514 REroRT — 1894. 



polar planimeter ; if T is moved round the boundary of an area wliich 

 may be denoted by (T), then the rod QT will sweep over an area equal to 

 (T) which has to be registered. This is done by the ' roll ' of a cylinder 

 EF, whose axis is parallel to the ' rod.' This cylinder does not roll on the 

 paper, but on a spherical surface S. The latter can revolve about a hori- 

 zontal diameter AD which is parallel to the arm OQ, and which has at B 

 a small toothed wheel gearing in the rim B'BB" of the fixed disc. Let 

 the cylinder EF touch this sjjhere at C in the horizontal great circle, 

 and let A be the centre of the sphere. If now the rod and with it the 

 axis of the cylinder make an angle a with the tangent QX to the circular 

 path of Q, then the distance of the point C from the axis of the sphere 

 will be CD=r sin a, where r is the radius of the surface S. Let now 

 the rod QT get a small translation to Q'T', sweeping over a small area 

 QTT'Q' =:lp. Of course, QQ' must be so small that it may be con- 

 sidered an arc of the circle in which Q moves. If QQ'=a;, then is p 

 ^jcsina, the areaQTT'Q'=/.r sin o. Atthe sametimeOQ will turn about O 

 through an angle proportional to x, and this will produce a rotation of the 

 sphere and a motion of C proportional to CD.v, i.e., proportional to x sin a 

 or to p. This will be communicated to the cylinder. Hence the roll of the 

 cylinder during a small motion of T will be proportional to the area swept 

 over by the rod during the translation of the rod. According to tlie 

 general theory of planimeters of Type III., the ' roll ' of the cylinder measures 

 the area enclosed by the path of T, provided that the rod does not com- 

 plete a whole rotation. If the point O should lie within the area, a certain 

 constant has to be added to the ' roll ' of the cylinder, as in Amsler's simple 

 planimeter. 



This very simple theory is, however, not quite correct. For if QT 

 turns about Q the cylinder will not always touch the sphere. To insure a 

 continuous contact the cylinder is mounted in a small rectangular frame, 

 which itself can rock to and fro about an axis parallel to that of the rod. 



Amsler, in the planimeter mentioned, places Q at the centre A of the 

 sphere and suspends the frame with the cylinder from above the sphere. 

 In this case a turning of the arm will make the cylinder slip along the 

 horizontal great circle. To avoid this slipping he makes the cylinder 

 movable along its axis, and gets thus an instrument without slipping. 

 Coradi places the point Q so that when the rod is in the direction of QX 

 the cylinder touches the sphere when its axis is vertically above Q. If now 

 the rod be turned about Q, the cylinder with its frame will be pushed side- 

 ways, whilst the point C will move below the horizontal great circle. This 

 up-and-down motion of C will produce a roll of the cylinder, the total 

 amount of which, however, will reduce to zero when the point C comes 

 back to its original position, and this it will do when the tracer has de- 

 scribed a closed curve. 



But this lowering of C also increases the radius CD. At the same 

 time the cylinder will not receive the full motion of the point C. In 

 fact, this motion will be decomposed into rolling and slipping. The 

 rolling will be just the same as if C was still on the horizontal great circle, 

 and will therefore give the area correctly. The slipping will always be 

 very small. There will also be slipping when the rod turns about Q, but 

 this again will be very small. Although the exact determination of these 

 quantities is not difficult, I prefer to refer the reader to A. Amsler's essay, 

 ' Ueber mechanische Integrationen,' in Dyck's 'Catalogue,' p. 107, where 

 a very elementary theory of this part of Coradi's planimeter is given, and 

 also (p. 105) a description of his father's planimeter-without-slipping. 



