644 Mr. A. 0. Allen on the 



It will, however, be found that certain portions of the 

 curves are not at all easy to draw empirically, and it seems 

 preferable to draw the curve from its equation, which is 

 deduced below. 



A knowledge of the chief (and easily obvious) geometrical 

 properties of the curve leads to a proof of the principle of 

 Amsler's Planimeter, differing from all those which are to 

 be found in the usual text-books, and superior to them (in 

 my opinion) in the way in which it combines simplicity 

 with true insight into the geometrical notions underlying 

 the use of the instrument. A knowledge of the equation 

 is not at all necessary for this purpose. Let AHP repre- 

 sent the essential parts of the planimeter, A being the pole, 

 H the hinge, P the tracing-point, and W the point of con- 

 tact of the recording wheel ; then it is required to find the 

 locus of P such that the point W shall always move in the 

 direction of the line PH. Or to state the same thing in 

 another way, we require the locus of P revolving about the 

 instantaneous centre C, CW being perpendicular to PH. 



A I" x 



Let AX be the initial position of AH ; put 0, <f> for the 

 angles XAH, AHP respectively ; also AH = a, HW = c, 

 HP = b, AP = r. Let the point P reach a consecutive point 

 on the slip-curve by two distinct motions ; first let the whole 

 instrument move as if rigid through an angle dO about A, 

 so that W moves off the line PH a perpendicular distance 

 (acos<£ — c)d6. Next bring it back on to the line PH by 

 a rotation d<f> of the arm HP, AH remaining fixed ; hence 

 cd<j> = (a cos <f> — c)dd, which is the simplest equation of the 



slip-curve. Put - = cos «, so that a is the value of <p when 



P is on the slip-circle referred to below ; integration then 



6 = ^wreA l0gsm —2 — logsm V )• 



