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A MEMOIR ON CUBIC SURFACES. 
365 
18. There are not, except as above, any common tangent planes of the two torses, 
that is, not only f = 0 as already mentioned, but also % =0. I do not at present 
account a priori for the values 6' = 16, 8, and 16, which present themselves in the 
Table. The cubic surface cannot have a plane of conic contact, and we have thus in 
every case C' — 0 ; but the value of B' is not in every case = 0. 
19. In what precedes we see how a discussion of the equation of the cubic 
surface should in the several cases respectively lead to the values b', t', p', <r’, /3', /, f, B', 
and how in the reciprocal surface the nodal curve of the order b' is known by means 
of the facultative lines of the original cubic surface. The cuspidal curve c might 
also be obtained as the reciprocal of the spinode-torse; but this would in general be 
a laborious process, and it is the less necessary, inasmuch as the equation of the 
reciprocal surface is in each case obtained in a form putting in evidence the 
cuspidal curve. 
Article Nos. 20 to 23. The Lines and Planes of a Cubic Surface; Facultative Lines; 
Explanation of Diagrams. 
20. In the general surface 1=12, we have 27 lines and 45 triple-tangent planes, 
or say simply, planes: through each line pass 5 planes, in each plane lie 3 lines. For 
the surfaces II to XXI (the present considerations do not of course apply to the 
Scrolls) several of the lines come to coincide with each other, and several of the 
planes also come to coincide with each other; but the number of the lines is always 
reckoned as 27, and that of the planes as 45. If we attend to the distinct lines 
and the distinct planes, each line has a multiplicity, and the sum of these is = 27; 
and so each plane has a multiplicity, and the sum of these is = 45. Again, attending 
to a particular line in a particular plane, the line has a frequency 1, 2, or 3, that is, 
it represents 1, 2, or 3 of the 3 lines in the plane (this is in fact the distinction 
of a scrolar, torsal, or oscular line); and similarly, the plane has a frequency 1, 2, 3, 4, 
or 5, according to the number which it represents of the 5 planes through the line. 
It requires only a little consideration to perceive that the multiplicity of the plane 
into its frequency in regard to the line is equal to the multiplicity of the line into 
its frequency in regard to the plane. Observe, further, that if M be the multiplicity 
of the plane, then, considering it in regard to the lines contained therein, we get the 
products (M, M, M), (2M, M), or 3M, according as the three lines are or are not 
distinct, but that the sum of the products is always = 3M, and that in regard to all 
the planes the total sum is 3 x 45, = 135. And so if M' be the multiplicity of the 
line, then, considering it in regard to the planes which pass through it, we get the 
products (M\ M', M', M', M'), (2M', M\ M', M'), ...(oM'), as the case may be, but that 
the sum of the products is = oM', and that in regard to all the lines the sum is 
5 x 27, = 135, as before. 
21. The mode of coincidence of the lines and planes, and the several distinct 
lines and planes which are situate in or pass through the several distinct planes and 
lines respectively, are shown in the annexed diagrams I to XXI ( x ): the multiplicity 
1 See the commencements of the several sections.