458 Comstock—Real Singularities of Harmonic Curves . 
for one double point to have as abcissa the negative of the ab- 
cissa of the other, 
cos 
r{kr — hs) 
rs 
it = cos £(2 /i 4 -l) ft ± 
r(k'r — h's)7t ~y 
rs J 
or 
r(fcr-ftg) rWr -hU 
rs ' rs 
which is easily seen to reduce to 
k ± k' = (2^ + 1 ?h' + h) 
s r 1 
a relation which is manifestly impossible since Jc > [s/2] and 
h > [r/2], and r and s are prime to each other. 
In a similar manner it can be shown that no two ordinates 
given by air/rs can be the negatives of each other. As the rea¬ 
soning for either abscissas or ordinates does not depend on 
both r and s being odd, these proofs hold equally well in the 
cases when r is even and s odd, and when r is odd and s even. 
In order that a? = 0, y — 0, may be a double point given by 
« 7 r/rs, cos r air/rs and cos san/rs must both equal zero, rair/rs 
and sair/rs must then be of the form mir/2. Call rair/rs, m^/rs 
and s air/rs, m$r/ 2; a must then both equal m^/2 and m 2 r/2, 
which is impossible since a Os and r and s are prime to each 
other. 
Therefore, when r and s are both odd, the double points given 
by 7 r ~ air/rs are all distinct from those given by air/rs and the 
number is then 2[r/2][s/2] which equals, 
{r —1) {s —!) 
2 
In case r is even and s odd it is easily seen from an exami¬ 
nation of equations (3) that aWrs and ir—air/rs can give rise to 
the same double point only when an/rs is such that y — 0, and 
that, whenever such is the case, air/rs and 7 r — a-rr/rs do give 
rise to the same double point. 
y = cos saTt/rs = cos s{kr — hs)Tt/rs == cos {kit — hsn/r) — 0. 
only when A = r/2, k having any value, k has (s—1)/2 possible 
values. Therefore, for (s—1)/2 values of an/rs, an/rs and * — air/rs 
