282 
therefore, restoring the values of a and /3, the required 
equation is 
a 2 (a? 2 + y- + % 2 - c 2 ) (m s + w 2 ) 
= ^ (# + mx + w?/) \/« s - c 3 — c \/ a 2 — x 2 — y 2 - z 2 | 2 . 
12. The locus of the normal to the surface y = x tan n% 
along a generating line is a hyperbolic paraboloid. 
Problems on Section VIII. 
The following are some examples of finding the equations 
to twisted and developable surfaces, and envelopes. 
1. To find the equation to the twisted surface of which 
the directrices are two vertical circles having the opposite sides 
of a horizontal parallelogram for diameters, and a straight line 
passing through the center of the parallelogram perpendicular 
to the planes of the circles. 
Let the center of the parallelogram be taken for origin and 
its plane for that of xy, and the rectilinear directrix for the 
axis of y; then the equations to the three directrices will be 
x — 0 % = 0, 
y = — b (¿v — a) 2 + z 2 = r 2 , 
y = + h (a? + a) 2 + z 2 = r 2 . 
Also let the equations to the moveable line be 
cc = a (y - /3), x = 7 (y - /3), 
so that it already fulfils the condition of meeting the axis of y, 
and one of the parameters consequently is eliminated; then, 
expressing that it passes through each of the circles, we have 
5a(6 + /3) + a} i + -/C+/3) : ’ = r ! l 
fa (6 - /3) + a}‘ + y (h - /3f = r*/-'’