400 



On Determinant Notation. 



in 1851 contains as its simplest special case the theorem of 

 Franke, which was not discovered until 1862. 



As another example in illustration of the block notation 

 take the determinant identities associated with the name of 

 Schweins. These are found, after (Jayley*, by expanding 

 the determinant 



( a pq) (W 

 (Crq) 



where p = l, . . .m ; g = 1, . . . n ; r= 1, . . . n—k ; 5 = 1,... m-k, 

 in two different ways and equating the results. This process 

 gives, by Laplace's theorem, 



2i(-i; 



i a e g ) 

 {c rq ) 



I h fft | = 2<p(— 1)* | {a p9 ){bps) | . | Crqi 



where 6, <£> are &-ads from 1, . . . m ; 1, . . . n respectively ; 

 6 f , (f>' are the sets comphmentary to 0, <j) respectively ; and 

 A, /u, are the sums of the numbers in the sets 0, </> respectively. 

 The general extensional of this identity is 



y.j— 



i) ; 



M 



(deu) 



• 



M 



(de'u) 



\prq) 



(d ru ) 





M 



{ftu) 



w 



(A) 









= 2<p(-lV 



(a p(p ) (b ps ) (d pu ) 

 M (ft*) (ftu) 



(e r<p >) (flru) 

 M {ftu) 



where t, u=l, . . . z. 



Muir f tells us that " the notation which in our time would 

 almost certainly be chosen for the statement of such identities " 

 as those of Schweins, " is the umbral notation of Sylvester ." 



The object of this short note is to show that the umbral 

 notation is not always the most convenient, and that in some 

 cases at all events the block notation conduces to brevity and 

 lucidity. 



Melbourne, July 31, 1897. 



* C. M. P. 9 and 676. 



t Phil. Mag. Nov. 1884, p. 422. 



