( 18 ) 



focal curve of a curve in space in function of the singularities of 

 this cnrve. 



Let the cnne be of degree u, of rank o, of class r; let its number 

 of stationary points be y., that of its stationary tangents 0. Suppose 

 the curve to have no real nodes or double tangents and no particular 

 position Avith respect to the plane at infinity or with respect to the 

 imaginarv circle at infinity. 



In that case the singularities of the evolute or of the cuspidal curve 

 of its focal developable (G. Darboux : Classe Remarqual)le etc. p. 19) 

 are the following: 



rank, ;• =: 2 (fi + 9). 



class, m = 2 Q. 



number of stationary planes, a = 2 (r -f 0). 



double osculating planes, G = (?' — Q — n — 3 (r + d). 



stationary tangents, r = 0. 



nodes, ^= x ^ 3 {(i — q) -\- v -\- 0. 



double tangents, to = 0. 



degree, ?i = 2 (3 ft -\- v -^ 0). 



degree nodal curve, .r = 2 (ft -j- qY — 10 (i — 2 ^ — 3 (r -f ^)- 



number of planes through two lines which pass through a gi\en 



point, i/=2 {h-\-qY — 4ft — 4o — (r4-^). 

 stationary points, ^ = 12 ft — 4 9 — Ö (r -{- 6). 



The chief singularities of the focal curve are : 

 degree, n = 2 ft' -1- 4 ft o -|- 9' — 11 ft — q — 3 (r -f- 0). 

 rank, r = 4 ft (? + ^^ — 4 n — 4 9. 

 number of stationary tangents, v = 0. 



class, m = (3 ft + 2 r + 2 0) (2 n + (>) + 3 n ^ — 36 ft + 12 o — (r-f 6»). 

 number of stationary points, ;> = 2 (3 ft -{- r -\~ d) (2 ft -)- o) — 57 ft + 



21 Q — 27 (r + 6). 

 „ „ planes, « = 6 (2 ft -f p + 0) (2 ft + ^) — 4 ft' — 



2 iiq — 2q' — 107 (i -\- ^7 Q — o7 {v -\- d). 



Wlien comparing these singularities with the values of the singu- 

 larities of the evolute and of the focal curve of a plane curve, w^e 

 see that they' differ only in the rank of the curve in space being 

 substituted for the class of the plane curve aiid in the number of 

 stationary tangents t being replaced by (v -|- 0). From this follo\vs 

 that the singularities of the evolute and of the focal curve of a curve 

 in space c are the same as those of a plane curve d, which is the 

 projection of c on an arbitrary plane from an arbitrary point. 



