Of GREATEST ATTRACTION. 211 



blem. This is evident, becaufe the attraction mult vanifh both 

 when y — o, and when x = o ; that is, both when u is nothing, 



and when it is infinite. This can only happen when V i + w 1 



is negative. 



Farther, the value of A is always pofitive (as it ought to 



be), 1 + u being greater than \/i + «*, becaufe it is the fquare- 



root of 1 -j- 2 u 4- « a . 



3. Perhaps the relation between A and u will be beft con- 

 ceived, by fuppofing A to be the ordinate of a curve in which 

 the abfcifTae are reprefented by the fucceflive values of u. Thus, 

 if OP (Fig. 7.) = u y and PM = A, the locus of M is a curve of 

 the figure OMM', which interfedts the axis at O, and has the or- 

 dinate PM a maximum, when OP = ° — ^ ' , beyond PM' the 



o 



curve has a point M' of contrary flexure, where it becomes convex 

 toward the axis OR, and afterwards approaches OR continually. 

 It has alfo another branch mm' n, correfponding to the af- 

 firmative values of / 1 -f- u 2 , which has the perpendicular OQ^ 

 for an afiymptote; and has the ordinate V m' a minimum, 



when u = 9 "*" — 7. After palling the point where P' m' is a 



o 



minimum, this branch of the curve recedes continually from 

 the axis OR. Befides thefe, there are other two branches of 

 the fame curve, on the oppofite fide of OQ^ anfwering to the 

 negative values of u. It is, however, only the firft -mentioned 

 of thefe four branches that is conne&ed with the mechanical 

 queftion confidered here. 



Vol. VI.— P. II. Dd The 



