412] 
A MEMOIR ON CUBIC SURFACES. 
371 
want of the equation of the reciprocal surface these analytical 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+ G'Z + D'W = 0, 
then the six coordinates of the line are 
a, b, c, f, g, h 
= AD'-A'D, BE-ED, CD'-CD, BG'-B'G, GA'-CA, AE - 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 £X, F, if, fF) = 0, 
-h, . /, b 
9> ~f> • c 
— a, —b, — c, . 
and that (reciprocating as usual in regard to A 2 + F 2 + Z 2 + W' 1 = 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, f fa, y, z, w) =0. 
-c, . a, g 
b, — a, . h 
-f, -g, -K • 
It is in some cases more convenient to consider a line as determined as the inter 
section of two planes rather than by means of its six coordinates; thus, for instance, 
to speak of the line X = 0, F= 0 rather than of the line (0, 0, 0, 1, 0, 0); and in 
some of the sections I have preferred not to give the expressions of the six coordinates 
of the several lines. 
Article Nos. 35 to 46. § 1 = 12, Equation (X, Y, Z, W) 3 = 0. 
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,” 
Gamb. and Dub. Math. Journ. vol. iv. 1849, pp. 118—132, [76], and which is shown in 
the right hand and lower margins of the diagram, starts from such an arrangement; but 
47—2