. Machinery and processes of the industrial arts, and apparatus of the exact sciences. easure of the sector. THEORY OF THE AMSLER PLANIMETER. 627 The constant in the above expression will be seen to be equal to thevalue which belongs to PS in the position represented in Fig. 149, whereQ is a right angle. Eepresent this radius by R, and we shall see that£R«> expresses the area of a sector in a circle WXYZ, of which theconstant radius is determined by the condition that the plane of thewheel shall be at right angles to the circumference of the circle WXYZ,which is the direction of its movemen
. Machinery and processes of the industrial arts, and apparatus of the exact sciences. easure of the sector. THEORY OF THE AMSLER PLANIMETER. 627 The constant in the above expression will be seen to be equal to thevalue which belongs to PS in the position represented in Fig. 149, whereQ is a right angle. Eepresent this radius by R, and we shall see that£R«> expresses the area of a sector in a circle WXYZ, of which theconstant radius is determined by the condition that the plane of thewheel shall be at right angles to the circumference of the circle WXYZ,which is the direction of its movement. While the wheel is in this posi-tion it slides without revolving. Hence, for the entire circle WXYZno record will be made; and if, in the measurement, the tracer is carriedentirely round the pole P, it is necessary to add a constant which isequal to this circle in value. If the figure measured lies wholly on one side of the instrument, as inFig. 144, the constant occurs in the subtractive part of the movement aswell as in the additive; and hence no account need be taken of Fig. 149. The term bo, which is the variable part of the value of S, measures theportion of the sector PST, Fig. 148, which is between the arc describedby the tracer and the circumference of the constant circle only. Takingin Fig. 149, PS equal to PR + RS, and sweeping with the tracer aroundan entire circumference, 1)0 ought therefore to be equal to the ring con-tained between the two circumferences WXYZ and ring equals 7r(PS/2-PS2)=^((a + &)2-((l2 + &2 + 2&c)^==7r(2(a-c)?>Y 628 PARIS UNIVERSAL EXPOSITION. In the position supposed, PR and ES being in a straight line, theplane of the roller coincides with the plane of its motion, cp=0°, cos p=l,and 6=0. Hence, in the entire revolution, Q taking the place of Q, &0=&(circumf.^ Which being equal to the area of the ring as just found, confirms theconclusion above deduced. When a figure lies entirely within the ring just mentioned, as ABODE,
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