452 
ON THE TRIPLE TANGENT PLANES OF 
[76 
where the last eighteen lines have been divided into two systems of nine each. The 
five planes through (ab. cd. ef\ may be considered as cutting the surface in 
ab; (ab.cf.ed\, 
cd; (af.cd.eb\, 
ef \ (ad.cb.ef\, 
(ad.cf.eb\; (af.cb.ed) 3 , 
(ad .cf. eb) 3 ; (af.cb.ed\, 
(which supposes however that the distinguishing suffixes 1, 2, 3, are added to the 
different planes according to a certain rule). And similarly for the lines in the planes 
through the other lines represented by symbols of the like form. The five planes 
through ab intersect the surface in the lines 
cb, 
eb, 
ad, 
af, 
(ab. cd. 
e /)i> 
(ab. cf. ed) } 
(ab. cd. 
e/) 2 , 
(ab. cf. ed)< 
(ab. cd. 
ef) s> 
(ab. cf. ed). 
and similarly for the planes through the other lines represented by symbols of a like 
form. 
Observing that rj, Ç, correspond to b, d, f and w, 6, Ü, to a, c, e respectively, 
ab corresponds to the intersection of w and f, i.e. to a^, &c. ; also (ab.cd.ef) 1 , 
(ab. cd. ef) 2 , (ab.cd.ef) 3 correspond to three lines meeting a 1 , b 2 , and c 3 , i.e. to 
a 5 , a 7 , a 9 , &c. ; and. the system of the twenty-seven lines as last written down corre 
sponds to the system, 
a 1} 
b 1} 
Ci ) 
«2, 
b 2 , 
c 2 , 
«3, 
bs, 
C3, 
a 7 , 
a g , 
b 5 , 
b 7 , 
b 9 , 
c 5 , 
c 7 , 
C9 y 
a 4 , 
Cts y 
b t , 
b 6 , 
b s , 
c 4 , 
c 6 > 
c 8 . 
The investigations last given are almost complete in themselves as the geometrical 
theory of the subject: there is however some difficulty in seeing d priori the nature 
of the correspondence between the planes which determines which are the planes which 
ought to be distinguished with the same one of the symbolic numbers, 1, 2, 3.