PHILLIPS. — THE INDETERMINATE PRODUCT. 309 



the sign being minus when 



bi . . . hjbk . . . bi 



is an odd permutation of hi . . . b^- 

 Multiplication by A gives 



(AB), = S ± (Ab, . . . b,) (b,. . . b,). 



This is the fundamental identity of Grassmann,^ many special cases of 

 which are used in geometry. 



Let A and B be two spaces intersecting in the space 



C = (Cx . . . 6-,). 



We can determine n-k points cii and m-k points b^ such that 



A = (cii . . . ttn-kCl . . . Cj), B — {bi . . . 6«_i. Cx . . . Cfc). 



We expand the product (AB) by the formula given above. If 

 p > m -{- n — k, there are two c's equal in each of the prefactors and 

 the result is zero. Up = m -\- n — k, there is just one term that does 

 not vanish, giving 



{AB)p = («!... a„_itCi . . . Ckbx . . . bm-k) (q • • • c^) 

 = DC, 



where D is the space containing both A and B and C the space com- 

 mon to them. Up < m + n — k, there are a number of terms in the 

 expansion. The determinants which are prefactors are all different. 

 The same is true of postfactors. Since the points in it are linearly 

 independent, the expression can not factor or vanish. Hence if 

 A ^ 0, B ^ 0, [AB]p = is the necessary and sufficient condition 

 that A and B lie in a space of order less than p. If the containing 

 space is of order p, \^AB^p is the product of that space and the 

 common space. 



Progressive products are used when the number of factors is equal 

 to or less than /?, regressive when that number is greater than p. 

 Hence it causes no confusion to use the same notation \^A B'\ for both. 



Expressions occurring in the discussions of linear geometry are 

 always homogeneous. Upon multiplying two such expressions regres- 

 sively, each term of the result is either zero or of the form DC where 

 D is the space in which we are working. Since it is a factor of all 



2 Ausdehnungslehre (1862), p. 83. 



VOL. XLVI. — 24 



