PROFESSOR CAYLEY ON CUBIC SURFACES. 
237 
18. There are not, except as above, any common tangent planes of the two torses, 
that is, not only y'=0 as already mentioned, but also i'—0. I do not at present account 
a priori for the values 0'=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 if, f j3 yj, B', and 
how in the reciprocal surface the nodal curve of the order V 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. 
The Lines and Planes of a Cubic Surface ; Facultative Lines ; Explanation of Diagrams. 
Article Nos. 20 to 23. 
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 mul- 
tiplicity, and the sum of these is=45. Again, attending to a particular line in a par- 
ticular 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 con- 
tained 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 3x 45, =135. And so if M' be the multipli- 
city of the line, then, considering it in regard to the planes which pass through it, we. get 
the products (M', M', M', M 7 , M'), (2M 7 , M', M', M') . . . (5M7), as the case may be, but 
that the sum of the products is =5M', and that in regard to all the lines the sum is 
5x27, =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 respec- 
tively, are shown in the annexed diagrams I to XXI * : the multiplicity of each line 
* See the commencements of the several sections. 
