422 



Dr. T. Muir on an overlooked Di 



tsco never 



Thus in the case where n = 4, g = 2, ra = 5 the identity is 

 written 



2± lAi A 2 B' x B' s B' s B' 4 B 



=S±| A x A 2 

 and this is meant to indicate that 



<*1 «2 "3 



: : I [ !i . 

 '5 



B x B 2 B 3 B 4 B. 



a 2 « 3 a 4 M A l 



Ax A 2 B, B 2 1 1 B 3 B 4 B t 



«2 «3 



A x A 2 B 2 B, 1 1 B 2 B 4 B 6 1 

 4- ... (10 terms) 



Aj A 2 



a 3 a 4 5] Aj 4^ 



B 1 B 2 B 3 B 4 B 5 



A 1 A 2 



a 2 a 4 *] b 2 b 3 



B x B 2 B 3 B 4 B 6 



+ . . . (6 terms), 



where the suffixes of the B's in the first factors of the terms 

 on the left-hand side are in order 



12, 13, 14, 15, 23, 24, 25, 34, 35, 45, 



and the sign of any term is determined by the number of in- 

 versions of order among the suffixes of all the B's mentioned 

 in the term ; while the suffixes of the a's in the first factors of 

 the terms on the right-hand side are 



12, 13, 14, 23, 24, 34; 



and the sign of any term is determined by the number of 

 inversions of order among the suffixes of all the a's of the 

 term. 



9. The notation which in our time would almost certainly 

 be chosen for the statement of such identities is the umbral 

 notation of Sylvester. In outward appearance it is not 

 unlike that of Schweins, although in essence it is different. 

 The superfixes of Schweins are, like the suffixes, appendages 

 of the capital letters A and B; whereas with Sylvester, and 

 indeed also with Leibnitz and Vandermonde, the capital letters 

 are dispensed with. Using then 



b 



c . 

 7- 



for | A aa A bfi A 



l<7 • 



• • 



and placing a line over or under the variables instead of 

 marking them with a dash, we may write Schweins' identity 

 thus : — 



2± 



= 2± 



a n -, 



&, . 









• 0L„ 



«!.. 



• dn-q 



A 





:.b, 



0L V . 



• *»- ? 



\*n- 



-q+1- 



• • CL n 



V 



>q¥ 1 



ft 



. . . . p m — q 



. . . b m 



• • • P m - q 



