WHICH SATISFY GIVEN CONDITIONS. 
79 
the d-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 d-fold locus, viz. the d-fold locus of the parametric point may 
break up into two distinct loci, the special d-fold locus, and another d-fold locus the several 
points whereof give the ordinary solutions : w T e can in this case get rid of the special 
solutions by attending exclusively to the last-mentioned d-fold locus and regarding it as 
the proper locus of the parametric point. But if the special locus be a more than d-fold 
locus, that is, if it be not a part of the d-folcl 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 d-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 it 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 d-fold condition, an 
/-fold condition, &c. of a total manifoldness —u. If the conditions are completely hide- 
pendent (that is, if the corresponding relations, ante , No. 5, are completely independent), 
we have a d-fold locus, an /-fold locus, See., 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 conditions 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 d-fold locus, the /-fold 
locus, &c. intersect in a common (u—j ) fold locus, and besides in a residual point-system. 
The several points of the (a — ^’)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 (d-j-/+&c.)fold, that is, a true cy-fold condition; the proper solutions are there- 
fore comprised among the solutions given by the residual point-system, and the number 
of them is as before equal to the order of thre 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 d-fold locus, the 
/-fold locus, &c., less a reduction depending on the nature of the common (a — ^‘).fold 
locus, and the difficulty is in general in the determination of the value of this reduction. 
11. In all that precedes, the number of the parameters has been taken to be a ; but if 
