INFLECTION. 



,., , ■ ,• 1 1 / . ,x, 1 J- •! ji ~, -1 ■ And now makinp; again the- fluxion of this quantity =; V. 



Xhich multiphcJ by {a" + x^]* and divided by 2 a' x\ gives ^^^^^ j^^^^_ o s h ; 



O = {a" + -v)' -^4 ''' -^■■- - 4 ■■'' i therefore, 4 .v^ + ^ x .i {a .v - x y) = (a' + 4^) (^.v - j- .i - x j) i 



j^a'x^ =^ {a' + x") ; which equation divided by x^' + a- ^.i^^^^^^ ^ „ ^ ^. ^ 



makes 4 k" = a' + *' ; therefore 3 .v^ — a , and .v = - = — x -- 



o ^/J ; and if this value be fubllituted for it in the given .>' «" — •»" a ~ y 



equation of the curve, we {liall have y, or the ordinate at And now equating tlicfe two exprcfiior.s, we hare 



ax"' 3 a' , f ^' + •''^ -■'•■ "^ -T- ■''■'' • 



t e poin o m ec ion _ -p^— — j^ — ^ a. a' — x^' a — y 2 .v a — y 



A C be equal to \ A E, and with the radius <: E, defcribe which being reduced, gives 2 .v' — a' — .r% or 3 .v' = . 



the arc E C ; then will C be the point from which the or- and x — a ,/i, the fame refnlt as before ; and this valuo 



dinate to the point of infleaion muit be drawn : for c C == § a; x fubllituted in the original equation of the curve, becon 

 and Ce-eA.-'= A C , i. e. 4 a - ia' = i a' = AC a x'- i a , , ,. , . 



= .v^ ; or.v = « V I . y = .^^^fn? = IT ""i"' *'" °'''^'"'"' "' '''' p"'"' 



Another method of finding the point of infleaion, or re- ^f infleftlon. 

 trogreffion, is as follows. From the nature of curvature it 'f [,j; f^cond method differs from the firft rather in enui- 



is evident, that while a curve is concave towards its axis, the ciation than in principle, the apphcation of it is thercfoic 



fluxion of the ordinate decreafes, or is m a'' xreafmg ratio, omitted. 



with regard to the fluxion of the abfcifs ; and on the con- ^_,._ 2.— Again, let it be propofed to Inid the point of in > 



trary, this fluxion increafes, or is in an inc ealmg ratio to fleaion in a curve wlioiV equation is iy — a .\-' — .v. 

 the fluxion of the abfcifs, when the curve is convex towards -phg fl^j^i^^ ^f ^1^:3 exprc-ffion gives 

 the axis : and iience it follows, that thefc two fluxions are . 2 a x — r, x^ 



in a conftant ratio at the point of infleaion, or retrogreffion, I- j — rm x x — 7, x- x ; cr — = ■ ^ — . 



where the curve is neither concave nor convex ; that is, -, Taking again the fluxion of this expreffion, and making it 



equal to o, we have 

 or -. , is a conftant quantity. But conftant quantities have 2 a h^ x — dh^ x x = o, ot 2 a V = G Ip- x, 



, , , • bccaufe j does not enter ; whence .r — ■' a, which is the 



no fluxion, or their fluxion is equal to o ; whence we derive ^^^^^^^ anfwering to the point of infleaion fought. 

 this general rule. Ex. 5. -Let it be propofed to find.the point of infleaion. 



Put the given equation of the curve into fluxions ; from or rctrogreflTion, in the cubic parabola, with the equation 



.V v J ^ ~ "^ + V(«' — 2a*.v 4- ax). 



which equation of the fluxions, find either -, or -7 , and gy taking the fluxion, we have 



take again the fluxion of this fraaion, and make it equal j = • ^1^ — , or-^ z= — ^SHJ!'-. ^— . 



to o, and from this laft equation find alfo the value of the 3 (a' — -2. a^ x + a x y •*■ (a' - 2 a\v + a xy 



— X V ,, .■ .u r » ^ ^„;) Taking again the fluxion of this expreffion, and making' 



fame expreffion ., or - ; and by equating thefe two, and j^ ,,j,,i ^^^ -q, according to the above rule, we have a I 



the given equation of the curve, x and y will be determined, ^^i _ ^ ^- ^, _(. ^ _^, ^i _ 2(— 2a x + 2.nx x) X {2 a x —lay 



being the abfcifs or ordinate anfwering to the point of inflec- :i, [a' — 2 a"' x ■]- a x')i 



r\ .u •/■ ..■ »i fl,„;„.,„f''^ _ r. .Lit ;^ becaufe j is not found in the fecond fluxion, which exprcfliion 



tion. Or, otherwife, putting the fluxion 01 - = o, tliat is , . , i • j .1 • 1 r i,a» ,» Vi f^,- . 



^1, unit.. 11.., ^^ J, ^^ being reduced, gives .\- := fl ; and this value fubltituted tor a-, 



..._-. _ gives alfo_y = a ; hence the point required is, that anfwering 



'U. Ji— =z o; or .iri — y .v = o, whence X- y = y x, to thefe conditions of the abfcil"s and ordinate, which is a 



■>" point of retrogreffion, as will appear from confidering the 



or, X : y :: X :y, that is, the fecond fluxions have the fame nature of the curve. 



ratio as the firft fluxions ; and therefore if x be conftant, or j,^, _Let it be propofed to inveftigate the point of 



x = o, then fliall >■ = o, which gives the following rule, ^.^_^^_.^,.+ ^^^^^^.^ -^ the curve comn^cnly called the luilch i 



'"'"• of which the equation is y x = a — a x. 



Take both the firft and fecond fluxions of the given cqua- Taking the fluxioa of this expreffion, we have 

 tion of the- curve ; in which make x = o, andj- — o, and . _ , , _^_ ,x 



the refulting equations will determine the values of x and 1, z y y x -f / .v = —ax; or 7 = ^ ; • 



or the abfcifs and ordinate anfwering to the required point. x ■ ^y -v 



rx,, • • u J ., „u ;„ ,„l,;^». Now, again, making tlie fluxion of this expreffion z=. C, 



Thus repealing again the preceding example, in w'hich ' b ' 6 r 



the ffiven equation is a A-' = a" V -f X' V ; anii of which the ^ '- ' .,,,., 



point of infleaion is required. - A f j >: + {2 xy f 2y .v) [a- + f) = o, or 



2 X {^a -Y y )y — 4 y xy 4- 2 v X [a -\- y J = o > 



By the f.vjl ruL; the fluxion of a x' = a- y + x"- y, ^^j^^_^^^ 



is 2 ax.i ~ a'j + 2xyx + x y ; whcHce :? = Z = =^ f ^^' + f) . 



•^ Ji' X 2 X (d' + y) - 4/ A' 



^ * "^ ^ And now equating this value of r with the foregoing, 



there 



