A MEMOIR ON ABSTRACT GEOMETRY. 
469 
413] 
69. We thus arrive at the notion of the double generation of a ¿-fold locus, viz. 
such locus is the locus of the points, or, say, of the ineunt-points 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 incomplete, 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 systematically 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 for which other reasons 
it may be expedient next to consider, the theory of Transformation, as depending on 
relations involving simultaneously the m+1 coordinates (%, y, ...) and the m! + 1 
coordinates (x' y,...).