The Hyperbolic Spiral—Its Properties 
and Uses. 
BY WALTER K. PALMER. 
There are many well known and useful varieties of spirals. Of 
these, one of the most interesting in its properties and uses is the 
Hyperbolic Spiral. 
It is called Aypferbolic because its equation is of just the same 
form as the rectangular equation of the common hyperbola, and 
because it may be plotted by transforming the rectangular co-ordi- 
nates of this hyperbola into polar co-ordinates. It is also called 
the ‘‘Reciprocal” or ‘‘Inverse’ Spiral, from the fact that one co- 
ordinate varies inversely as the other, or equals a constant times the 
reciprocal of the other. These facts are shown fully by the general 
; Be, Cc : 
form of the curve’s equation, which 1s I=] , where r is one co-or- 
dinate, the radius vector, 6 the other, as shown in Fig. 1, and c, a 
numerical constant, the magnitude of which determines the size of 
the particular spiral in question. 
Bigs 4, 
Now this relation between r and @ andc is the same at every 
point along the curve as well as at P. That is, at any other point 
QO, the particular length of r there, (r) equals the number c, divided 
by the particular angular value of 6 (6’) corresponding to r’. 
Bearing this in mind a practical drawing board construction for 
the spiral may be derived readily and many interesting properties 
discovered. 
(155) KAN. UNIV. QUAR.,. VOL. VII. NO. 3, JULY, 1898, SERIES A. 
