Slip- Curves of an Amsler Planimeter 



647 



the base-circle corresponds to the infinitesimal convolutions 

 round the pole. The line joining corresponding points on 

 the two curves is of course normal to the slip-curve. The 

 equation to the locus of W only differs from that of P in 

 having c instead of b. The locus of C is easily found ; the 

 vectorial angle of any point is the 6 given above; the radius 

 vector is a — csec<£, and on eliminating <j> we obtain 



<■ 



+ c cosh 



vV 





The asymptote is rsin (ft -0) = c, where 



£tana=log,cot^-*j 



in this diagram /3=18° l'"5. 



With a general knowledge of the character of the slip- 

 curves we can easily establish the validity of the method by 

 which the instrument substitutes a line-integral for a surface- 

 integral, as follows. 



Consider first an area PQRS bounded by two circular 



arcs PQ and RS with the pole 

 as centre, and by portions of 

 two slip-curves QR, PS ; and let 

 PQ, PS each subtend an angle e 

 at the pole. Then while PQ is 

 traced there is no change in 

 the angle at H ; the wheel 

 moves over a circular arc of 

 length e . AW, but only records 

 the component e . AW. cos AWP, i.e. e (AH cos ^-HW) or 

 eCacos^— c), where 6 X is the angle at H during this part 

 of the cycle. From Q to R nothing is recorded, but 

 the angle at H alters to <f> 2 ; from R to S the record is 

 — e(a cos 2 — c) ; from S to P nothing is added. The whole 

 record is ae(cos t^-cos 2 ). If the radii of the arcs PQ, RS 

 are rj, r 2i this is the same as 



ae 





But the area is 





ih 



e. Uv-rV, 



so that the area is b times the line-integral recorded, 



