On the USE of NEGATIVE ^UANflfJES. 145 

 And this again is reducible to the form, 



N . . . 



tfc. = o. 



THIS equation, therefore, is another expreflion of the condi- 

 tions of the problem, upon the fuppofition that AB, AC are 

 taken to the right of A. 



Now, by reafoning in the very fame manner as before, it is evi- 

 dent, that when AB or AC is taken to the left, the conditions 

 of the problem will ftill be exprefled by an equation, which will, 

 in all refpecls, be the fame with /,, except only, that the quan- 

 tity which has changed its fituation, will change its fign. 



FROM the whole, the two following conclufions feem to be 

 <iemonftrated. 



1. WHERE the problem allows us to confider x, any of the 

 unknown or indeterminate quantities, as capable of exifting in 

 two oppofite fituations, which may be reprefented by addition 

 and fubtraclion ; then the equation, which exprefles the condi- 

 tions required of x in one of thefe fituations, and whofe pofitive 

 roots determine the magnitudes of x in that fituation ; the fame 

 equation, by its negative roots, will determine the magnitudes 

 of x in the oppofite fituation. 



2. WHERE the problem allows us to confider a, any of the 

 given quantities, as capable of exifting in two fuch oppofite fi- 

 tuations j then the equation which exprefles the conditions of 

 the problem, upon the fuppofition that a is in one of thefe 

 fituations, will be reduced to the equation expreffing the con- 

 ditions of the problem on the contrary fuppofition \ by fimply 

 changing the fign of , or, in other words, the fign of the 

 terms involving the odd powers of a. 



V. 



