376 Transactions of the South African Philosoiohical Society. 



the columns other than the first and second having the non-zero 

 elements ^o, a^, a^, ..., and the last s — r — l of them being '' lowered " 

 from the position of persymmetry. 



With the help of this result | N,.Ds \ may be expressed as a series 

 of terms arranged according to powers of cto, the lowest power being 

 the rth and the highest the (r + s- 2)th. The term containing the 

 former is readily seen to be 



a\~ 



a, . 

 ao a. 



e,. a. 



a.._ 



.aL 



(xviii) 



and it can be shown that the term containing the latter is 



(-r 





.al- 



(xix) 



The determinant in (XVIII.) manifestly remains the same for all 

 values of s : and when s is taken equal to r + 1 we are brought back 

 to (XII.). 



6. It may not be amiss to recall the fact that determinants in 

 which every element is zero whose column-number exceeds its row- 

 number by more than 1 — the class to which belong the N's, the D's, 

 the M's, and others occurring above — made their appearance com- 

 paratively early in the history of the subject. They are to be found 

 first in Wronski's books, namely, in the Befutation cle la theorie cles 

 fonctions analytiques de Lagrange, published in 1812, and in the 

 second section of the Philosophic de la technie algorithmique, pub- 

 lished in 1817. They next turn up in Scherk's Mathematische 

 Ahhandlungen, which bears the date 1825. Both writers are led to 

 them by having to solve a set of linear equations each of which has 

 one more unknown than the equation preceding it : and the latter 

 writer actually uses his result to find the 4th Bernoulli number." 



Since the determinant of the 7^th order partitions itself into two 

 similar determinants of the {n — l)th. order, for example, 



a. 



a^ 







= a, h^ 



^ 





- a^ 



h 



b. 



b. 



b. 



K 





c^ 



c. 



^A 





Ct 



c. 



Ci 



c^ 



^s 



^4 



\d. 



d. 



d, 





^T 



d. 



d. 



ck 



d. 



^^4 















* For details see chap. xvi. of The History of Determinants in the Historical Order 

 of Development, London, 1906. 



