On the USE of NE GAT WE $UAN?mES. 139 



BY comparing this equation with M and L, it will be evi- 

 dent that it may be reduced to the form 



L' . . . a bx+cx* tc. o. 



THIS equation L', therefore, exprefTes the conditions required 

 of AB, when it lies to the left of A j and, confequently, the po- 

 fitive roots of L determine the different magnitudes of AB in 

 that fituation. 



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

 the terms involving the odd powers of x. Therefore L and L 

 have the fame roots, except only that the pofitive roots of the 

 one are the negative roots of the other. 



THEREFORE, the negative roots of the equation L, to which 

 the problem is reduced, upon the fuppofition that AB lies to 

 the right of A, are the pofitive roots of the equation to which it 

 would be reduced, upon the fuppofition that AB lies to the left 

 of A. Therefore, thefe negative roots are the determinations 

 ofAB in this laft fituation. 



AGAIN: Let it be required to determine, not only the di~ 

 Jlance of time between a certain event and a given inflant ; but 

 alfo, whether the event happened after or before the given in- 

 flant. Then reduce the problem to an equation, Z,, upon either 

 fuppofition, as, for inflance, that it happened after ', and by af- 

 fuming X t a period of time immediately before the given in- 

 flant, we can fhew, as above, that the negative roots of L will 

 be the determination of the epoch, when it happened before 

 the given inflant. 



IN like manner, let it be required to determine, not only the 

 momentum of the force which afts in the right line 

 AB, but alfo, whether it acls in the direHion AB, ^ % 



or in the direction BA. Then, if we reduce the 

 problem to an equation, upon the fuppofition that it ads in the 

 direction AB, we are to aflume X, an additional force adling in 

 the fame direction. 



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