376 



HITCHCOCK. 



ai2>n n , «i2'"i2 



biinn, bnrin 

 bunn, b 12 ni 2 



(66) 



which develops into 



aiimn&i2?h2 — ai2Wu6uWi2 — anWi2&i2»n + «i2»?i2&n??ii (67) 



There are six of these groups of terms, or 2-4 in all. The scheme of 

 each group is given by 



\ClrsMpq, drs''lrs 



pq1lpq> 



'pg" rs 



r s'rl pqj r s 7l r £ 



(68) 



which develops into 



(IpqfdpqO rs^rs (Irs^^pqOpqflrs d pqW rs® rsWpq ~T~ ClrsMrsVpqtlpq \t)JJ 



and the summation is performed by giving to the number-pairs pq 

 and rs unlike values chosen from the pairs 11, 12, 21, and 22. Com- 

 paring with (64) it is easy to verify that the two methods agree. 



If we use the method of polyadic determinants and adopt the dyads 

 ii, ij, etc. as operands we first form the dyadics AM : ii, AM : ij, etc. 

 treating BN in a similar manner. We thus have two rows of a de- 

 terminant of the fourth order whose elements are dyadics. The other 

 two rows are alike and given by I : ii, / : ij, etc. that is by ii, ij, etc. 

 The result 



Aran, Ara.12, Amii, A»^2 



Bn u , B//12, B»2i, B«22 



ii , ij , ji , jj 



ii ij ji jj 



(70) 



should by (56) be the double of the required scalar. Developing by 

 the definition of Art. 8 the leading term is [A.m n , Bri r2 , ji, jj] which 

 denotes the ordinary determinant of the fourth order 



a n m u , ai 2 ran, «2i»'ii, «22'»n 



&11"»12 > &127&12 , b»l)l 12 , ^22»22 



, , 1 , 



, , , 1 



(71) 



which develops into an?»n^i2»i2 — «i2'»ii^n"i2- The next term may be 

 taken as — [Amu, Bn i2 , jj, jii which doubles the two scalar terms 

 already found. It is easy to see that developing (70) by two-row 

 minors after Laplace's method yields the scalar terms in a fashion 

 similar to the method of cubic determinants. 



