REMARKS ON SOME GENERAL PROPERTIES OF CURVES. 279 
P in S cuts the curve S’ at a constant angle in P’, tangent to evolute 
of S makes with evolute of S’ a constant angle. 
6. If S and S’ intersect in the point O, the arc OP’ bears a con- 
stant ratio to the difference between the arc OP and the tangent PP’. 
The logarithmic spiral will serve to illustrate the two last theorems. 
7. If right lines drawn from any point #& in the curve § to touch 
the curves S$‘ and S” in the points P and Q are equal, the product of 
the tangents of the halves of the angles which the lines RP, RQ make 
with the tangent to S at the point # is constant. 
As particular examples of this theorem we may take, firstly, the 
case of tangents drawn to a circle from any point in a line given in 
position. 
Secondly, tangents drawn to two given circles from any point in 
their radical axis. 
8. In the same figure as the last, if instead of having the tangents 
equal we have the angle PRQ constant, the circle passing through the 
three points P, R, Q, touches the curve S at the point R, and the 
normals to the three curves at the points P, 2, Q, meet in a point. 
9, If right lines drawn from any point & in the curve S touching 
the curve S’ in the points P and Q contain with the are PQ a con- 
stant area, tangent at R is parallel to the right line joining P and Q. 
10. If the vertex of a constant angle is at the point O, and the 
sides of the angle cut the curve 8 in the points P and Q, and the 
curve S’ in P’ and Q’, area of the figure PQ P’Q’ is a maximum when 
difference of squares of OP and OP’ is equal the difference of squares 
of OQ and OQ’. 
Hence if from a point O outside a circle it is required to draw two 
secants containing a given angle, so that the area of the figure con- 
tained by the secants and the circumference of the circle may be a 
maximum, it is when the secants make equal angles with the diameter 
passing through the point O. 
11. If the vertex of a constant angle is at the point O, and the 
sides of the angle cut the curve § in P and Q, the sum of OP 
and OQ is a minimum when the ratio of OP to OQ is equal to the 
ratio of the tangents of the angles which the sides of the given angle 
make with the curve. 
