460 Comstock—Real Singularities of Harmonic Curves. 
that the curvature can change but once in one of these inter¬ 
vals, since as t takes on values from mr/r to (n +1 )ir/r 9 the co¬ 
sines of multiples of these values of t present a continuously de¬ 
creasing or increasing series of values in this interval. 
There are two ways in which the second derivative, and hence 
the curvature, may change sign. Either the curve passes 
through a point of inflexion or else is perpendicular to the axis 
of y. 
It can become perpendicular to the axis of y only when 
dx/dy = oo. For this case 
dx — r sin rt _ 
dy —s sin st 
when sin st = 0. 
Now sin st equals zero only when t — mir/s and there are evi¬ 
dently only s — 1 values of t of the form mic/s between 0 and tt. 
If r is greater than s , let r = s + a. There will then be 5 +a — % 
intervals between successive tfs of the form mr/r and (n + 1 )ir/r. 
Now the intervals between mr/r and (n + l)v/r are shorter than 
those between mir/s and (wi + l)w/«, so that no two values of 
mn/s can coexist in an interval from mr/r to (n + Y)ir/r. 
Now since there are s + a —2 changes of sign in the second 
derivative and but s — 1 of these are caused by the curve be¬ 
coming perpendicular to the axis of y , there must be a—1 points 
of inflexion. Therefore there are (r — s — l) points of inflexion 
when r is greater than s. 
Similarly it can be shown that if s is greater than r the num¬ 
ber is (s — r — 1). 
§ 4. The conditions of symmetry .— 
Three cases can arise, r and s both odd, r even and s odd, or 
r odd and s even. 
In the first place let t' be a value of t giving rise to the points 
x\ y on the curve. If we use tt — t' we obtain — x\ — y, as the 
co-ordinates of the resulting point. Therefore the curve is 
symmetrical with respect to the origin. 
In the second case if t' gives rise to the point x\ y\ ir — t' will 
give rise to the point x\ — y\ so that the curve is symmetrical 
with respect to the axis of y. 
