192 Transactions of the South African Philosophical Society. 



product of the remaining factors. Doing this, and writing A for 

 a-\-b — r, we arrive at the identity 



A + -•- 



Ill 



m + l_ 



7 



III 



1 - 111 111 - 1 



111 



a ,1+ra 2 ra-2 

 377Z- 3 



771 



4 ra+1 A .2 — m hi- 2 

 3 7;t dm m 



in + 2 . . 2 + ?» 4 in - 3 



-r A- + -— r 5 . - — r 



771 Dill D 111 



6 771 + 2 . 3 - 777 



r A-)-—- r 



O ??i 0772, 



A - 2r + - rV f A+ 2r - -r") = J A 



in ) \ ill ) 



A-2r+ 4 rY( / A + 2-- V) 

 ill J \ ill j 





- W ^.3rYl--lA" 



777, 



2m -2 



) ( 



1.2, 



m 



(IX.) 



Here the general expressions for the elements are 



A20-1 



72^-1 



A , — 777, 



_h (2^1)m r ' 



771 - 



T, 



111 



m+0 



111 



■1 > 



A 2( > = 



e+ m x 



^(20 + 1)™ ' 

 20 m - - 1 



20 + 1 ' " m 



2(0 + 1) 777, + 0_. 



20 + 1 ~~m 



but as the set of column-multipliers necessary for effecting the 

 resolution is easily ascertained to be 



1? — 1) A> ~ A 3, — 3, 

 1, 1, 3, 3, 6, 



that is to say, the set (S) deprived of its first line, a more general 

 theorem than (IX.) is obtainable by means of the procedure of 

 §§ 3, 4. 



When r is put equal to 111 in (IX.) we obtain a result which on 

 one side closely resembles Sylvester's. 



