462 Comstock—Real Singularities of Harmonic Curves. 
Since t' cannot be greater than 7 r for a single tracing of the 
curve 
(2) h > [n/2], k > [r/2] and 3 > [s/2], 
unless all signs in ( 1 ) before air/b are negative. If all the 
signs of air/b in ( 1 ) are the same we have 
2 h/n — %k/r — 2//«, 
which can only be true for A, k, and j equaling respectively 
ft, r, and s, which is impossible by ( 1 ), as € is less than 7 r. 
The sign before a 7 r/& in one of the members must then differ 
from the other two, giving rise to the three cases: 
2 hit/n =F aic/b — %kit/r ± ait/b — 2jn/r ± ait/b 
(3) 2 hit/n ± «7T/6 = ‘Zkit/r =f ait/b = 2j7r/s ± atf/fr 
2 hit/n ± ait/b == 2&7r/r ± a?r/6 = 2^’^r/s T a?r/6 
Let us consider the first case. Putting the first member equal 
separately to the second and third and simplifying, we obtain 
the relation 
a _ kn — hr _ jn — hs 
~b nr ns 
Since ft, r, and s have no common factor, b must equal nrs in 
order for this relation to be satisfied. 
Putting b = nrs we obtain the relation ± a — kns — hrs 
=jnr — hrs or Jc/j — r/s. For this to be true it is necessary that 
k = l x r/y and j=l l s/y where l x > [y/2]. Then ± a will be given 
by the relation ± a — l x srn/y — hrs in which l x > [y/2] and 
[n/2]. Using a method exactly similar to that used in §(3) 
Plane Curves, it can be shown that, except for (y — 1)/2 values 
when ft is even and (ft — 1)/2 values when y is even, double 
points also arise when t — ir— an/nrs , so that in all there will be 
(y — l)(ft — l)/2 double points. 
If we use the second equation of (3) we obtain in the same 
way (8 — 1 )(?— 1)/2 double points and from the third we get 
(s — l)( e —■ l)/2. Some of the values, counted in the first case, 
in the (y— l)(ft — l )/2 values, were also counted in the second 
case, in the (8 — l)(r — l )/2 values. 
In the value for ± a — l x nrs/y — hrs we should note that as l x 
