MATHEMATICAL SECTION. 131 
at ones (21,) 
A 21 
ange be ye 
1 : 
Now B+ bist a ai + Am ale = 0 
and (21,) or (21,) can always be satisfied for some value of z; 
therefore (20,’) intersects the chord. In the same manner it may 
be proved that (20,’) and (20,*) intersect the chord. It follows, 
then, that (20,), (20,), and (20,) cannot intersect the chord, and 
hence belong to the same system of generation. 
The equations of a pair of lines intersecting in a given point of 
the hyperboloid and belonging to different systems of generation 
can be easily found by the condition that one of them must inter- 
sect (20) and the other (20). I omit this, but give a remarkable 
symmetrical form of the equation of the hyperboloid : 
0=(e@—%,) (y—4,) (2-4) — (@— &) (Y — Ya) @—%) (22) 
TF @(Y,2,— Ya%y) + y (2,2, Ta 2, X,) = 2(2, Yq aah x, Y,) 
— xy (2, —%) — y2(%, — 2.) — 2e(Y, —Y,), because 2,4, 2, = %, Y,% 
by (18) and (18'). 
It is immediately evident that this equation is satisfied by equa- 
tions (20). Itis not uninteresting to prove that it also satisfies (21), 
or that it contains the chord, since it shows the remarkable plia- 
bility of these forms by virtue of the relations (16), (17), (18), 
fo"), (177), (18°). 
The points (2, Yes 2,)s (®er Yor 2v)» (Lp Yor 2) (Lvs Yar 20)» (ys Yar Za)» 
(&,, Y,, 2,) form a warped hexagon, which lies wholly in the hyper- 
boloid, and its sides may be considered six intersecting edges of a 
characteristic parallelopipedon. These edges are: 
1 1 1 
A= 5%, —2,)3 B= > y.—9.); C= > @—4,) 3) 
and the co-ordinates of its center are: 
1 1 1 
t= 5 (% + %)3 Yor va iy “U3 Se OY (2, + %) (24) 
and these must be those of the center of the hyperboloid also. 
Transferring the origin of co-ordinates to this center, we have 
the equation of the hyperboloid regarding (23) : 
0= (@— A) (y— B) (@— C)—(@+ A) V+ B+) (25) 
