559 
(ava: (ta). (11K 
And these represent, respectively, groups of six, of sez, and of tliree new 
lines, and therefore on the whole'a system of fifteen new lines, each passing 
through four points P,, and consequently counting as s¢x combinations ; 
for example, as symbols, they denote the three following lines : 
(210) (211) (021) (231), or c’a™ a’ B,"™; 
(210) (211) (021) (281), or c’o™ a” aX; 
(201) (110) (021) (111), or 3B," 0" A" cp. 
But 6.15 = 90; we are therefore entitled to say, that all the 1326 binary 
combinations [59.], of the 52 points Po, P,, Pz [53.] i the plane axzc, 
have now been fully accounted for. 
[65.] Collecting the results, respecting the collineations in the plane 
Axe, it has been found that there are 261 lines Az, whereof each connects 
two, but only two, of the 52 ports in that plane; and that these lines, 
which at the present stage of the construction are not properly cases of 
collinearity at all, are represented by a system of 44 ternary types. 
[66.] There are 126 other lines A;, each connecting three (but only 
three) points ; they are represented by a system of 25 types ; and account 
for 378 binary combinations. 
[67.] There are 15 lines A;, each connecting four points Pp, ; they are 
represented by a system of 3 types, and account for 90 combinations. 
[68.]| There are 12 lines A;, each connecting one point Pp) with four 
points Pp, ; they are represented by 2 types, and represent 120 combina- 
tions. 
[69.] There are 13 other lines As, namely the traces of planes 11, or 
II,, whereof each connects six points, namely a point Py or P, with five 
points P,, or else six points P, with each other; they are represented by 
4 types, and account for 195 combinations. 
[70.] There are 3 lines A,,., each connecting two points P, with 
eight points P.; they have one common type, and represent 135 combi- 
nations. 
[71.] There are, in like manner, 3 lines A,,,, each connecting one 
point Py with two points P,, and with four points P,, but having only one 
common type; and they represent 63 combinations. 
[72.] Finally, there are (in the same plane) 3 lines A,, each con- 
necting two points Py with one point P,, and with five pornts P,; these 
lines also have all one type; and they account for 84 combinations : with 
the arithmetical verification, that 
261 +378+90+ 1204 195+135+ 63 + 84=1326 =26.51 ; 
which proves that the enwmeration 1s complete. 
