226 LIMITS AND FLUXIONS 



sequently have not the least Regard to Time or 

 Motion, which are necessarily implied in a Fluxion. 

 And the essential Property of a Fluxion is certainly 

 excluded, after the most singular Manner, in the 

 Idea of Quantity considered at its Ne plus ultra : 

 that is, in other Terms, when it is in a State, 

 where ali Possibility of such imaginary Flux is 

 taken from it. So that the Term Fluxion, when 

 used to this Purpose, if it have any Meaning at ali, 

 is as contrary to the true, as Darkness is to Light." 

 He takes an algebraic equation A^o±B^'it:C<3:2jt 

 D<23± . . . ±Z<3:" = o, assumes the coefficients Y 

 and Z of the two highest terms as fixed, and 

 declares (without proof) that the absolute term A 

 is a maximum when the n roots of the equation are 

 equal. When such an equality exists, the equation 

 is reduced '*to its ultimate." When the roots are 

 equal he represents them by +<: or —e. To reduce 

 the trinomial A±Brt±Z^ =0 to its ultimate, " we 

 must make B / Z = -:^nc''~^ in the n Power of ^zh^ 

 = o. That is (because ^=^) B / Z. = na**~'^. There- 

 fore the Ultimate required is ^■:^nZa''-'^ = o, or 

 B^"zb^Z<2'*" 1^:0 = 0." To be observed here is that 

 Kirkby connects, though only in an obscure way, 

 his ultimate with the coefficient of a in the second 

 term of the binomial expansion of {c-±:,ay\ He then 

 pretends to prove ''that the Ultimate of the Sum 

 of never so many Equations is the same with the 

 Sum of their respective Ultimates " ; hence, the 

 Ultimate of the above general equation is f?±B^®it 

 2Caa^-^lDa^a^-±z • • • ^nZa"~^a^ = o. He gives 



