PROFESSOR CAYLEY ON CUBIC SURFACES. 
24.3 
and planes, although from the want of the equation of the reciprocal surface these ana- 
lytical expressions have no present application. And so in some of the next following 
sections, no application is made of the analytical expressions of the lines and planes. 
34. I call to mind that if a line be given as the intersection of the two planes 
AX+BY+CZ+DW=0, A'X+B'Y+C'Z+D'W=0, 
then the six coordinates of the line are 
a, b, c , /, g, h 
=AD'-A'D, BD'-B'D, CD'-C'D, BC-B'C, CA'-C'A, AB'-A'B, 
and that in terms of its six coordinates the line is given as the common intersection of 
the four planes 
( . h, -g, a JX, Y,Z, W) = 0, 
—h, . /, b 
9> -/> • p 
— < 2 , — b, —c, 
and that (reciprocating as usual, in regard to X 2 -j- Y 2 + Z 2 -f- W 2 = 0) the coordinates .of 
the reciprocal line are (f, g, h , a, b , c) ; that is, this is the common intersection of the 
four planes 
( • 
c, 
-b, 
/ 
— c 
a, 
9 
b, 
— a -> 
h 
-9> 
-h, 
w)=0. 
It is in some cases more convenient to consider a line as determined as the intersection 
of two planes rather than by means of its six coordinates ; thus, for instance, to speak of 
the line X=0, Y=0 rather than of the line (0, 0, 0, 1, 0, 0) ; and in some of the sec- 
tions I have preferred not to give the expressions of the six coordinates of the several 
lines. 
§ 1=12, Equation (X, Y, Z, W) 3 =0. Article Nos. 35 to 46. 
35. There is in the system of the 27 lines and the 45 planes a complicated and many- 
sided symmetry which precludes the existence of any unique notation : the notation can 
only be obtained by starting from some arrangement which is not unique, but one of a 
system of several like arrangements. The notation employed in my original paper 
“ On the Simple Tangent Planes of Surfaces of the Third Order,” Camb. and Dub. Math. 
Journ. vol. iv. 1849, pp. 118-132, and which is shown in the right hand and lower margins 
of the diagram, starts from such an arrangement ; but it is so complicated that it can 
hardly be considered as at all putting in evidence the relations of the lines and planes ; 
that of Dr. Hart (Salmon “ On the Triple Tangent Planes of a Surface of the Third Order,” 
same volume, pp. 252-260), depending on an arrangement of the 27 lines according to 
a cube of 3 each way, is a singularly elegant one, and will be presently reproduced. 
