456 ON THE TRIPLE TANGENT PLANES OE SURFACES OF THE THIRD ORDER. [76 
which lie upon two of the three lines (the order being determinate), the systems of 
four lines which intersect in the eight points of the third line are 
(aa, 
6b, 
cc , 
dd), 
(a'a', 
6'b', 
c'c, 
d!d') ; 
(ah', 
6a', 
c'd, 
d! c), 
(a'b, 
6'a, 
cd', 
dc' ) ; 
(ac', 
ca', 
d'h, 
6'd), 
(a'c, 
c'a , 
6d', 
rfb'); 
(ad', 
d&, 
6'c, 
c'b), 
(a'd, 
d' a, 
6c', 
cb'): 
the principle of symmetry made use of in this notation (which however represents the 
actual symmetry of the system very imperfectly) being obviously entirely different from 
that of the case of an arbitrary intersecting plane. The transition case where the 
intersecting plane passes through one of the lines upon the surface (and is thus a double 
tangent plane) would be worth examining. It should be remarked that the preceding 
theory is very materially modified when the surface of the third order has one or 
more conical points; and in the case of a double line (for which the surface becomes 
a ruled surface) the theory entirely ceases to be applicable. I may mention in con 
clusion that the whole subject of this memoir was developed in a correspondence with 
Mr Salmon, and in particular, that I am indebted to him for the determination of 
the number of lines upon the surface and for the investigations connected with the 
representation of the twenty-seven lines by means of the letters a, c, e, 6, d, f as 
developed above.