PROFESSOR CAYLEY OX ABSTRACT GEOMETRY. 
57 
36. [In illustration of the foregoing Nos. 29 to 35, I remark that, for the functions or 
equations P = 0, Q = 0, R = 0, 6cc., if we have identically AP+BQ+ CP =0, where 
the factors A, B, C, . . . are integral functions of the coordinates, and where some one of 
these factors, say, A, is a constant (or if we please =1), then the system of functions or 
equations is in involution; or, to speak more accurately, the function or equation P = 0 
is in involution with the remaining functions or equations Q=0, 11=0, . . . But when 
the factors A, B, C, . . . are no one of them constant, then we have a convolution. If 
P = 0 is in involution with the remaining equations Q=0, B = 0, . . ., then P=0 is im- 
plied in these equations, and the relations (Q=0, R=0, . . .) and(P=0, Q=0, R=0, ...) 
are equivalent to each other. But in the case of a convolution where 
AP+BQ + CR+ ... =0, 
then the relation the equations Q=0, R=0, . . . imply AP=0, that is, A=0 or else P = 0 ; 
or, what is the same thing, the relation (Q = 0, R = 0, . . .) is a relation composed of the 
two relations (A = 0, Q = 0, R=0, ...)and (P = 0, Q=0, R=0, . . .). In the k-fo\<l 
relation expressed by the more than k equations (P=0, Q = 0, R=0, . . .), selecting 
any k of these equations which are not in convolution, and uniting thereto any one of 
the remaining equations, we have a convolution of k -\- 1 equations; and when a /r-fold 
relation is precisely expressed by means of a system of k or more equations (P = 0, 
Q=0, . . .), then every equation 0 = 0 implied in the given relation, or, what is the same 
thing, the equation of any onefold locus passing through the locus given by the X'-fold 
relation is in involution with the equations P=0, Q=0, . . ., that is, we have identically 
0=AP + BQ+CR+ . . ., A, B, C, . . . being integral functions of the coordinates.] 
Omal Relation ; Order. Article Nos. 37 to 42. 
37. A /c-fold relation may be linear or omal. If k=m , the corresponding locus is a 
point; if k<m the locus is a /r-fold, or (m— ^dimensional omaloid; the expression, 
omaloid used absolutely denotes the onefold or (in — l)dimensional omaloid; the point 
may be considered as a m-fold omaloid. 
38. A m-fold relation which is not linear or omal is of necessity composite, composed 
of a certain number M of m-fold linear or omal relations ; viz. the m-fold locus corre- 
sponding to the m-fold relation is a point-system of M points, each of which may be con- 
sidered as given by a separate m-fold linear or omal relation ; each which relation is a 
tactor of the original m-fold relation. The given m-fold relation, and the point-system 
corresponding thereto, are respectively said to be of the order M. 
39. The order of a point-system of M points is thus =M, but it is of course to be 
borne in mind that the points may be single or multiple points ; and that if the system 
consists of a point taken a times, another point taken ft times, &c., then the number of 
points and therefore the order M of the system is considered to be =a-|-/3-f- . . . 
40. It to a given /c-fold relation (k<m) we unite an absolutely arbitrary (m— &)fold 
linear relation, so as to obtain for the aggregate a m-fold relation, then the order M of this 
m-fold relation (or, what is the same thing, the number M of points in the corresponding; 
MDCCCLXX. I 
