370 Transactions of the South African Philosophical Society. 



Pascal, having been led to take up the problem, published a solution 

 early in the present year," his mode of attack being to consider the 

 ratio of the said determinant to another of simpler formation, 

 namely, the persymmetric determinant, D^ say, of which the first 

 three instances are 



«., !»■ «H 



^ lag <^i > 



a. 



ao 



. 



a^ 



a. 



«0 



a. 



a^ 



ttl 



The object of the present short paper is to make known a corre- 

 sponding investigation undertaken with a view to restate and amplify 

 Pascal's results. 



2. At the outset it is manifest that a necessary and sufficient con- 

 dition for the equality of the n fractions 



a. 



^0 



-^a„ 



b. 

 b.. 

 a. 



a, 

 b._ 





-^ 



a. 



a. 



K 



a.^ 



, 















h. 



^<i 



<^o 





a. 



«o 









«4 



a,. 



a, a, 

 h.. b. 



~r- 



a. 



a. 



tto 











&3 





a. 



a._ 



a, 



J 





is the vanishing of the n~l determinants 



|N,D,I, |N,D3|, ... jNA.|. 



(I) 



In the next place it is readily seen that the co-factor of the second- 

 highest power of - a? in a determinant of the form 



CloT 22 * 



ttgi a^2 «33 X 



a„j a^3 a„3 a, 



is got by multiplying the first column by the last row after striking 

 out the common element a^, ; that is to say, the term in x'"~- is 



(-If (a„, a,„ ..., a,_,,;jja,„ a,,, ..., a,„)x''-\ 



(") 



* Pascal, E., I determinanti riccorrenti e loro proprieta. Rendic. . . . 1st. Lom- 

 bardo . . . (Milano) (2) xi. pp. 293-305. See also pp. 462-475, and Feriodico di 

 Mat., xxii. 



