fht Infinitejlmal Calcuhis, 49 



t^« Tub tangent TP, anfwering to M, wliicli reprefents any 

 given point whatever of that curve. 



. Let AB be the tranfverfe axis of the curve, for one half of 

 which put a', and for one half of the conjugate put /'^ and 

 Jet V reprefent the abfcils AP^ andj/ the ordinate PM. We 

 fliali tlicn have, 



M 



This being laid down, let a new ordinate NO be drawn infi- 

 nitely near to MP; that is, let this auxiliary line NQ be at 

 .firft drawn at any arbitrary dillance w^hatever from MP, and 

 let the former then be fuppofed to approach the latter conti- 

 nually, fo that their ultimate ratio may be a ratio of equality. 

 The lines MO and NO will then be the refpcclive differen- 

 tials of j; and y (by article 49). Now the fimilar triangles 



TPM and MZO, give ^- = --• = _.^-^, . But it 



is evident, that the more NQ approaches to MP, the more 



will ;ZiV diminifli in comparifon with NOy and that their 



ultimate ratio will be o. Therefore- ZN is infinitely fmall 



' TP MO 

 compared with JVOj and confequently -,,,7, = -vf/^ is an 



imperfe6l equation (by article 31); that is, = -■- is an 



imperfe^l equation. 



Farther, the equation of the propofed curve being^, 



U , . 



yy = {%ax — XX), 



we fhall thcnre have this other imperfe<Sl differential equa- 

 lion, 



hh 



J 



>ii 



xy ~ {adx — xdx) , 



aa 

 In this laft equation, then, fubftituting the value of dy, 



namely, - — -^ , found by the fird, and reducing, we have 



the required fubtangent TP — -, .- X -^^,, 



an equation free from infinitefimal quantities, and which is 

 weceflarily and rigoroufly accurate. 



* For. in the eriipfe, u^ : ^^ : : (2^ — .v) . x • v^.— \V. D. 



' Vol. IX. * G 6q. Oiher- 



