232 UNIVERSITY OF VIRGINIA PUBLICATIONS 



obvious that in making the transformation there may be substituted for 

 any or all of the numbers x, y, . . . their differentials of any orders 

 obtainedfrom (1) regarding the m's constant and the numbers a;,-, t/,-, ... as 

 variables, regard being had as to homogeneity of order and degree in the 

 resulting equation. The form (2) and that which it takes when the k's 

 have been replaced by their values in to's 



2 "ii Qi = MQ+^^ ^i ^i Qiu (3) 



constitute the general theorem which is the object of this note. 



If p = n there is a unique and reciprocal correspondence between the 

 point set x, y, . . . and the point set hi,k2,. . . , when the discriminant 

 of the m's in equations (1) does not vanish. In this case the numbers 

 Xi, y,, . . . may be taken to be the orthogonal cartesian coordinates with 

 respect to arbitrary axes of the n corners Pi of a simplicissimum in (n — 1)- 

 space, then the k's are the content coordinates with respect to the simplicis- 

 simum of the point P whose orthogonal coordinates are x, y, . . . The 

 m's may be regarded as the mass-charges of the points Pi respectively, 

 in which case P is their center of inertia or mass-center. Otherwise P 

 is a point of mean position which may be said to divide the system of points 

 Pi in the ratio of the numbers rrii : jn2 : ?ns : . . . , the numbers m's are 

 then called the ratio coordinates of P with respect to P,-. 



When p>n the correspondence between x, y, . . . and the k's is not 

 unique. However a given set of m's determine P uniquely. In this case 

 there may always be assigned in addition to the equations (1) a sufficient 

 number of additional independent equations in the m's, in which the other 

 numbers are arbitrarily assigned constants such that the variable m's are 

 uniquely determined in terms of the variables x, y, . . . , z. In this con- 

 dition the point P is confined to an {n — l)-space as a section of a higher 

 space, the relation between the x,y, . . . , m.- or ki coordinates of P now being 

 1:1, and the k's being interpretable as content coordinates. 



When the configuration of points Pi is displaced to a new configuration 

 Pi', and any point P whose coordinates are fc; with respect to P, is dis- 

 placed to a point P' whose coordinates are the same ki unchanged with 

 respect to P/, the displacement is homogeneous. We give now some theo- 

 rems corresponding to different forms of the quadratic Q. 



3. Let Q = rr^ + 2/2 + . . . . 



Represent an arbitrary origin by 0. Then (2) gives 



p2 = Zkipi'- SkikiPii", (4) 



