^^Z Carnoi on the Theory nj 



both towards A and towards B, it is evident that the tangent 

 to the curve at the point M^ ought to be parallel to the 

 line AB. Then by putting (as in article 62) Q for the angle 

 formed by the tangent and the ordinate, we ftiall have, at the 



point M, Cot, Q =:Oy or Lim, (~r-) — o. I .find, there- 

 fore, the ditferential of the equation of the curve, and I get 

 the imperfect equation, 



, , - ^y a ~ X 



ydy zzz aax — xax\ or — -- = ; 



"^ "^ dx y 



and therefore the rigoroufly accurate equation will be 

 li„. (^) = Ulf ' or Cot. = "-:^. 



But we ought to have Cot. Q = o ; therefore — — = o, or, 



laftly, a :=! Xy which was to be found. 



64. The procefs, therefore, for difcovcring the greatefl: or- 

 dinate of any curve whatever, is to find the differential of its 



equation, and thence the value of /iw. (-j-J which mud be 



made equal to nothing. This rule is commonly enunciated 

 by faying fimply, Find the differential of jv, and make dy 

 = o. But what this enunciation gains in brevity, it lofes in 

 accuracy. 



6^. Problem ITT. To determine the abfciffe and ordinate, 

 anfwering to the point of inflection, in a propofed curve. 



Let ABMN (fig. 5.) be the propofed curve; AP the 

 abfcifTe, and MP the ordinate correfponding to M, the point 

 of inflection fought, and let MK^ a tangent at that point, be 

 drawn. It is plain that the angle KMP is a maximum, tha^ 

 is, greater than the angle LNQ, formed by any other tangent 

 whatever NL, and the correfponding ordinate NQ. The 

 tangent, therefore, of the angle KMP is alfo a maximum^, 

 and its cotangent a minimum. But the cotangent (by ar- 

 ticle 62) is, in general, lim.(--r-'j: and confequcntly (by 



lim\ — — -^iL_y ^.0, w: 



article 63) we have lim.K — - — J - o, wliich wa& to 



be found. 



For 



