Slip- Curves of an Amsler Planimeter. 143 



The theory of the instrument having been established by 

 any one of the various methods when the pole is outside 

 the area to be measured, it can be applied at once to the 

 case in which the pole is inside the area by a simple 

 method which I have described in the 'Mathematical Gazette' 

 (July 1911). 



This may be briefly indicated by reference to the figure, 

 in which, for simplicity, the curve is shown wholly outside 

 the base-circle. 



Draw any arbitrary curve PS from the curve to the base- 

 circle. Starting at P, take the tracer round the boundary 

 PQ RP, in the positive (clockwise) direction, along PlS, 

 round the base-circle SU TS in the positive (this time, anti- 

 clockwise) direction, and finally back along SP. 



The record is clearly the area of: the portion of the curve 

 outside the base-circle. But this record is merely that which 

 would be obtained by taking the tracer round the boundary 

 PQ RP, for nothing is recorded round the base-circle and 

 the records along PS and SP cancel. Hence by suppressing 

 these motions and taking the tracer once round the boundary 

 in the usual way, the instrument gives the excess of the 

 area of the curve over that of the base-circle. 



Similar reasoning can be applied when the curve lies 

 wholly inside or partly outside and partly inside the base- 

 circle. 



Yours faithfully, 

 Technical College, Bradford. J. A. ToMKINS. 



