ON THE CURVES WHICH SATISFY GIVEN CONDITIONS.
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406]
8. Considering the general case where the condition, and therefore also the locus,
is ¿-fold, it is to be observed that every solution whatever, and therefore each special
solution (if any), corresponds to some point on the ¿-fold locus; we may therefore have
on the ¿-fold locus what may be termed “ special loci,” viz. a special locus is a locus
such that to each point thereof corresponds a special solution. A special locus may of
course be a point-system, viz. there are in this case a determinate number of special
solutions corresponding to the several points of this point-system. We may consider
the other extreme case of a special ¿-fold locus, viz. the /c-fold locus of the parametric
point may break up into two distinct loci, the special ¿-fold locus, and another ¿-fold
locus the several points whereof give the ordinary solutions: we can in this case get
rid of the special solutions by attending exclusively to the last-mentioned ¿-fold locus
and regarding it as the proper locus of the parametric point. But if the special locus
be a more than ¿-fold locus, that is, if it be not a part of the ¿-fold locus itself, but
(as supposed in the first instance) a locus on this locus, then the special solutions cannot
be thus got rid of: we have the ¿-fold locus of the parametric point, a locus such
that to every point thereof there corresponds a proper solution, save and except that
to the points lying on the special locus there correspond special or improper solutions.
It is to be noticed that the special locus may be, but that is not in every case, a
singular locus on the ¿-fold locus.
9. Suppose that the conditions to be satisfied by the curve are a ¿-fold condition,
an ¿-fold condition, &c. of a total manifoldness = co. If the conditions are completely
independent (that is, if the corresponding relations, ante, No. 5, are completely indepen
dent), we have a ¿-fold locus, an ¿-fold locus, &c., having no common locus other than
the point-system of intersection, and the number of curves which satisfy the given
conditions, or (as this has been before expressed) the number of solutions, is equal to
the number of points of the point-system, or to the order of the point-system, viz. it
is equal to the product of the orders of the loci which correspond to the several con
ditions respectively; among these we may however have special solutions, corresponding
to points situate on the special loci upon any of the given loci; but when this is
the case the number of these special solutions can be separately calculated, and the
number of proper solutions is equal to the number obtained as above, less the number
of the special solutions.
10. If, however, the given conditions are not completely independent (that is, if
the corresponding relations are not completely independent), then the ¿-fold locus,
the ¿-fold locus, &c. intersect in a common (<w —j) fold locus, and besides in a residual
point-system. The several points of the (co —j) fold locus give special solutions—in fact
the very notion of the conditions being properly satisfied by a curve implies that the
curve shall satisfy a true (k + l + &c.) fold, that is, a true co-fold condition ; the proper
solutions are therefore comprised among the solutions given by the residual point-
system, and the number of them is as before equal to the order of the point-system,
or number of the points thereof, less the number of points which give special solutions:
the order of the point-system is, as has been seen, equal to the product of the orders
of the ¿-fold locus, the ¿-fold locus, &c., less a reduction depending on the nature of
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