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NOTES AND REFERENCES. 
nodal lines of the cone, and besides through 0 lines of the cone: the complete inter 
section of the cone and monoid surface is thus made up of the curve once, the 
h nodal lines each twice, and the 0 lines each once; and if * as before the order of 
the curve is =d, then we thus have (k + 1) d = d + 1h 4- 6, viz. we must have 
kd = 2h 4- 0, as the condition to be satisfied in order that the curve of the order d 
may be the partial intersection of the cone and monoid surface. 
In my papers in the Comptes Rendus I endeavoured to find, and Halphen and 
Nother both endeavour to find, the surfaces of lowest order which have the curve of 
order d for their complete or partial intersection. This (although, as will presently 
appear, the theory may be considered in a more complete form) is an important and 
interesting question; but upon further reflection it appears to me that it is a 
question beside that which first presents itself and ought to be in the first instance 
considered, viz. this is the question of the classification of curves in space according 
to the foregoing representation of any such curve as the partial intersection of a cone 
and monoid surface. Supposing it effected, and a kind of curve completely defined 
according to this mode of representation, then there arises the further question to which I 
have referred (Salmon’s Solid Geometry, Ed. 3, (1874), p. 285, and Ed. 4, (1882), p. 281), 
viz. we may have passing through any given curve a complete system of surfaces, 
that is a system 17=0, V = 0, W = 0,... where these functions are not connected by 
any such equation as TJ=NV+ PW +..., and where every other surface which passes 
through the curve is expressible in the form MU+ NV+ PW + ... = 0. It is not easy 
to prove (but as to this see Hilbert “ Zur Theorie der algebraischen Gebilde,” 
Gottingen Nachrichten, 1888, p. 454), but it may be safely assumed that for a curve of 
any given order whatever, the number of equations in such a complete system is 
finite, and we have thus the representation of a curve in space by means of a 
complete system of surfaces passing through it. Obviously the curve is here the 
partial (or if the system consists of only two surfaces then the complete) intersection 
say of the two surfaces U= 0, V = 0 of lowest order passing through it, which is 
the question above referred to. 
Reverting to the representation by the cone and monoid surface, Halphen gives 
the capital theorem, that if we have any particular inferior cone P = 0 passing through 
the curve, then we may without loss of generality take the equation of the monoid 
surface to be w = ~: viz. if instead hereof the equation of the monoid surface is 
taken to be w = ~, then this equation in virtue of the equation U = 0 of the cone 
is always reducible to the first mentioned form w = ; that is in virtue of the 
equation 17=0, we have u> = ~ = ~, or what is the same thing, j>' = ~p in virtue of 
17 = 0, that is Q'P — QP' = MU, where M is a rational and integral function (x, y, z) K 
of the degree X, = k + n +1 — d, if k be the degree of P' and n that of P. 
It thus appears that if n be the order of the cone of lowest order which passes 
through the h nodal lines of the cone U =0, then we have always functions Q, P