490 Dr. D. M. Y. Sommerville on the 



H on a given curve while the other end W moves always- 

 along WH (fig. 2). Let WH be slightly displaced to WH'. 

 The instantaneous centre C is found by drawing WC perpen- 

 dicular to WH to cut the normal at H to the guide-curve. 

 W is moving instantaneously in a circle with centre 0, 

 hence C is the centre of curvature for the locus of "W 

 at W. Thus the locus of C is the evolute of the locus o/'W : 

 this is the space-centrode. The body-centrode is the straight 

 line WC. Hence the motion can be produced by rolling a 

 straight line on a fixed curve. Further, the locus of W at W 

 cuts orthogonally the circle with centre H and radius HW. 

 Hence the locus of W is an orthogonal trajectory of the system 

 of circles ivith radius equal to H\V and centres on the guide- 

 curve. 



In the Amsler planimeter, while the wheel traces a u slip- 

 curve," the tracing-point P describes another curve. It will 

 be convenient to distinguish these respectively by the names 

 primary and secondary slip-curves. It is stated somewhat 

 loosely that in order to obtain the most accurate measure- 

 ments the tracer should as far as possible cut the secondary 

 slip-curves orthogonally. The essential requirement, how- 

 ever, is that the wheel should as far as possible cut the 

 primary slip-curves orthogonally, and this is expressed more 

 simjly by the condition that the wheel should as far as 

 possible move in a circle with centre H. At the same time 

 the tracer would also move nearly in a circle with centre H. 

 The curves which the tracer should as far as possible cut 

 orthogonally are therefore not the secondary slip-curves for 

 the point W, but the primary slip-curves for the point P 

 itself. It may often happen, in fact, when the tracer is 

 moved at right angles to the secondary slip-curve that the 

 wheel moves nearly along and not nearly at right angles to 

 the primary slip-curve. 



It is hardly necessary to use these curves at all since their 

 function can be replaced by circles, but it is of interest to 

 note that, since now HP can exceed AH, we may get slip- 

 curves of entirely different character from those of the wheel 

 for which HW is alwavs less than AH. 



Taking as initial line the line AX in which AH and WH 

 initially coincide, the angle A.HW being then zero, let 

 ZXAH = #, ZAHW = </), both measured in the same sense ; 

 also let AH = a, HW = c, then when W describes a primary 

 slip-curve we have, with Mr. Allen, 



m=- cd t ■ 



a cos <p — c 



