DE. PLUCKEE ON A NEW GEOMETEY OE SPACE. 
27. If, in the second instance, the trace (45) is parallel to OY and intersects OZ' 
in E, OE being equal to the equation (44) becomes 
ws=j3 2 w, 
representing a point of OZ', the distance of which from O is 
OE'=-j = -i 2 “ = pOE. (47) 
The hyperbolic cylinder therefore degenerates into a right line (EX 0 ) within xz' 
parallel to OX and passing through E'. Hence 
All refracted rays of the complex confined within a plane intersecting yz' along a trace 
(EY 0 ) parallel to OY converge into a fixed point of the plane xz'. If the plane turns 
round its trace, that point describes, within xz', a right line EX 0 parallel to OX. Each 
ray meeting both lines EY 0 and E'X 0 is a ray of the complex. 
28. The axes of coordinates OX and OY may be interchanged by writing a 0 instead 
of b 0 , and reciprocally. Then we get analogous results if, instead of traces within YZ', 
we consider traces within XZ'. Especially we may immediately conclude from the last 
equation "written thus, 
^.OE'=«;-.OE, (48) 
that the relation between the two right lines E'X 0 and EY 0 is a mutual one. 
29. All rays intersecting two fixed right lines constitute a linear congruency , the 
fixed right lines being its directrices (Sect. I., 55). Consequently the complex of 
refracted rays may be generated in three different ways by a variable linear congruency. 
In each case the two directrices of the congruency move parallel to any two of the three 
axes of coordinates OX, OY, OZ', intersecting the third axis in two points, the distances 
of which from O are in a given ratio. 
30. Hitherto we have supposed that the plane xy is any section whatever of the 
crystal. Let us now, in particularizing again, admit that the crystal is cut along one of 
the two circular sections of the third auxiliary ellipsoid E, then represented by 
A(x 2 +y 2 )+Fz 2 =l; 
/3 being equal to unity, the equation of the complex becomes 
ra—sg (49) 
In this peculiar case therefore all rays of the complex meet the diameter OZ', conju- 
gate with regard to E to its circular section xy. Hence all refracted rays of the com- 
plex intersect OZ' as all corresponding incident rays start from OZ. 
Both the diameter of the third auxiliary ellipsoid E perpendicular to its circular section 
xy, and its diameter conjugate to that section, fall within a principal section of the ellip- 
soid containing its greatest and least axis, and consequently also its two optic axes. The 
rectangular axes of coordinates OX and OY may, without changing the equation of the 
complex, turn round O within the section xy. If one of them, OX for instance, become 
