UNIVERSITY OF VIRGINIA PUBLICATIONS 



BULLETIN OF THE PHILOSOPHICAL SOCIETY 



SCIENTIFIC SECTION 



Vol. I, No. 9, pp. 231-242 March, 1912 



ON A CERTAIN QUADRATIC FORM WITH ITS GEOMETRIC 

 AND KINEMATIC INTERPRETATIONS.* 



WILLIAM H. ECHOLS. 



1. The writer has for years naade use of a linear transformation of the 

 quadric in a certain form for the purpose of exhibiting certain geometric 

 and kinematic relations, many of the particular results of which are well 

 knoAATi but as the writer has not seen them presented as all being but par- 

 ticular cases of one general permanent form, it appears worth while, on 

 account of its extreme simplicity, to call attention to it in a brief note. 



2. The theorem is as follows. Let x, y, . . . , z he n — 1 numbers 

 determined by the relations 



Mx = :SmiXi, My = Smj^,-, . . . , Mz = XniiZi, ... (1) 



wherein m,-, (i = 1, 2, . . . , p), have the relation 2mi = M. Also let 

 nii = Mki, whence Sfci = 1. Then it can be easily shown that any homo- 

 geneous quadratic function oi x,y, . . . , z such as Q(x, y, . . .) can be writ- 

 ten identically equal to 



^kiQ{Xi,yi, . . . ) - ^kiki Q(AXi,; Ay a, . . . ), (2) 



or as we shall write more briefly 



Q = -Lki Qi - Xki kj Qij, (2) 



where Axij = Xi — Xj, Ayij= yi — yj, etc., and i, j = 1, 2, . . . , p. The 

 result (2) is obtained by making the substitution (1) in Q and in the result 

 where ki^ occurs write ki times 1 minus the sum of the other k's. It is also 



*Read before the Scientific Section, February IS, 1912. 

 231 



