the general Equation in Differences. 437 



diagonal line, with the exception of those in the oblique line 

 immediately under the diagonal, are occupied by zeros, but of 

 which all the other places are or may be occupied by finite quan- 

 tities. For instance, supposing n to be 4, such a determinant 

 would be 



K 



C 4 



d 4 



e* 



«8 



b* 



C 3 



d 3 







a. 2 



K 



C 2 











a l 



*i 



Let us for a moment consider more particularly this determi- 

 nant. If, using double indices to denote each coefficient, we 

 were to write the above according to the usual method of nota- 

 tion as below, 



4.4 



4.3 



4.2 



4.1 



3.4 



3.3 



3.2 



3.1 







2.3 



2.2 



2.1 











1.2 



l.l 



the law of formation of the general term would be very far from 

 becoming evident on a cursory inspection ; but a slight change, 

 suggested by the very system of equations in which the determi- 

 nant originates, makes the law at once obvious. Nothing is more 

 natural than that we should use r.sors.r, where r > s, to denote 

 the coefficient of u 8 in the equation of which r is the highest 

 subindex of u\ with this modification, the above determinant 

 changes into the following : — 



4.3 4.2 



4.1 



4.0 



3.3 3.2 



3.1 



3.0 



2.2 



2.1 



2.0 



• • 



1.1 



1.0 



(the terms with equal indices appearing not now in the diagonal, 

 but in the oblique line below it). With this notation it becomes 

 apparent (and the reason of the rule may be deduced by the 

 most simple reasoning from following the course of the succes- 

 sive substitutions in the system of equations giving rise to the 

 determinant) that to find the general term we must write all the 

 descending series of integers which can be formed, beginning 

 with 4 and ending with zero, viz. 



43210 



4310 



4210 



4820 



430 



420 



410 



40 



