430 Prof. Sylvester on Rational Derivaiion 



^1, 7^2, ... kn, the roots of the other. The general derivative of 

 the r^^ degree is represented by 



% ( SR {hy Iiq /iq ... h,) {x — k^ . x—h^ ••• x—hr} 



-{-t'-X".'' -''}) = ' 



SR (^1 . 7/2 . ^3 ... hr) denoting any symmetrical rational (in- 

 tegral or fractional) function of 7ii h^ ... 7?^; 



preted as above explained, and iS of course including as many 

 terms as there are ways of putting n things r and ?• together*. 

 A form tantamount to the above, and which may be sub- 

 stituted for it is its analogue, 



2 f SR (^1 . Jcc^ ... ICj) . {x — Atj . x—kci^ ... x—kr] 



f^r+l • h+l "' K l ^ _ « 



^1-7/1.-^2... -M J ~ 

 When r — the theorem gives simply 



J ^1 h '•' m y—o and is coincident 



with that given by Bezout in his Theory o^ Elimination. 



Subsidiary/ theorem (A). 

 If ^1 7^2 ... /i„t be the roots of the equation 



and ife'^ + «! e*^-' + a^ e"*-^ + +«„ — «< = then 



^ (^1-7^ • (7/, -7^3) ... {h,-k„,)'==7ri •'57/^^^^')'" 

 being made zero after differentiation. 



Cor. If R 7^1 denote any integral rational function of Tii, 



^ ^ R (h,) . , . , , . 



then 2, 77 ttTt t~\ ti r~\ is always mtegral and is 



;?ero when the dimensions of R (k) fall short of (m — 1.) 



* The general derivative may clearly be expressed also by the sum of 

 any two particular derivatives affected respectively vvitli arbitrary rational 

 coefficients. The equivalency of an arbitrary /e«jc^/oH to two arbitrary imd- 

 tipliers is very remarkable, and analogous to what occurs in the solution of 

 certain differential equations. 



