228 



LIMITS AND FLUXIONS 



I conce i ve may be each fìtly called the Peculiar Unii 

 of its respective Scale of Powers. Hence every 

 Ultimator may be defined to be, The proper Reference 

 of each Subject Ì7i a given Equation to the Peculiar 

 Units of the Powers of ali its VariantSy in Order to 

 discover the Ratios of those Variants to one another in 

 their Ultimate State. Which 1 take to be the true 

 Defìnition of what has been hìtherto most impro- 

 perly and unintelligibly called a 

 Fluxion by some, and a Differ- 

 ential by others " (page 49). 



199. Kirkby's doctrine may 

 perhaps become plainer by the 

 study of one of his applications. 

 In any curve with the concave 

 side to TO, the greater abscissa 

 VP (or v) has always the greater 

 "semi-ordinate" PM (or s), 

 "and each are the greatest 

 that they possibly can be to 

 the same Arch VM, or to the 

 same intercepted axis VR. Therefore the Sub- 

 normal PR (or r — v), and consequently the 

 Normal MR ( = c) are each the least that they 

 possibly can be to the same Arch VM, or the 

 same intercepted Axis VR (or r). Therefore, if 

 in the last Equation [r^ — c'^=2rv — v^—s^\ e and 

 r be invariable, v^e have r^ — c^ an Ultimum, Con- 

 sequently, the Ultimate of that Equation . . . is 

 2;'z;" — 2vv° — 2j^° = o, or (dividing by 2) j^° = r — z^ X t'°. 

 Whence V" \ s° = s : r — v. That is in ali Curves, as 



FlG. Il 



