704. Mr. W. Bennett on 
Then, since (9, Yo, 20) lies on this line, the ray becomes 
~ 
~ 


vas =F (mmzn+h) | 
Reece ap shear | 
Ye ge —, (neo+k) J 
Equating the values of 2) obtained from these, we get for 
the equation of the ruled surface : 
abt Mec Pa sah eee 
x—m(z—b)” y—n(z—a) SO? 
(a—b)ay—(am+h)(z<—b)y + (bn +k) (e—a)e 
+ (nh—mk) (s—a)(e—b) = 0. 
This represents a ruled quadric whose section by a plane 
z=const. is the rectangular hyperbola whose equation is 
obtained by substituting for z its constant value. 
The asymptotes of this section lie in the planes 


am+h 
. a—b ae, 
sy bn-+h 
y= ae ae) 
respectively, each of which passes through one of the focal 
lines. 
The two sets of generators are given by the two pairs of 
equations : 
vb+h(z—b) _ sa ‘7 



v—m(<—b) \ 
yatkhe—a) Su 
y—-n(e—-a) sid 
and 
DU se) a 
yatk(z—a) { 
“—m(e—d)- | 
y—n(z—a) They, 5 
respectively. 
The first set is the rays, and A=z. The object-line and 
the focal lines are members of the second set. The two focal 
lines are given by w=0 and w= respectively, and the 
mb -+ h 
ons This case is illustrated in model 6. 

object-line by Dee 

