432 Prof. Sylvester on Rational Derivation 



Theorem (3.) 



And by virtue of the subsidiary theorem (C), the above two 

 equations are the '' Prime Integer Derivatives," and are exact- 

 ly identical with each other. 



Cor. 1. The leading coefficient of the "prime derivative" 

 of the r*^ degree is always of (»z — r) («— r) dimensions. 



Cor. 2. If P^ be called the prime derivative of the r^^ de- 

 gree and if (X= Y = 0) be the two equations of coexist- 

 ence, and Kr jit^the two " prime constituents of multiplication " 

 to the said derivative, i. e. if X^ and j«,^ satisfy the equation Kr • X 

 + jtA;. Y = P„ then the coefficient of the leading terms in A^ 

 and in /x^ is of (w — r—\) {n—r — 1 ) dimensions. 



Theorem (4.) 



The " Prime Derivative " of any given degree is an exact 

 factor of the " derivative by succession," of the same degree. 

 The quotient resulting from striking out this factor is called 

 ** the quotient of succession." 



Theorem (5.) 



If Li Lg L3 , &c. be the leading coefficients of the deriva- 

 tives occurring first, second, third, &c., in order after the 

 equations of coexistence, and if Qi Qg Q3, &c. represent the 

 first, secondj third, " quotients of succession " reckoned in the 

 same order, then 



-'t. 



* That the appearance cf the index 4 may not startle, let my reader 

 bear in mind that there are what may be termed secondary derivatives of 

 succession for every degree appearing in the process of successive division. 



t The prime derivatives must be capable of yielding an internal &V\Aqwc& 

 of the truth of Sturm's theorem. In fact, for the case of all the roots be- 



