56 PROCEEDINGS OF THE AMERICAN ACADEMY. 



When p-\- q < 71 this determinant gives 



[ap+i ...an ^g+i . . ./3J = 2 (7 ^^^j . . .^J S . ' . (4) 



where 7 is any combination of q as and S the remaining ones so 

 arranged that 



Symbolic notation. 



8. The determinants in the matrix representing a space S are 

 the coordinates Si of S. Of o" is a space comphmentary to S, we 

 consider it as represented by a matrix of the same kind as S. It has 

 then a hke number of coordniates o"j (algebraic compHments of Si 

 in the determinant | ^o" | ) . Then 



This is a linear function of the coordinates Si and by a proper choice 

 of cr (perhaps complex) can be made any linear function of those 

 coordinates. To obtain a bilinear function of the coordinates ri, s^ 

 of two spaces R and S we take matrices p and o" complimentary to 

 R and 6'. Then 



{R p) {S CT) = ^ p. a, r, sj,. . . . (5) 



In order to obtain the most general bilinear function 



2 Oik Vi Sk 



we consider the above as a symbolic representation in which pj cr^ 

 is to be replaced by Oj^. Thus {R p) {S cr) represents symbolically 

 any bilinear function of the coordinates r^, Sk- Any linear relation 

 connecting the symbolic cjuantities {R p) {S o") will be satisfied by the 

 bilinear functions 2 fl,/.. r^ s^ . This is the symbolic representa- 

 tion so much used by Clebsch. 



We can consider {R p) (.S o") as resulting from an expression p o" 

 by operating on the first factor with R and on the second with S. 

 This product /a o" is the dyadic of Gibbs.^ It may be considered as a 

 distributive product of p and o". It is called the indeterminate ^ 

 product. In it the order of factors must be preserved. In fact there 

 is no general functional relation between p cr and o" p. The dyadic 



8 Vector Analysis, Gibbs-Wilson, page 265. 



9 Cf. H. B. Phillips, loc. cit. 



