ON rLANIMETERS. o07 



and became, as executed by Ausfeld in that town, instruments of very 

 great accuracy, known as the Wetli-Hansen Planimeters. A description 

 with figure is given by Bauernfeind in the ' Proceedings ' of the ' Poly- 

 technischer Verein fur das Konigreich Bayern,' 1853, pp. 130-147, and 

 213-244. The ligure is copied in Prof. Dyck's 'Catalogue of the Mathe- 

 matical Exhibition,' Nachtrag, p. 32, Munich, 1893. 



The disc is turned by aid of a silver wire which is attached to the rod 

 QT and slung round a pulley on the axis of the disc, so that here rod and 

 carriage have interchanged their role. 



Among Hansen's improvements one may be mentioned, as I have 

 recently heard of it as a new American improvement. The tracer T is 

 replaced by a piece of glass with a small circular mark on it which is 

 guided along the curve. A lens above the glass serves to make the guiding 

 more accurate. This form of a tracer is, however, straining to the eye, and 

 seems only of use where areas have to be determined with great accuracy. 



In England, John Sang made a planimeter which is described in the 

 'Civil Engineer and Architect's Journal,' 1851, p. 505, and also in the 

 * Transactions ' of the Ptoyal Scottish Society of Arts, vol. iv. Bauernfeind 

 says that it differs from that by Ernst 'fast gar nicht.' 



It was exhibited at the Great Exhibition in 1851, and obtained 

 honourable mention, whilst Gonnella received a Council medal and Wetli 

 a prize medal. Ausfeld also exhibited a planimeter, which is described 

 as invented by Dr. Flaussen (probably an error for Hansen), of the 

 Observatory of Seeberg. Of this instrument, which received honourable 

 mention, I have not found any further description than that it had a 

 wheel rolling on a disc. It was most likely the modification of Wetli's, 

 referred to above. 



These exhibits are of importance in the history of planimeters, because 

 they drew Maxwell's attention to the subject, on which he communicated 

 early in 1855 a paper to the Royal Scottish Society of Arts ('Trans- 

 actions,' vol. iv., and 'Collected Papers,' vol. i. p. 230). In this he 

 points out the deleterious effect of the slipping of the recording wheel 

 on the cone or disc, and then proceeds to the description of a new recording 

 apparatus, in which the wheel rolling on a cone or disc is replaced by a 

 sphere, which rolls without slipping on an equal sphere. It is a most beautiful 

 conti'ivance, but it has never been carried out. It may, however, be con- 

 sidered as the starting-point of all integrators which work without slipping. 



Maxwell describes two planimeters having this new contrivance, the 

 one belonging to Type I., the other to Type II. 



About this time Josef Stadler, of Eisenerz, in Styria, constructed a 

 planimeter of peculiar character. It works without slipping, and is in many 

 respects of interest on account of the peculiar integrating apparatus used. 



The tracer T is fastened to a bar whose motion is constrained to remain 

 parallel to the axis of x. Rigidly connected with this is a thin cylinder, 

 which can turn about an axis parallel to the bar, hence to the axis of x. 

 This cylinder rests on a surface of revolution movable about its axis a, 

 which is parallel to the axis of y. If the tracer be moved in the direction 

 of the y, the cylinder will roll along a meridian curve of the surface of 

 re\ olution, but the latter will remain at rest. A motion of the tracer in 

 the direction of the x will pull the cylinder across the surface of revolution, 

 and therefore make it revolve to an extent which is proportional to the 

 distance of the point of contact from its axis a. If ;; denotes this distance, 

 and y the distance of the tracer from the axis of x, i.e., the generating line 



