Curves of Double Curvature . 
463 
passed through all values to [y/2], h passed through values of 
the form l 2 n/8. The results for ± a given by these [y/2] [S/2] 
values were evidently again given when l 2 passed through val¬ 
ues to [S/2], and k was of form l x r/ y. The number of double 
points must then be diminished by (S — l)(y —1)/2 if 8 and y 
are both odd, as in that case w — an/nrs also gives rise to 
(8 — l)(y— l)/4 double points counted twice. 
If, however, 8 is even, then (y — 1)/2 values were not counted 
twice, for these (y — 1)/2 values were thrown out above with 
(r — 1)/2 — (y — l)/2 others. If y is even, (8 —- 1)/2 values were 
not counted twice. In either case this will give either 
(y - 1)8/2 - (y - l)/2 - (y - 1)(8 - l)/2 or (8 - l)y/2 - (8 - l)/2 
= (y-l)(S-l)/2. 
Similarly in the first and third cases (y — l)( e —1)/2 values 
must be thrown out, and in the second and third cases 
(8—l)(c—1)/2 values must be thrown out. This accounts for 
all double points counted twice, so the total number of double 
points in the curve of double curvature x — cos nt f y = cos rt, 
z — cos st is equal to 
(n-l)(r-l) , (r-D(d-l) («— 1) (e — 1) (y~l)(8~l) 
2 ' 2 ' 2 2 
(«—l) (* — 1) (r-D(e-i) 
2 2 
y being highest common factor of r and s, 8 being highest com¬ 
mon factor of n and s, e being highest common factor of n and r. 
1 3. For the special cases when either y, 8, or e or any two or 
all three are equal to unity, we can derive the formulae either 
by independent proofs or we can deduce them from the results 
obtained in the last paragraph. If y equals unity the formula 
for the number of double points becomes 
(r— 1) (#— 1) , (« — 1) (a — 1) (* — )(« —1) 
2^2 2 
If 8 and y are both unity the formula becomes 
(«—•!) (g-2) 
2 
and if all three are unity, that is if n, r and s are all prime to 
each other the number of double points in the curve reduces to 
zero. 
