142 On the USE of NEGATIVE QUANTITIES. 



ditions required of thefe quantities in an equation of the com- 

 mon form, as, 



L . . . a + bx + c+dx t + ex+ t + &c. = o. 



Then, in determining the correfponding magnitudes of AB 

 and EC in thefe directions, the pofitive roots are to be em- 

 ployed. 



Now, on the other fide of A y in the line AB, take MA, 

 which we mall denote by X, not lefs than any of the negative 

 values of x j and in the line AC, below A B t take AL, which 

 we mall denote by T, not lefs than any of the negative values 

 of y. It is evident, that the equation L may be reduced to the 

 form 



A 

 + BX + Bx 



cr + . . . + Cy 



M . . . ^ + DX 1 + zDXx +....+ JDat' 



ETx + EXy -\ ---- + Exy 



+ 

 + Vc. 



And this equation may be reduced to the form 

 N . . . A+B(X+x)+C(r+y) + D(X+x 



= o. 



= o. 



+ FT* +....+ iFTy -i ---- + . + Fy* 



} 



THIS equation N, therefore, is another expreffion of the con- 

 ditions which are required of the variable quantities, upon the 

 fuppofition that x lies to the right of A, and that y lies above 

 AB. 



BUT if we make the fuppofition that x lies ftill to the right of 

 A, but that y lies below AB ; then, fince this is the only al- 

 teration in the conditions which the equation N requires of x 

 and y \ therefore the conditions required of them, upon this 

 new fuppofition, will be exprefled in the equation 



N' . . . A+B (JSr+) + C (Ty) +D (X+xy +E (X+x~) .(r j) + 



We. = o. 



Now, 



