On the USE of NEGATIVE QUANTITIES. 143 



Now, by reducing this equation to the form of M, and com- 

 paring it with L, it will be evident, that it is the fame with the 

 following equation j 



L' . . . a-\-bx cy+dx* exy+fy* -j- &c. o. 



THIS equation L\ therefore, exprefles the conditions re- 

 quired of the variable quantities, when x lies to the right of A t 

 and when y lies below AB. And, consequently, the corre- 

 fponding pofitive roots of L' are the determinations of the cor- 

 refponding magnitudes of AB and JBC in this fituation. 



BUT L' differs from L, only by having changed the figns of 

 the terms involving the odd powers of y. Therefore, L' and L 

 have the fame roots, except only that the pofitive values of y in 

 the one, are its negative values in the other. 



THEREFORE, in the equation L, the pofitive values of x, and 

 the correfponding negative values of j, are the determinations 

 of the correfponding magnitudes of AB and BC, when AB 

 lies to the right of A, and when BC lies below AB. 



IN the fame manner, it may be fhewn, that the negative va- 

 lues of x, and the correfponding pofitive values of y, are the de- 

 terminations of AB, BC, when AB lies to the left, and BC lies 

 above AB. 



AND, lajlly, That the correfponding negative values of x and 

 y, are the determinations of AB, BC, when AB lies to the left, 

 and BC lies below AB. 



THE obfervations which have been made under the head 

 of determinate equations, are equally applicable to thofe which 

 are indeterminate. And the foregoing demonftration may be 

 eafily extended to any number of indeterminate quantities. 



II, Of 



