464 Comstock—Real Singularities of Harmonic Curves. 
I 4. To find the points, axes and planes of symmetry for the curve 
x — cos nt y — cos rt z = cos st. 
Divide the curves into three classes: First, those with n , r , 
and s all odd; second, those with two odd and one even; third, 
those with one odd and two even. If n, r, and s are all odd, 
substituting (?r — t ) for t in the equations gives the x for (tt — t) 
as the negative of that for t , the y for (tt — t) as the negative of 
that for t , and the 2 for (7 r — t) as the negative of that for t, so 
that the origin is a center of symmetry for the curve. 
If two of the numbers n, r and s are odd, say, for conven¬ 
ience, n and r , and the other even, substituting (tt — l) for t, 
gives the x and y for ( 7 r — t) as negatives of the x and y for t 
but the 2 for (?r— t) is the same as the 2 for t. Thus the curve 
is symmetrical with respect to the 2 axis. 
Similarly when n and s or r and s are odd, and r or n even, 
the curves are symmetrical about the y or x axis respectively. 
When one number is odd and the other two even, say n odd 
and r and s even, the curve will be symmetrical with respect 
to the yz plane. In this case the y and 2 , coordinates for ( 7 r — t) 
will equal those for t, while the x for the (tt — t) will be the neg¬ 
ative of that for t. 
University of Wisconsin, June 1 , 1897. 
