6±$ The Slip- Curves of an Amsler Planimeter. 



Hence the area is 2-73- R& for each revolution of the wheel, 

 R being the radius of the wheel ; it is generally so chosen 

 that 27rR?> = 10 sq. ins., e.g. in this case the diameter of the 

 wheel is almost exactly £ of an inch. 



It has been tacitly assumed that P and S lie on the same 

 branch of the slip-curve ; if they are separated by the base- 

 circle, a different expression of the facts may be adopted. 

 The journey from P to Q, then out on to the base-circle by 

 a slip-curve and back along another slip-curve to P, will 

 record simply (a cos </>i — <?) e. Then from P back along the 

 slip-curve on to the base-circle, so along the slip-curve to R, 

 then to S, and finally back by a slip-curve, the record will 

 be — (a cos <f> 2 — c)e, and this will be added to the previous 

 one. The only difference from the last case is that the 

 numerical value of this second term is now positive, whereas 

 it was negative before. 



It is easy now to see how an area of any shape may be 

 split up into elements of this kind and the proof then 

 generalized. 



A possibility not yet considered is that the pole may lie 

 within the area to be measured (but of course at not more 

 than a + b nor less than a—b from any 

 point of the perimeter). In this case 

 let LMN be part of the base-circle, 

 LRSM and MQPN being portions of the 

 perimeter respectively external and in- 

 ternal to this circle. Then the record 

 during the journey LRSM evidently gives 

 the area of the figure LRSM, positively ; 

 during the journey along MPQN we are 

 travelling along the perimeter in the same 

 sense, but inside the base-circle instead 

 of outside, so that the wheel will be rota- 

 ting in the opposite sense, and we shall 

 therefore have the area MQPN recorded 

 negatively. Proceeding thus all round 

 the perimeter, it is clear that what is recorded is merely the 

 excess of the area of the curve above that of the base-circle ; 

 it is therefore necessary to add the latter, viz. nr{a 2 -\-b 2 — c 2 ). 

 This area is generally engraved on the weight which keeps 

 the foot of the instrument pressed into the paper ; in the 

 present instance it is 179*54 sq. ins. 



