545 
which represents immediately the variable plane 11, may be regarded as 
being also a Tangential Symbol (or Plane-Symbol) jor the Line A. For 
example, the line zc may thus be represented, not only by the local sym- 
bol (99), but also by the tangential symbol, 
[s00¢u], ifo =t+u, and o=-— a. (100) 
In fact, this last symbol can be derived, by linear combinations, from the 
symbols (94) for the two planes Bcp, pcr, which intersect in the line Bc; 
and if any particular value be assigned to the ratio ¢: u, a particular 
plane through that line results. But it is time to apply these general prin- 
ciples to the Geometrical Net in Space. 
Part IT.— Applications to the Net in Space: Enumeration and Classifica- 
tion of the Lines, Planes, and Points of that Net, to the end . the 
Second Construction. 
[22.] The data of the Geometrical Net are, by [1.], the five points 
ABCDE, Or Py; of which the guinary symbols (27) have been assigned, and 
shown to be s yntypical [9.|; and also the ternary symbols (92) of the 
three first of them. Of these the symbol 
4 = (100) 
may be taken as the type; and the point a itself may be said to be a 
Ltrst Typical Point. 
[23.] The derived lines A,, of First Construction [1.], are the éen 
following, 
Bc, &c.; DA, &c.; Ba, &c.; and DE; 
the ‘‘ &c.”’ being<interpreted as in [20.]; and each line A, connecting, 
by its construction, two points Pp. Among these the line Bc may be se- 
lected, as a First qT ypical Line ; and its. symbols [21.], namely, 
(Oyz), and [c00¢u], 
whereof the former represents this line Bc considered as the Jocus of a 
variable point, while the latter represents the same line considered as the 
hinge of a variable plane, may be taken as types (the point-type and the 
plane-type) of the group of the ten lines Ay. 
 [24.] The derived planes M1, of first construction are in like manner 
ten; namely, 
ADE, &¢.; BcE, &.; BoD, &c.; and Azc, 
each obtained by connecting three points Pp). Of these the last has, by 
[20.] the quinary symbol, 
anc = [00011], 
which may be taken as a type of the group 11,; and the plane azc itself 
PROC. R. I. A.— VOL. VII. 46 
