ON PLANIMETERS, 501 



But if the wheel is made to turn faster in proportion to the length QT, we 

 should have what is wanted. This was obtained in the first planimeter by 

 Hiaking the wheel run on a cone. 



The diagrammatic drawing (hg. 6) will simplify the description. 



A frame FF is made movable in the direction of the line OX, either 

 by letting it run on rails or by giving the wheels milled rims. On it 

 rests a rod T'T perpendicular to OX, which can slide to and fro in its 

 own direction. It ends at T in a tracer which can be made to follow 

 any curve. The frame also carries a cone VCC, whose axis is inclined to 

 the paper, so that the upper edge is horizontal. Its rim CC rests on the 

 paper, so that it turns when the frame moves, and this turning will be pro- 

 portional to the forward motion. Mounted on the rod, or, as in the figure, 

 on an axis parallel to the rod, is a wheel W resting on the upper edge of 

 the cone. 



If the rod is pushed back till the wheel W comes to V, the vertex of 

 the cone, then T will come to Q on the line OX. On moving T along this 

 line the wheel will not turn. But if T is pulled out a distance QT=2/, 

 then the wheel will assume a position W in the figure such that VW 

 ^QT^y. On moving the frame forward the roll of the wheel will be 

 proportional to y and to the forward motion. Hence if the tracer T is 

 guided along a curve from A to B, then the roll will be proportional to the 

 area between the curve and the axis OX. 



Taking rectangular co-ordinates OQ=.x' and QT=y, we have in the 



symbols of the Integral Calculus the above area= ydx, and the roll of 



the wheel will be proportional to this. 



This planimeter follows in its construction the determination of an 

 area in the Integral Calculus using rectangular co-ordinates. Planimeters 

 of this type have therefore been called orthogonal planimeters (Dr. A. 

 Amsler). 



If the tracer is moved round a closed curve the instrument gives the 

 enclosed area, but it is worthy of notice that the reading has also a mean- 

 ing if the curve is not closed. 



Second Mode of Generating an Area. 



If a line QT of constant length turns in the plane of the paper about 

 its fixed point Q, it will sweep over a sector of a circle. If this generating 

 line has, as before, a positive sense from Q t? " 



to T, it will be seen at once that the area ' ' ' 



swept over will be positive or negative ac- 

 cording as it turns in one or in the opposite 

 sense. It will also be seen that our rule of 

 sign will again serve to determine the sense, 

 and that this will still hold if the length of 

 QT is variable. 



From this it follows : 



If a line QT of variable length turns 

 about its fixed end Q, then the area generated 

 by it whilst T describes a closed curve tvill 

 he equal to the area bounded by this curve. 



For our rule shows at once that every 

 point within the curve, if this does not cut itself, will be passed over once 

 (fig. 7), every point without the curve either not at all or as often in one 



