of equal length to it, then, whilst T describes the boundary 

 of the area A, the point T' will describe another closed curve. The area 

 bounded by this curve is SZ. This area will often be small, but it is easy 

 to draw curves for which it is considerable, by guiding T' round an area 

 of considerable size. If T circumscribes an area A, and T' an area A', 

 then the error S in A will be A sin t, and that in A' will be A sin f. 



This error has been investigated by Herr Wilski in ' Zeitschrift fiir 

 Vermessungswesen,' 1892, p. f>10. 



In the same journal for 1891, Herr Lang has shown how to eliminate 

 it. About this more will be said presently in connection with the Lang- 

 Coradi Planimeter. 



Third error : The axis of the joint at Q between the arm OQ and the 

 rod QT is not perpendicular to the paper — i.e., not parallel to the axis about 

 which the instrument turns at O. See Wilski's paper. 



