489 ] 



XLVII. The " Slip- Curves" of an Amsler Plani meter. By 

 D. M. Y. Sommeeville, M.A., D.Sc, Victoria University 



College, Wellington, N.Z* 



IX a paper with the above title f Mr. A. 0. Allen has 

 discussed the loci of the tracing-point and the recording- 

 wheel of an Amsler planimeter when the wheel slips without 

 rolling. There are certain very simple geometrical properties 

 connected with these curves which do not appear to have 

 been noticed. It is the object of this paper to indicate some 

 of these, and also to point out what appears to the present 

 writer to be a mistaken idea regarding the role of these 

 curves. 



Consider, first, the variety of planimeter in which the radial 

 arm AH becomes infinitely long, so that the hinge H is 

 guided along a straight line XX' instead of in a circle. The 

 slip-curve for the wheel W is then a tractrix with asymptote 

 XX' (fig. 1). The locus of C, the instantaneous centre for 



Fisr. 1. 



Fig-. 2. 



X H 



HW, is a catenary which is the evolute of the tractrix, and 

 the motion of the rod HW can be produced by rolling the 

 line CW on the catenary. Also it is well known that the 

 tractrix is the orthogonal trajectory of a system of equal 

 circles whose centres lie on XX'. 



These results can be extended to the general case in which 

 a line-segment HW of constant length moves with one end 



* Communicated by the Author. 



t Phil. Mag-, vol. xxvii. p. 643 (1914). 



