Complex-cone. 
Complex 
cylinder. 
Complex conic. 
Equatorial 
surface. 
Or 
bo 
mS 
PROFESSOR CHRYSTAL ON 
On the Complex of Central Axes. 
From the equation 
pay t+ Ppp thy? 
we get at once for the locus of all the rays passing through a given point abe 
P'(a—ay +9 (y—b)' + (z—ey 
= (bz —cy)? + (cv—az)’ + (ay—ba)’ , . , (0) 
a cone of the second degree. The complex is, therefore, of the second order. 
As a particular case of this we see that all the rays of the complex parallel 
to a given line (/mn) lie on the right circular cylinder where radius is 
SPE +9?m? + hn’ . 
The locus of the feet of the perpendiculars on the generators of the com- 
plex-cone is the curve of the fourth degree, common to (10), and the 
sphere 
P=antby+cez.  . : : ; 
In the case of the complex cylinder this locus is, of course, a circle. 
Since any plane through the vertex of the complex-cone cuts it in two 
generators, we see that through every point of a given plane there pass two 
rays of the complex. Hence all the rays in a given plane envelop a conic. 
If we consider all the conics in planes parallel to a given plane, the complex 
conics generate PLUcKER’s equatorial surface. 
Let the equation to one of the planes be 
lat+uyt+tnz=p . : ‘ | tiay 
Putting E=2—a n=y = C=z—¢, 
we have from (10) 
A&’+ Br’? +Cl— 2Dnl—2ElZE— 2F&#=0, . fi (TS) 
and lE+myn+nl=0, . ; , : : .* (ae 
where A=r—#—/f’, &. D=yz, &c. 
The resulting envelope is given by the equation of the fourth degree 
A—F —E / |=0 
—F B—D m 
—E —D me! a hig : ; : . (15) 
Ll sm nm 0 
