T 
"•«■S 
790] 
GEOMETRY. 
579' 
having the common transverse axis zOz ; the section by any plane z=± y parallel to 
that of xy, y being in absolute magnitude >c, is the ellipse 
^ . V' _ t _ i . 
a 2 + b 2 c 2 ’ 
and the surface, consisting of two distinct portions or sheets, may be considered as 
Fig. 25. 
generated by a variable ellipse moving parallel to itself along the hyperbolas as 
directrices. 
The hyperbolic paraboloid is such (and it is easy from the figure to understand 
how this may be the case) that there exist upon it two singly infinite series of right 
lines. The same is the case with the hyperboloid of one sheet (ruled or skew hyper 
boloid, as with reference to this property it is termed). If we imagine two equal 
and parallel circular disks, their points connected by strings of equal length, so that 
these are the generating lines of a right circular cylinder, then by turning one of 
the disks about its centre through the same angle in one or the other direction, 
the strings will in each case generate one and the same hyperboloid, and will in 
regard to it be the two systems of lines on the surface, or say the two systems of 
generating lines; and the general configuration is the same when instead of circles we 
vu '2/ Z^ 
have ellipses. It has been already shown analytically that the equation — + — — = 1 
CiZ 0“ c~ 
is satisfied by each of two pairs of linear relations between the coordinates. 
Curves ; Tangent, Osculating Plane, Curvature, cfie. 
38. It will be convenient to consider the coordinates (x, y, z) of the point on 
the curve as given in terms of a parameter 6, so that dx, dy, dz, d?x, &c., will be 
proportional to ~^, &c. But only a part of the analytical formulae will 
be given; in them £, rj, £ are used as current coordinates. 
73—2