ON PLANIMETERS. 511 



Fourth error : Due to slipping of the wheel. This is the error on which 

 Maxwell dwelt strongly. 



This error increases with the amount of slipping, with the friction 

 resisting the slipping, and lastly with the resistance to the turning of the 

 wheel. 



In order to reduce this error, the above three causes have to be dimin- 

 ished. This has led to a number of new constructions, which have to be 

 discussed presently. It may here be stated that the third has led to the 

 introduction of a ' disc,' as used in the planimeters of Type I., on which 

 the wheel rolls. The rough paper is, therefore, replaced by a smooth pre- 

 pared surface. 



To reduce the resistance to the turning of the wheel, mechanical skill 

 alone can help. 



But to reduce the first cause we have to investigate the amount of slip- 

 ping at different positions of the rod of the instrument relative to the arm. 

 If it be placed in such a position that the plane of the wheel passes through 

 the pole O, and if now the instrument be moved as a rigid body, then the 

 tracer will describe a circle. At the same time, the wheel will not roll, but 

 only slip. This circle, formerly referred to as the base-circle, will be the 

 locus of greatest slipping. 



If the tracer be moved along this circle there should be no turning, and 

 there will be none. Hence, on this circle itself, the error due to slipping 

 will be zero. If, however, the tracer be moved on a concentric circle a 

 little inside the base-circle, then rolling will take place in one sense, but 

 on a circle a little outside in the opposite sense. In either case there will 

 be much slipping, causing an error. This error will be pi'oportional to 

 the length of the path of the wheel, i.e., to the roll ; hence, on moving 

 the tracer parallel to the hasec-ircle, but at an increased distance from it, 

 the error will increase on account of the increased roll, and decrease on 

 account of the decreased slipping. It will soon reach a maximum, and 

 then diminish. The same is true outside the base-circle, only the error 

 will be of the opposite sense.' 



It becomes thus apparent that the error due to slipping is dependent 

 on the position of the pole relative to the area, and also that with changing 

 the position of the pole the eiTors change in an apparently haphazard 

 manner. 



It is of interest to compare these results with practical tests. 

 Wilski points out that there are certain positions of the pole for which 

 tlie record of the wheel becomes a maximum, others for which it is a 

 minimum. This is explained by supposing that in these cases a part of 

 the boundary of the area is parallel and near the base-circle either on the 



' I have just (Octoher 30) received from Coradi a paper by Lang (reprint from 

 the Allgemehie Yermcsmnqs-Naeliricliten, 1894) in which he discusse.'* the errors 

 of an Amsler Planimeter, and remarks that the tracer T can be moved from any 

 position so that the wheel will only slip without rolling. This is obvious when 

 pointed out. These paths are spirals, one through every point, which approach the 

 base circle asymptotically either from within or from without, so that there are 

 two sets of these curves, all of one set being congruent. My reasoning in the text 

 must therefore be extended to these spirals also. 



To reduce the error due to slipping no part of the boundary of the area to be 

 measured should therpfore be parallel to one of these curves or to the base-circle. 

 Lang recommends to draw one of each of these spirals by trial and to cut it out 

 in stiff paper By aid of these templets it is then easy to place the planimeter 

 so on the paper that the boundary cuts the base circle and these spirals nearly at 

 right angles. 



