900 
A; the focal planes, however, do not coincide, for one is t,; and the 
other connects the tangent to 7; 
22. Order and class of the focal surface can be immediately 
determined by means of two dualistically opposite equations of 
SCHUBERT, viz. 
cop* = opge + Ophe — ope, 
and 
80e —. eq, + deh, — pe *). 
We conjugate to each ray g of the congruence all other rays as 
rays A, we then obtain a set of o* pairs of rays and we can apply 
to these the two equations just quoted. The symbol o indicates 
that the two rays of a pair must intersect each other, ¢ that they 
lie at infinitesimal distance and p’ that the point of intersection p 
must lie in two planes at a time, thus on an indicated line; so 
cop? is evidently tbe order of the focal surface. The condition opge 
indicates the number of pairs which cut each other, whilst the point 
of intersection p lies in a given plane and the ray g likewise in a 
given plane; now there lie in a given plane 14 rays of our congru- 
ence, thus 14 rays g; each of these intersects the plane of the con- 
dition p in one point and through each of these pass 5 more rays 
of the congruence; 6pg, is therefore 14 X 5 = 70, and oph, means 
the same and is thus likewise = 70. 
With ope we must pay more attention to the point of intersection 
of the two rays and to the connecting plane than to the rays them- 
selves; ope indicates namely the number of pairs of rays which cut 
each other and where the point of intersection lies on a given line 
and at the same time the connecting plane passes through that line; 
this number is evidently the third of the three characteristics of the 
congruence, thus ‘the rank, however multiplied by 2 because each 
pair of rays of the congruence represents 2 pairs gh: so ope is == OU: 
so that the order of the focal surface is equal to 70 + 70—89 = 60. 
soe’ indicates the number of pairs of rays at infinitesimal distance 
whose connecting plane passes through 2 given points, so through 
a given line, i.e. the class of the focal surface. Now oeg, indicates 
the number of pairs of rays whose connecting plane passes through 
a given point, whilst also the ray g passes through a given point. 
So tkere are 6 rays g and in the plane through one of those rays 
and the point of the condition e lie besides g still 13 others; ceg, 
1) ScHuBERT |. c. page 62. 
