VOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 385 



of which are a and |3 ; suppose x = ■ , then we shall have 1 — py + qy' = O, 



the roots of which equation are -, -. Let the root of the equation 2qy — /> = 



be equal to-, then- will always lie between the quantities -, -, and therefore one 

 would think at fii-st sight that the quantity a must always lie between x and j3. 

 But this would be contrary to what is proved in art. 7. In the present case a 

 can never lie between a and p, unless these two quantities have the same sign, 

 and it is obvious, that the same reasoning holds in equations of higher 

 dimensions. 



These observations, as far as I know, are entirely new. The fundamental 

 proposition (^'4) was, in the year 1775, couinmnicated to Dr. Waring, 

 Lucasian professor of mathematics in this university, and by him inserted among 

 the additions to his Meditationes Algebraicae.* 



^11. M. Euler, at the conclusion of his 13th chap. Calcul. Different, has 

 given a demonstration of Des Cartes's rules for finding the number ot affirmative 

 and of negative roots in any equation, the roots of which are real. From what 

 I have already said, his reasonings will appear inconclusive, though I freely own, 

 that what he has done suggesced the following different method. Suppose (d) 

 L + mx + nx"^ + i*-^''- ■ . ■ + x" := O, then the roots of the equation (e) ?« + 



<2,nx + nx" ~ ' = will be limits of the roots of the equation (d) ; and 



therefore there must be at least as many positive roots in the equation (d) as 

 there are in the equation (e). The same may be said of the negative roots: for 

 since every root of the equation (e) lies between the different roots of the equa- 

 tion (d), it is impossible that the number of roots should be less in either case. 

 Suppose L and mx to be both positive; then since the last term in any equation 

 is always the product of all the roots with their signs changed, the number of 

 positive roots in each of the equations (d) and (e) must be even: therefore the 

 number of positive roots in (d) cannot exceed the number of those in (e) by 

 unity; but there is in (d) one root more than in (e), and consequently it must 

 be negative. 



If both the terms l and mx are negative; because then the number of posi- 

 tive roots in (e) and (d) are even, it follows in the same way, that there is one 

 negative root more in (d) than there is in (e). And lastly, if the terms l and 

 mx have different signs, for the same reasons there must be one positive root 

 more in the equation (d) than there is in (e). 



Des Cartes's rule is, that there are as many positive roots in any equation as 

 there are changes in the signs of the terms from -f- to — , or from — to-f , and 

 that the remaining roots are negative. From what has been demonstrated it 



* See the end of Pioprietates Curvarum, — Orig. 

 VOL. xiv. 3 D 



