540 



PHILOSOPHICAL TRANSACTIONS. 



[anno 1799. 



and a — b be the first difference, a — c, a — d, a — e &c a n, having all 



the same antecedent letter, whose number will always be n — 1 , must be the rest ; 

 then, the table will be as follows : 



Table of the simple Representation of the Roots of Equations of progressive 



Indexes. 



In quadratics, 





In biquadratics, 



" A + B + C 



In cubics, 



A + B 





*= < 



A — B 



B + C 



f + 



^3 3 



_1_ A — c 1 B ~* c 



3 .4 



< + 

 I 



A + B 

 3 



3 



A + c 



In the 5th degree, 



A + B + C + D 



A — • B 



2 



A + B 



2 





— A— B— C— D. 



+ 

 + 



+ 



A — B 



B -f- C 



B -f D 



In the (nth) degree, or generally, 



r A + B-t-c + D + E &c (» — 1) in number 



X = * 



4- ~ n — r — Z — T &c - ( n ~ 2 ) m number 



n — 1 n — 1 x ' 



— A — B— C— D— E &C. 



(n — 1) x » 



+ ^ + * 



+ 



n — 1 



&C. 



n — 1 



+ ^-&c. 



h^4,-fc 



n - 1 



B — E 



-&c. 



n — 1 ' n — 1 



20. The inspection of the table shows us, that in all cases, to construct a gene- 

 ral simple representation of any number of quantities, and consequently to construct 

 a direct resolution of their equation, we must first find a certain number of their 

 differences ; but we have no general means of separating particular differences from 

 the rest ; and the whole number of differences increases in a proportion so much 

 greater than the number of quantities, that the former difficulty recurs, the pre- 



