573 
Part V.—Applications to the Net, continued: Distribution of the Given or 
Derived Points, in a Plane of Second Construction, and of First or 
Second Group. 
[109.] It will be necessary to be much more concise, in our remarks 
on the distribution of the net-points in planes of second construction; but 
a few general remarks may here be offered, from which it will appear 
that each plane II,, , contains forty-seven of the 305 points P,,P,,P,; and 
that each plane II,,, contains forty-three of those points; with many 
cases of collineation for each. 
[110.] We saw in [33. ], that each plane II,,, contains two lines A,,,, 
which intersect in a point P,, and may be regarded as the diagonals of a 
quadrilateral, of which the four sides are ifnce Wen ca Yale conteanel there- 
fore, as has been seen, one point P,, and four points P,; but it is found 
to contain also 42 points p,, arranged in six groups, as follows. 
[111.] There are 2 points P,,,, namely the intersections of opposite 
sides of the quadrilateral ; thus, in what we have called the second typr- 
cal plane [33.], the sides B,c,, ¢,B, intersect in-the point a’; and the 
sides ¢,C,, BB, In D,’ (62). - 
[112.] The plane contains also 8 points P,,,; namely, two on each 
of the two diagonals, and one on each of the four sides; and it contains 
4 points P,,;, namely two on each diagonal: but it contains no point of 
either of the two groups, P., 4, Ps,5, aS a comparison of their types suffi- 
ciently proves, or as may be inferred from the Jaws of their construction 
[46.] [47.]. 
[113.] The same plane contains 12 points P,,,; namely two on each 
side of the quadrilateral ; and four others, in which the plane is inter- 
sected by four lines A,,,; as the types sufficiently prove. But to show, 
geometrically, why there should be only four such intersections, conduct- 
ing thus to new points p,,, in the ps let the five inscribed pyramids 
[28.] be denoted by the symbols a’..’; then ‘the six edges of the 
pyramid a’ are found to intersect the present plane II,,, in points already 
considered, namely in the two points P,,,, of meetings of opposite sides, 
and in those four points P,, 2, Which are situated on the diagonals of the 
quadrilateral ; they give therefore xo new points. Also, each side of the 
same quadrilateral i is an edge of one of the four other pyramids, 3 sipRts 
but there remains, for each such pyramid, an opposite edge: and these 
are the four lines, out of the plane, which intersect it in the four points 
P.,, additional to the ecght points P,,,, which are ranged, two by two, 
_ upon the sides. There are thus twelve points of the group P,,,, im any 
one plane [,,,; and we have now exhausted the intersections of that 
plane with lines A,,,; and also, as it will be found, with the lines A,,,, 
and A,. 
