"Slip-Curves" of an Amsler Planimeter. 491 



(1) When c<a, say c = acosu, we get the solution which 

 he gives : 



either 6 = cot a{ log sin J (« + $) — log sin \ {a — </>)}, 



or #=cot «{log sin i(</>-f «)— log sin ^ ((/>—»)}, 



according as initially </> = or 180° wdien 6 = 0. 

 These may also be written 



tan \§=- tan \ol tanh {^6 tan a), 



tan J<£= tan \a coth (-J0 tan a). 



These give the inner and outer branches of the slip-curve. 

 The radius-vector is given by 



r 2 = a 2 -f c 2 — 2ac cos (f>. 



In each case as #>co , <£>#, and r> \/a 2 — c 2 . Both branches 

 are therefore spirals with the same circular asymptote, the 

 Ci base-circle." Each also has a cusp the tangent at which is 

 the initial line. (Only half of each curve is drawn in the 

 article cited, p. 646.) 



(2) Consider, now, the case c > a, and put c = a cosh a. The 

 integral is then 



Q— — 2 coth a tan -1 (tan \$ coth \a) , 



i.e., tan J</>= — tanh \ol tan (\Q tanh a), 



where = 0, $ = initially. There is no restriction in this 

 case on the value of <f> ; when cf> = 180°, 6 = — 7rcoth a. The 

 distinction between inner and outer branches thus disappears. 



Fiar. 3. 



The circular asymptote has become imaginary, and the curve 

 consists of a succession of continuous branches all lying 

 between the circles r = c + a and r = c — a, and 

 on these circles. (Fig. 3.) 



having cusps 



