ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
195 
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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