NOTES AND REFERENCES. 
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thereof as regards the line-singularities of the inflexion and the double tangent. We 
are thus led to consider as ordinary singularities in the theory the above-mentioned 
four singularities of the inflexion, the double tangent, the node and the cusp: and we 
know further that any other singularity whatever of a plane curve is compounded in 
a definite manner of a certain number of some or all of these singularities. 
But in the theory of surfaces, starting in like manner with the general equation 
(x, y, z, w) 11 = 0, such a surface has torse-singularities, the node-couple torse, and the 
spinode-torse; each of these is in general an indecomposable torse of a certain kind 
(but there is the new cause of complication that it may break into two or more 
separate torses), but we do not know the analytical expression of these singularities, 
nor consequently the analytical expression of the curve-singularities which correspond 
to them, the nodal curve and the cuspidal curve. Thus if we attempt to start with 
a surface (x, y, z, w) n = 0 having a nodal curve, we can indeed write down the equation 
in its most general form, viz. if the nodal curve has for its complete expression the k 
equations P = 0, Q = 0, R = 0, &c. (viz. if the curve is such that every surface whatever 
through the curve is of the form ft, = AP + BQ + GR + ..., =0) then the most general 
equation of the surface having this curve for a nodal curve is (A, B, G, ...$P, Q, R, ...) 2 = 0, 
but this form is far too complicated to be worked with ; and if for simplicity we take 
the nodal curve to be a complete intersection P = 0, Q = 0, and consequently the 
equation of the surface to be (A, B, CQP, Q) 2 =0, then it is by no means clear that 
we do not in this way introduce limitations extraneous to the general theory. The 
same difficulty applies of course, and with yet greater force, to the cuspidal curve; 
and even if we could deal separately with the cases of a surface having a given 
nodal curve, and a given cuspidal curve, this would in no wise solve the problem for 
the more general case of a surface having a given nodal curve and a given cuspidal 
curve. It is to be added that the general surface of the order n has no plane- or 
point-singularities, and thus that such singularities (which correspond most nearly to 
the singularities considered in the theory of reciprocal curves) present themselves in 
the theory of reciprocal surfaces as extraordinary singularities. 
END OF VOL. VI. 
C. VI. 
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