456 Comstock—Real Singularities of Harmonic Curves. 
Now h and k cannot be greater than [r/ 2] and [s/2] respect¬ 
ively, as, in case they should, (2 h/r ± a/b)ir or (2 k/s ± a/b)ir 
would be greater than ir. 
It was shown above that in the equation 
(2 h/r ± a/b ) 7C = (2 k/s ± a/b) it 
the signs before a/b in each term must differ. Therefore we 
can change the form of the equation to 
If we assume that a/b is in its lowest terms, b must then equal 
rs, since hs — kr is a whole number. This then gives rise to 
the conditions for the entrance of double points, viz.: 
( 2 ) 
± a = kr — hs. 
in which k > [s/2] and A > [r/2], and k can have [s/2] values 
and h can have [r/2] values, provided no two values for ia so 
given are equal. 
To show that no two values of ± a given by (2) can be equal 
to each other, we write (2) in the form of a congruence, 
kr = ± a (mod s). 
Multiplying each side of this congruence by r^ (s)_1 and sim¬ 
plifying by means of Fermats’ theorem, we find that 
k = ±ar ep &- 1 (mod s) 
which shows that for a given value of ± a , there is an interval 
of s units between successive values of k giving rise to the same 
value of ± a. As was shown above, the values of k in (2) must 
all be less than [s/2], so that no two values of k can differ by 
s units. Therefore in the series of values for ± a given by (2) 
no two can be equal. 
As a result of the ambiguous sign in (2), there will be a posi¬ 
tive and a negative value of a for every set of values of h and 
k. Since we are examining the curves for values of t between 
