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A MEMOIR ON CUBIC SURFACES. 
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In B 4 , the tangent plane is distinct from each of the biplanes: 
In B 5 , the tangent plane coincides with one of the biplanes; we have thus an 
ordinary biplane, and a torsal biplane: 
In B e , the tangent plane coinciding with one of the biplanes becomes oscular; we 
have thus an ordinary biplane, and an oscular biplane. 
U (= U s , U 7 or U s ) is a uniplanar-node, where the quadric cone becomes a coincident 
plane-pair; say, the plane is the uniplane. It is to be observed that there is not in 
this case any edge. The uniplane meets the cubic surface in three lines, or say “ rays,” 
passing through the uniplanar-node, viz. 
In TJg, the rays are three distinct lines : 
In U 7 , two of them coincide : 
In U 8 , they all three coincide. 
4. To connect these singular points with the theory of the preceding Memoir, it 
is to be observed that they are respectively equivalent to a certain number of the 
cnicnodes (7 (= C 2 ) and binodes B (= B 3 ), viz. we have 
C. 2 = G, 
B 3 = B, 
B 4 = 2(7, 
B, = C + B, 
(Bg = sc, 
\u 6 = SG, 
U 7 = 2(7 + B, 
U 8 = C + 2 B. 
5. I take the opportunity of remarking that although the expressions cnicnode and 
binode properly refer to the simple singularities G and B, yet as C. 2 = G, C 2 is properly 
spoken of as a cnicnode, and we may (using the term binode as an abbreviation for 
biplanar-node) speak of any of the singularities B 3 , B 4 , B 5 , B 6 as a binode. Thus the 
surface X = 12 — B 4 — C 2 has a binode B 4 and a cnicnode C 2 ; although theoretically the 
binode B 4 is equivalent to two cnicnodes, and the surface belongs to those with three 
cnicnodes, or for which G = S. I use also the expression unode for shortness, instead 
of uniplanar-node, to denote any of the singularities U 6 , U 7 , U a . 
6. The foregoing equations (substantially the same as Schlafli’s) are Canonical 
forms; the reduction of the equation of any case of surface to the above form is not 
always obvious. It would appear that each equation is from its simplicity in the form 
best adapted to the separate discussion of the surface to which it belongs; there is the 
disadvantage that the equations do not always (when from the geometrical connexion of 
the surfaces they ought to do so) lead the one to the other; for instance, V = 12 — B 4 
includes VII = 12 — B 5 , but we cannot from the equation WXZ + (X + Z)(Y 2 — aX 2 — bZ' 2 ) = 0 
of the former pass to the equation WXZ+ Y 2 Z + YX-— X s = 0 of the latter. This would 
be a serious imperfection if the object were to form a theory of the quaternary 
function (X, Y, Z, IT) 3 ; but the equations are in the present Memoir used only as 
means to an end, the establishment of the geometrical theory of the surfaces to which 
they respectively belong, and the imperfection is not material. 
C. VI. 
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