138 On the USE of NEGATIVE QUANTITIES. 



we exprefs the conditions required of AB in an equation, as, 



L . . . a + bx + ex* + ISc. o. 



Then, in determining the magnitude of AB in this direction, 

 the politive roots are to be employed. 



Now, on the other fide of A, take MA t which we fhall de- 

 note by X ; and let it be of any 



magnitude, not lefs than any of M ^ _ ^ _ f 

 the negative roots of L. It is evi- 



dent that L may be reduced to the form 







A 



+ BX +B X 



+ CX* 4- iCXx + Cx* 



* * - ' 



And this again may be reduced to the form 



N . . . . A+BX+x + CX+x+Vc. - o. 



THE equation A r , therefore, is another expreffion of the con- 

 ditions required of x, upon the fuppofition that it lies to the 

 right of A. 



BUT if we make the contrary fuppofition, that AB lies to 

 the left of A ; then fince this is the only alteration in the con- 

 ditions which the equation N requires of x ; therefore, the con- 

 ditions required of it, when it lies to the left of A y will be ex- 

 prefled in the equation 



N' . . . A+B(Xx) + C(Xxy+tfc. - o. 



Now, this equation may be reduced to the form 



A 



By 



