144 On tie USE of NEGATIVE 



II. Of the Negative Quantities which are not the Roots of the 

 Equation. 



BESIDES the roots of the equation, or the quantities to "be 

 determined, problems frequently occur, which require us to 

 confider fome of the given quantities, as capable, confiflently 

 with the conditions of the problem, of exifting in one or other 

 of two oppofite fituations, which may be reprefented by addi- 

 tion and fubtraction. 



SUPPOSE, for inftance, that the given lines AB a, AC b t 

 may be taken either on the right 

 or left of the given point A ; and 

 let the problem be reduced to an 

 equation, upon the fuppofition 

 that they are each taken to the 

 right. This equation, when or- 

 dered according to the powers of 

 a and b, will be reprefented by 

 .the following general formula : 



L . . . 



+ &c. = O. 



IN the lines AB, AC, and on the left of the point A, take 

 AM = A, AN B t of any magnitudes, not lefs refpedively 

 than a, b. 



THEN the equation L may be reduced to the form, 

 P 



M . . . 



+ RB +'...+ Rb 



+ SA 1 + 



+ SA l 4- iSAa 4- ... + Sfl* 



4- TAB + TBa + TAb + . . . + Tab 



4- F.S* 4- . ,4- zV r Bb+ . . 4- . 





, 4- VT> % 



= 0. 



And 



