628 DR MUIR ON A DEVELOPMENT OF A DETERMINANT OF THE MN™ ORDER. 

 In the case of the simplest alternant of the 6th order we have — 



d»c l P | | rf-V/ 6 



= 2, l«W*l -UW/ 2 ! • I 1 aW 



| 1 dV/ 3 



= 2 | a°8V • | dV/« | ■ | (a&c) (defy \, 



where it should be noted that each term of the development consists of three factors, 

 each of which is a simple alternant, and each of which is, therefore, expressible as a 

 product of binomial factors. Thus the specimen term here given after 2 is equal to 



(c-b) (c-a) (b-a) (f-r) (f-d) (e-d) (d 3 « 3 / 3 -aW). 



Similarly we have 



| aWcW/fyW \ = Y, \ a°& x c 2 1 | d°e l f 2 I | g hH z \ ■ I (abef (deff (ghif I , 



where each term under the sign of summation is expressible as a product of twelve 

 binomial factors, the first, for example, being equal to 



( c -b) (c-a) (b-a) (f-e) (f-d) (e-d) (i-h) (i~g) (h*-g) 



x (fhW-dWf*) (ffhW-aWc 3 ) <W/ 3 -aW). 



