380 HITCHCOCK. 



which brings out the "skew" character of the star proi* 110 ^ J t is 

 formed by the multiplication of unlike elements. If the s'P a ' ar e ' e " 

 ments of <pi and <p 2 are a rs and b rs in the style of (63), we shah have 



P1V2 = («nii + «i2ij + «2iji + 02 2 jj)*(&iiii + 6uij + &21J1 + b 2 ^ 



The result is of course the same as if the dyadics had been expand"" in 

 any other manner and we had applied one of the three method s °* 

 Art. 10. 



By contrast the double dot product <p\ : ^2 yields a multiplication 

 table having +1 along the main diagonal and elsewhere zero, giving 



<pi : <p2 = an&ii + «i2^i2 + 021^21 + 022&22 (79) 



The concept of <p*ip2 as a product is justified by the fact that it is 

 in every case possible to set up systems of double polyadics <p u 

 <p2, ' • , <p» such that <p T *<p s = 1 if subscripts are equal, otherwise zero. 

 Here v = ?i 2 = N 2K for this is the number of linearly independent 

 double polyadics in any case. In the example (77) if we make 



.i.ffl+JJU.O^ffi, ,._«+£,,. -«^» 90)' 



V+2 V — 2 V-2 v+2 t 



we shall have F r *F s = 1 if subscripts are equal, otherwise zero. 



It is clear that, just as the dyads ii, ij, ji, jj could be formally 

 treated with reference to double dot product as would be vectors in 

 4-space with reference to ordinary dot product, so the dyadics F can 

 be similarly treated with respect to star product. 



A correspondence may be thus set up between invariant properties 

 of the dyadics or polyadics or double polyadics and the geometry of 

 the space in which these magnitudes behave formally like vectors. 



A special case of much interest has been treated in great detail, with 

 some differences of terminology, by E. Waelsch. 11 By his very in- 

 genious system the whole of the well-worked theory of invariants of 

 binary forms is employed as algorism for phenomena of various sorts 

 in ordinary space. In the language of the present discussion his 

 system would arise by making up three fundamental units 



E 1= ii, Ea= 0H-P) , E 3 = jj (81) 



V2 



so that E b E 2) E 3 form a normal orthogonal system with respect to 

 double dot product. A binary quadric may be written 



