508 REPORT— 1891.. 



QT, then the turning of the surface of revolution will measure the area, 

 provided that zy remains constant. In order tliat this condition may be 

 satisfied, the meridian curve of the surface of revolution must be a solution 

 of a differential equation of the first order and the second degree. An 

 integral of this equation cannot be found in a finite form, but an equi- 

 lateral hyperbola revolving about an asymptote satisfies the condition very 

 nearly. In the instrument made this hyperbola was corrected by actual 

 trial. It will be seen that there is no slipping. The difficult construction 

 of the surface of revolution makes it doubtful whether a very great accu- 

 racy is obtainable. But as Dr. A. Amsler, who called my attention at the 

 Munich Exhibition to this model, poihted out to me, the principle might be 

 useful in the construction of integrators designed for special purposes. 



Stadler had a first, and as it seems an only instrument made in 1855, 

 which is now in the possession of the Technische Hochschule in Gratz. It 

 was exhibited in Munich. Stadler published an account of it in the journal 

 ' Erfahrungen im berg- und liiittenmannischen Bau- und Aufbereitungs- 

 wesen,' edited by Rittinger, 1857. It seems to have remained practically 

 unknown till the Munich Exhibition. In Dyck's ' Catalogue ' is a full 

 account of it by Professor Lichtenfels, of Gratz. 



It appears that Stadler, in the out-of-the-way place where he lived, had 

 iseard of planimeters, but was not acquainted with their construction. 



Planimeters of Type II. 



Of Polar-co-ordinate planimeters Amsler mentions three (Appendix to 

 his paper of 1856) proposed by Gierer, of Fiirth,' by Bouniakovsky, of 

 St. Petersburg, and by Decker, of Augsburg (the last two described in 

 Dingler's ' Pol. Journ.,' vol. cxl.). Each has a recording wheel rolling on 

 the paper whose axis passes through a fixed point Q, whilst its distance 

 from Q is always proportional to the square of the distance of the tracer 

 T from Q. In Gierer's instrument the wheel is kept in the required 

 position by aid of a guiding curve, in the other two by aid of link-work. 



One of Maxwell's planimeters belongs to this type, and so does one by 

 C V. Boys.- Both were invented with the object of avoiding all slijjpiiig 

 in the recording apparatus, which is too complicated for an instrument 

 designed simply for the determination of areas. 



Lastly, there is an instrument described by W. E. Bousfield in the 

 discussion of Hele-Shaw's paper ' On Mechanical Integrators.' ^ In it 

 guide curves are used. 



None of these planimeters has, so far as I know, ever been made. 

 Of Boys' a model is to be found in the Science collection at South Ken- 

 sington. 



Amsler's Planimeter and its Development. 



Whilst planimeters of Type I. were gradually reaching a state of 

 ■great perfection, Amsler invented his polar planimeter, which, in conse- 

 <quence of its simplicity, handiness in use, and low price, soon drove all the 



' Programm der Ginrerhs- vnd Handelsschttle :u Fiirth, ISff. 



= Phil. Mag., 1882, p. 83. 



' Proc. Just. Civ. Eng., vol. Ixxxii. part iv. 



