158 



.=«.--.-"•" "'• --^ &c. r^TTlTf-i-l."-"^! +&C. Now the 11. 



mit oi a-ne is a for the limiting ratio of — : e (limiting ratio of arc to fine) is the ra- 

 tio of equality, therefore the limit of the equal ratio a : a-ne is the ratio of equality, that is 

 as a is fixed, the limit of a-ne is a. Hence it readily appears, taking the limits of the 

 terms of the above feries, j =: a — - - &c. 



This equation is not deduced by negleding quantities as infinitely fmall ; but becaufe the 

 firft equation is true when n is any affigned number however great, the lafl: muft neceffarily 

 be true alfo for otherwife it is eafily Ihewn, that the firft could not be generally true, when n 

 is any afiigned number however great. 



3. The ratio of the centripetal force in two points of a curve is the limiting ratio of the 

 fagittE. The limiting ratio of the fagittae in the ellipfe, the force tending to the focus, is the 

 inverfe duplicate ratio of the diftances. And as the ratios which are the limits of the fame va- 

 riable ratio muft be equal to each other, the ratio of the forces is equal to the inverfe dupli- 

 cate ratio of the diftance. 



The application of limits to phyfical enquiries ftrikingly illuftrate the value of the method. 

 The beautiful inftances in the Principia Mathematica of Newton are too well known to 

 dwell upon. The flighteft comparifon will in thofe inftances, fliew the fuperiority of this roe- 

 rliod over thofe that have lately been brought forward. 



