PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
03 
meters, the locus of the point (x, y, . . .) is a Mold omal locus : it is to be observed that, 
by what precedes, the linear Mold relation is satisfied by writing therein the values 
x : y, . . .=x : y, . . . , that is, the Mold omal locus passes through the original point 
tx, y, . . .) ; the Mold omal locus is said to be the tangent-omal of the original Mold 
locus at the (point x , y, . . .), which point is said to be the point of contact. 
68. If in the original Mold locus we replace (x, y, . . .) by (x, y, . . .), and combine 
therewith the Mold linear relation, we have between the coordinates (x, y, . . .) a 2Mold 
relation (containing as parameters the coordinates (x, y, ...)); these parameters satisfy 
the original Mold relation, and in virtue hereof the 2Mold relation (whether 2 k is or is 
not greater than m) is satisfied by the values x, y, . . .=x : y : . . . ; and not only so, but 
the point in question is a nodal or double point on the 2Mold locus. It also follows 
that the tangent-omal locus, considering in the Mold linear relation ( x , y, . . .) as para- 
meters satisfying the original Mold relation, has for its envelope the Mold locus. 
69. We thus arrive at the notion of the double generation of a Mold locus, viz. such 
locus is the locus of the points, or, say, of the ineunt-jyoints thereof ; and it is also the 
envelope of the tangent-omals thereof. We have thus a theory of duality; I do not at 
present attempt to develope the theory, but it is necessary to refer to it, in order to 
remark that this theory is essential to the systematic development of a m-dimensional 
geometry; the original classification of loci as onefold, twofold, . . .(m — l)fold is incom- 
plete, and must be supplemented with the loci reciprocally connected with these loci 
respectively. And moreover the theory of the singularities of a locus can only be sys- 
tematically established by means of the same theory of duality; the singularities in 
regard to the ineunt-point must be treated of in connexion with the singularities in 
regard to the tangent-omal. These theories (that is, the classification of loci, and the 
establishment and discussion of the singularities of each kind of locus), vast as their 
extent is, should in the logical order precede that which for other reasons it may be 
expedient next to consider, the theory of Transformation, as depending on relations 
involving simultaneously the m-\- 1 coordinates (x, y, . . .) and the ml -\- 1 coordinates 
(T, y, . . .). 
