42 UNIVERSITY OF COLOBADO STUDIES 



two nappes of the focal surface are formed by the two given conies, 

 and the developable surfaces consist of the cones having their centers 

 on these conies and passing through these. The singular surface of 

 the congruence, or the envelope of these cones, is itself developable 

 and is generally of the 4th class and 8th order. If the plane of the 

 conic is inclined 45°, the envelope of the circles is of the 4th class 

 and the 6th order. 



8. Parallel Curves. 



In case that the curve is parallel to the plane of reference, we 

 may write the equation of the curve in the form 



so that the equation of the representative system of circles becomes 



[x-f{t)Y + [y-g {t)Y^r {const.) 

 By differentiation 



From these equations the co-ordinates x and y of a point of the 

 envelope are found. 



x=f {t)± r g^ it) , 



^f{ir^g\tY 



y-^g{t)± rf[t) 



i7" W + / iff 



These are, however, also the values of the co-ordinates of a point Q 

 on the normal through P at a distance ± /■ from P. Hence the 

 theorem : 



The envelope of a system of equal circles having their centers 

 on a plane curve and heing situated i?i the plane of the curve is 

 identical with the curve described hy the center of an equal circle 

 which rolls upon the given curve. 



