INFLECTION. 



there is obtained 



■or, 4J.' 

 •whence - 



I y ("' + /•) . _ 



2 X la"- -i-y') -4/-V 

 -2A- {a- -i- y) - ^y' 



a' + /' 



-yx . 

 or 4 y' = rt" 



t ; which being fubllitutcd in the original 

 equation, gives ,v - ^ a ; which arc therefore the values of x 

 and J' anfwerin^to the required point of infledion. 



At prefcnt we have only confidered thofe curves that have 

 parailtl ordinares, and which are refi nvd to one common 

 axis ; but it is neceflary to employ a difForent method when 

 the curves are referred to a common focus ; it will therefore 

 be proper now to attend to this latter cafe, obferving only, 

 \vith regard to the form.er, that in order to know whether 

 any curve be concave or convex towards any point alTigned in 

 the axis, find the value of j' at that point ; tlien if this value 

 be pofuive, that curv;; will be convex towards the axis, and 

 if it be ne;^ative, it will be concave. The rule for deter- 

 mining the points of contrary fl-xure and retrogreffion of 

 curves, which fuppofes the fecond fluxion of the ordinate to 



be nothmg or uitinite. 



o or CO, is liable to fe 



exceptions, as is (htrwn Very fully and clearly by Maclaiirin 

 in his Treatife of Fluxions, book i. chap. 9. bookii. chap. j. 

 art. 866 and S67. 



The ordinate jj paiTes through a point of contrary flexure 

 when, the curve being continued on both fides of the 

 ordinate, is a niaxmuim or minimum. But this does 

 not always happen when y = o, or oo. Maclaurin ob- 

 ferves, in general, that if ',■, y, j, &c. vanifh, the number 

 of thefe fluxions being odd, and the fl;ixions of the next or- 

 der to them having a real and finite value, then y pafles 

 through a point of cDntrary flexure; but if the number of 

 thefe fluxions that vanilh be even, it cannot be faid to 

 pafs through fuch a point ; unlefs it fliould be allowed that 

 a double infinitely fmall flexure can be formed at one 

 point. 



The curve being fuppofed to be continued from the or- 

 dinate y, on both fides, if jf be infinite, the extremity of 

 the ordinate is not therefore always a point of contrary 

 flexure, as > is not always, in this cafe, a maximum or 

 minimum ; and the curve may have its concavity turned 

 the fame way on both fides of the ordinate. But thefe cafes 

 may be diflinguiflied by compai-ijig the f.gns of } on the 

 diSerent fides of the ordinate ; for when thefe figns are dif- 

 ferent, the extremity of y meeting the curve is a point of 

 contrary flexure. 



The .fuppofitions of )'. = o, or 00, and of v = o, 

 or ce, ft-Tve to direft us where we are to fearch for the 

 maxima and minima, and for points of contrary flexure ; 

 but we are not always fure of finding them. For though 

 an ordinate or fluxion that is pofitive, never becomes ne- 

 gative at once, but by increafing and decreafing gradually, 

 yet after it has decreafed till it vanilb, it may th. reafter in- 

 creafe, continuing dill- pofitive ; or after increafing till it 

 becomes infinite, it may thereafter decreafe without changing 

 its fign. 



Of th: points of iiifleclhn and retrogrejfton of curves referred 

 fo a common focus. 



Let AD E (fgs. 4. and j.) reprefent a curve referred 

 to the focus Q, from whence the ordinates Q D, Q D 

 proceed ; and let Q d be indefinitely near to Q D. l)r;.w 

 Q T perpendicular to Q D, and Q / pi vp'rndicuhr to Q^/. 

 Draw D T a tangent to the point D, and d t a tangent in 

 the point d. Let Q / (produced if necen"ary) meet D T 

 in the point o. Now it is plain, that as the ordinates in- 

 creafcj if the curve be concave towards the focus Q (jT^. 4.), 



Q i will be gi-eater than Q T ; but if the curve be convex 

 towards the focus Q (fg. 5.) Q t wiUbc lefsthan Q T ; and 

 therefore, as the curve changes from being concave to con- 

 vex, or -vice verfd, that is, in the point of inficftion, or retro. 

 grclTion, the line or quantity /, from being pofitive, ought 

 to become negative, or the contrary ; and therefore mull 

 pafs through nothing or infinity. 



Therefore make Q D y, D M = .v, and with the 

 centre Q, let the indefinitely fmall arcs D M, T H, be 

 defcribed: then the two triangles ^Z M D, </ Q T will 

 be fimilar, as alfo d Q 0, T H 0, and therefore it will be 



</M:MD:^r/Q(orDQ): Q T, or , : .v :: y : :y ^ QT. 



But the two fcclors D Q M, T Q H are alfo fimilar; whence 



D O : D M :: Q T : T H,or^ : .v : : ^'- : '^.i = T H ; 



and becaufe of the fimilar triangles d Q 0, and T H 0, it 

 willber/Q (or D Q) : Q (or Q T) :: T H : H ; that 



is,y:-^-^ ::-'-: ±- ^ H 0. But H t {fg. 5.) re- 



prefents the fluxion of Q T, that is, H / = -^ — '^^-Ul^ 

 by taking x as conftant. Therefore to z= /H -f-Ho = 

 , which muft be equal to o, or to «, 

 infinity ; and confequently multiplying by jr% and dividing 

 byi.it willbe>= - y y + x' = o, or yt. 



■^" .^o 4- ' becomes negative, and tht^rcfore t =z 



■ ■ Tz — , and multiplying and dividing this 



expreffion as above, we have again 



y" —yy + *■' = c, or co, 



which therefore is the general formula for finding the point 

 of infleftion or retrogreffion in curves that are referred to a 

 focus. 



Let us now make an application of this formiJa to an- 

 example, by propofing to find the point of contrary flexure 

 in the curve, which is generated as follows : 



A E D being a circle defcribed from the centre B, let 

 A F K be fuch a curve, that, drawing any radius B F E, 

 the fquare of F E may be always equal to the rettangle of 

 the correfponding arc A E, 'into a right line b. See 



Let the arc A E = a, B A = B K = «, B F = y, and 

 F G = jr, drawing B E' indefinitely near to B E, and with the 

 centre B, and radius B F delcr;be the fmall arc F G ; then 

 by the nature of the curve, it will he b ■:. — a' — 2 a y -j- y- ; 

 and taking the fluxions we have l> ~ 

 i y i — 2 a 

 b 

 fimilar fedors B E E', and B F G, ft will be as B E : B F : j 



whence i. .= 



"y + 2JKJ, 

 = B E'. But becaufe of the 



E E' : F < 



that 



: X ; -whence 



■ay. 



; and now taking the fluxions agaia 

 and making x conftant, we \\a,te /^yy'' ^. 2 y''^ — 2 ay' 



, r .: a i'^ — 3 y v^ 

 thereiore v y = -^ — -. 



■Z ayy 



■yy 



yy 



Now the general formula above is x' 

 and therefore fubftituting for thefe quantities llielr refpec- 

 live values, th# expreffion becomes 



4>^ 



