ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 375 



ants, also due to Cayley, but these two equivalent formulations differ 

 more in appearance than in reality from such expressions as Hamilton's 



— Di-2 = Si<pj<pk + Sj<pk<pi + Sk\piipj 



which in Gibbs' notation would be 



vio — 2(i<p*j«£*k) 



where i, j, k denote rectangular unit vectors. 



It will be of some value for the sake of later applications if we take 

 account of these scalars as they appear when our polyadics are de- 

 termined by Zv-dimensional matrices. Consider for a moment the 

 case ((AM, BN))g where A, M, B, and N are dyadics in two dimen- 

 sions, — evidently the simplest case which can be regarded as a gener- 

 alization of the usual vector analysis. With unit vectors i and j 

 we may either write 



A = a u ii + «i2ij + «2iji + a 22 jj 

 or may regard A as an abbreviation for the matrix 



an, «ia 



(hit G&22 



(62) 



(63) 



with similar notation for the other three dyadics. If we form the 

 required scalar directly from the definition, namely as A: MB: N — . 

 A: NB: M we obtain 



((AM, BM)) S = (an7?in + ai 2 mi 2 + « 2 im 2 i + ao 2 m^ 2 ) (b n n n + bv,n n 



+ &21W21 + 6 2 2^22) 



— (aii'/*n + «i 2 ft.i 2 + a 2 i?2>>i + a 22 w 22 ) (6117^11612^12 + 



6 2 iw 2 i + 6 22 >m 22 ) (64) 



If instead we use the method of cubic determinants we shall first 

 form the square array of the fourth order belonging to the double 

 dyadic AM, which is 



aii.mii, a u wi 2 , a n ?» 2 i, anw 22 



Ol 2 /»H, «i 2 ?M 12 , 012W21, Ol2?»22 



«21»'ll, «2]?»12, 02177121, 02i??7 22 



a22mn, a 22 ?»)2, 022m21, a 2 2^»22 



(65) 



with a similar array for BN. We then take the sum of all cubic 

 determinants along the main diagonal, using the two arrays as simi- 

 larly placed layers. The first of these will be 



