Slip-Curves of an Amsler Planimeter. 645 



Taking AB as #-axis, the coordinates of P are 



# = acos — b cos (# + </>), y = asin 6 — b sin (6 + </>), 



from which the curve may easily be drawn by assigning 

 values to <j> between the limits and it. If 6 is required in 

 degrees, and if tabular logarithms of sines are used, the 

 equation is 



„ 180 cW,10 /, . 6 + a T . (b^a\ 



7T v« 2 -c 2 v 2 2 ; 



The diagram accompanying this paper shows one of the 

 curves for a planimeter in which a= 6*305 in., 6 = 4*225, 

 o = 0*674, a = 83° 52'. The points are denoted by Roman 

 numerals, the successive values selected for <£ being 0° 0', 

 7° 57', 17° 23', 27° 15', 35° 54', 46° 51', 57° 44', 68° 55', 

 76° 12', 80° 42', 82° 14', 85° 21', $6° 56', 91° 49', 102° 14', 

 116° 3', 130° 25', 146° 27', 161° 36', 180° 0'. These have 

 merely been chosen so as to give convenient points on the 

 locus. When </> = 83° 52', 6 is infinite, and x and y are 

 indeterminate, giving any point whatever on the slip-circle 

 or so-called " base -circle." Its radius is *Ja 2 — c 2 , i.e. 

 7*20 in., and it obviously has the property that if the tracer 

 be taken round it, the wheel will only slip, since </> will be 

 constant and AW will be always normal to PH. The slip- 

 curves are a family of similar curves, each of which 

 approaches the base-circle asymptotically from within and 

 without ; it will be noticed that the approach is very rapid, 

 being almost complete before a single convolution about the 

 pole has been described. 



The diagram also shows the corresponding points on the 

 locus of W (Arabic numerals, primed) and on the locus of C 

 (Arabic numerals). There is no practical value attaching 

 to the latter, but from the geometric standpoint it is in- 

 teresting. It consists of two long flat branches ; the first 

 starts from the point M (a — c, 0) at right angles to the 

 ,i*-axis, approaches the pole O and performs an infinite number 

 of convolutions about it in the anti-clockwise direction ; it 

 then performs convolutions in the opposite sense and goes 

 to — oo along the asymptote. The other branch returns 

 from + co along the asymptote, on the opposite side of it, 

 and finally runs into the point N(a + c, 0) at right angles to 

 the A'-axis. The correspondence between various portions of 

 this curve and those of the slip-curve are interesting ; the 

 inflexion on the outer branch of the slip-curve of course 

 corresponds to the points at infinity on the C-locus, while 



