1904.] 



BROOKS — ORTHIC CURVES. 821 



we seek the curve enveloped by this line, as / runs around the unit 

 circle. 



For the sake of simplicity, let us refer this equation to a new 

 system of coordinates, so chosen that the centre circle of the pencil 

 becomes the new unit circle. The equation becomes 



x — t-\- f^t (x — r')= o. 



If now we take an axis of reals which makes <t^ = i, and also put 

 T^ for /, we have 



XT~^ — r^ -]- X — z'^ = o. 



The map equation of the curve enveloped by this line is obtained 

 by equating to zero the result of differentiating with respect to r. 

 It is 



This is a curve of double circular motion. The curve is of order 

 six, for it meets any line, 



{7 



where 



^— 



or 



2-' — 27° — ^r- + 3r _ 3 = O. 



This gives six r's, and, therefore, the curve is of the sixth order 

 In order to determine the class of the curve, we must examine the 

 equation of a tangent, 



XT~'^ — '^'^ -f- -^v — "^"^ = o- 



This is of the fifth degree in the parameter, and there are, therefore, 

 five tangents from any point x. 



The stationary points, or cusps, are the points where the velocity 

 of X is zero. For such a point we must have Z)tx = o, and at the 

 same time | r | = i . Both these conditions are satisfied by 



The curve has, therefore, five real cusps ; one when r is each of 

 the fifth roots of minus one. 



PROC. AMER PHILOS. SOC. XLIII. 177. U. PRINTED OCT. 19, 1904. 



