598 
[61.] Considering next the new lines which connect a point of the 
first construction, with one.or more points of the second, we find these 
five new types, 
[811]; [122]; [123]; [133]; and [134]; 
which as symbols denote the five lines, 
(011) (121) (112); (011) (201) (210); (111) (210) (121); 
(011) (312); (111) (sa); } 
or a’ BY A," ; a’ BY cS Dic G73 A’ 45 and D, co: 
but as ¢ypes represent each a group of six lines. We thus get 18 new 
lines, each passing through 1 point P,, and 2 points P,; and 12 other 
lines, each connecting a point P, with only one point Pp, And these 
thirty lines A, account for 54 + 12 = 66 binary combinations of points ; 
leaving however 621 such combinations to be accounted for, by new 
lines Az, of which each must connect at least two points P,, without 
passing through any point P, or P,, and without being any one of the 
traces already considered. 
[62.] The symbol [283], which denotes a line passing through two 
points P,, namely, (011) and (811), ora” and 4,”, but through xo other 
point, represents, when considered as a type, a group of three such lines; 
and 40 other types, as for example [134], which as a symbol denotes the 
line (111) (182), or a) 8", are found to exist, representing each a group 
of siz lines, whereof each connects in like manner ¢wo points P,, but 
only those two points. We have thus asystem of 243 new lines, which 
represent only so many binary combinations: and there remain 378 such 
combinations to be accounted for, by new lines A;, whereof each must 
connect at least three points Px. 
[63.] For lines connecting three such points, and no more, itis found 
that there are twenty types; whereof evght, as for instance the type 
[311], which as a symbol denotes the line (011) (121) (112), or a” B/” 
o’”, represent each a group of ¢hree such lines; while each of the twelve 
others, like [123], which as a symbol denotes the line (111) (121) 
(210), or 4, B/’” o”, represents a group of siz lines. We have thus 96 
new lines, counting as 288 binary combinations: but we must still ac- 
count for 90 other combinations, by new lines A;, connecting each more 
than three points Pp. 
[64.] Accordingly, we find three new types of lines, which alone 
remain, when all those which have been above exhibited, or alluded* 
to, are set aside: namely 
* It has been thought that it could not be interesting to set down all the types of 
lines, above referred to; especially as those which relate to lines o¢ passing through at 
least four points give rise, at the present stage of the construction, to no theorems of har- 
monic (or anharmonic) ratio. 
