(1087) 
and 5,/ in common with it. When revolving about Oz we find that 
9, remains in C, and so the section of C, with 2 forms a congruence 
of revolution (2,2). If A, describes a right line p,*, then s describes 
a regulus @,*, which generates by revolution the same congruence’ 
as .Q,. 
The singular rays starting from the points A, of O,? form a suchlike 
congruence. The singular rays of 8 form therefore two congruences 
of revolution (2, 2). 
§ 5. A congruence of revolution C' (2,2) is generated when @ is 
cut by an arbitrary linear complex having Oz as axis: 
Per ee ndr a we 
If we replace in (1) p, by —mp,, we get: 
ANB Pe ye et pS a Pe) Oer rde (13) 
where B’ = B—2Cm + Dm’. 
We easily find for the focal surface : 
AE («7 + y*)? 4+ B («* + y’) (A + Be) + m? (A 4+ £27)? =0 (14) 
So it consists of two quadratic surfaces of revolution #,* and F,?. 
which touch each other in the same points B, and B, as O,? and 
O0. Intersection of p,— imp, =O with the invariable complex (13) 
furnishes a congruence with the same focal surfaces F'? and #*. 
The common tangents of #,* and #,? form therefore two quadratic 
congruences of revolution, which are each other’s images with respect 
to wOy. 
§ 6. Let a ray s of the congruence cut the planes 8, and @,, which 
‘ ‘ A 
are in 4, and B, perpendicular to Oz, in P, (anil 
Ld “ss t B 
—A 
ane Gz - Ve } If we express the coordinates of s in 
ty, Wy, v, and y, and if we then substitute them into (12) and (13), 
we arrive at: 
LY, — V,y, = 2m een ee en ES), 
and 
Det Ble ae es ck ay ae OL) 
The rays s of the congruence therefore determine a correspondence 
(1,1) between the points of the planes 8, and 8,. If P, describes 
a ray of the pencil (£,,9,) then P, describes a ray of the pencil 
je 
