RECTIFICATION. 



radius CN (or C B) = a, Sec. (fee Quaduature of and thi.' fluent of the lad of the two terms is = i a x In-p 



Curves, Cafi 2.) we {hsilhavez:}:: y {CK) : I (RV); , v 



■ log. of^ -1- da' +/)•; therefore 2 = A ^ (^a- -f. y^'- 



and confequently, z = "^ ; whence the vahie of z will be . " 



, , .^ , , . -' . . + i^i X hyp. log. of^ + (^a'+/)i. But when z and 



found, it the relation ot jj and i is given. y vanifli, or become equal o, as they do at the vertex, this 



I. To find the length of the femi-cubical parabola, of fluent becomes = ^e x hyp. log. of i a ; and, therefore, 



j-J the faid fluent being correfted, gives the true value of z, or 

 which the equation is a x'' = y; or jc = — . 



a'- the length of the curve A M = -- x (^ a + jr")/ 4. i. a 



^V' y 9yy^ 



Here.v= --^, or i^ = -^--- ; fubftituting, therefore, ^ j.^p. j^g. of ,-. + (^ a' + ;■): - ^ a x hyp. log. 



this value of .»■' in the general expreHion 

 - ~ = ^/ [i'' + )■'), wc have 



\ 4 '2 / 2 a' 



ch is 



X (9>' +- 4^)- -r C, correftion. 



tlie fluent of which is 

 I 



27 ai 



of U « = — X (t «' 4- /)'■ + i a X hyp. log. of 



ia 



Hence, if A C and D C (fig. 12.) be the conjugate femi- 

 axes of an equilateral hyperbola; and A C = a, MP'=: 

 2y, Q M = X, then will A P = a; _ a ; and a-' - a' = 

 4 y'' ; therefore x' = 4 y' + a- ; confequeiitly x z= ^ 

 (4>' + "1- If then qm be fuppofed infinitely near Q M, 

 we fliall have Q y =j, and therefore the element of the area 

 C Q M A ~_y x ^/ (a' + 4^1 ). Whence it appears, that 

 the reftification of the parabola depends on the quadrature 

 of the hyperbolic fpace C Q M A. 



III. To determine the length of an are of the common hyper- 

 bola. Let the femitranfverfe axis be reprefented by b, 



which is the length of the cui've,"anfwering to any length ^^"'^ '''^ femiconjugate by ., and we fliall have -^ = zb >: 



of the ordinate V. , ■> r .1 ^ r .1 /r tt 



n-r a-r ^1 i II . c J .1 1 lU +• ■»" J irom the nature or the curve (fee HyperbolvI • 



. To reSify the common parabola ; o~ to nnd the length ' ^" w ^i<.c iiirtKuui^.i; , 



of any parabolic arc AM { fi?. II.) Let the parameter , .u r I' \' {<:' + y^) 



.V. ^^ -r a d <■ 1, £• ..u 'i"d therefore x — — ^ — ^ =— ^ — b : hence i = 



— a, the abfcifs = A P = x. Sec. as above. From the c "t.m.e x — 



Now when the arc = o, then j- = o ; therefore 



4 a- _ 

 27 ai 



-)-C=o, orC= — 



27 ai 

 whence the complete fluent is 



(gy + 4«)^' - 4a 



27 ai 



V 



{. 



well known property of this curve, a .v = y'' ; and ax = 



2y y ; confequently x = —-—, and .i" = —-7- > which fub- ^ -/ (^' 4- >") 



ilituted for x' in the general expreffion for the lengfth of the -; r-. rr t = J' a' ( ' + - — - = — I ; which, by co 



6 ^ ^ f' X (f - 4- /) J ' <:* + cy / ' 





■ve, makes z = ( X^^ h 7 ) = - X (a" + 4j'-)' 



\ a- /a 



ich, thrown into an infinite fe 



/ 2/ 2^ 4/ ^ \ 



(a + --^ --4 +if , &c. ) 

 \ a a' a' / 



b' 



into an infinite feries, becomes y ^/ [1 -f- 



vertinj? -:- 



y ^ + '^'J'' 



which, thrown into an infinite feries, becomes = — x ai 1 i- i z: « zi 3 



''^ ~f ^+-f--^, &c.) But ftill we have the 



I. e. % — y -X- _— ^^ 



a' fquare root to extraft : in order to which, let it be affumed 

 z= I + A/ + B;-' 4- C/ -^pf, &c. Then, by fquar- 

 ing, and tranfpoling, there arifes 



^, , , f,,,- r ■ • , ='->'' 2/ 4y I + 2A/4-2B/+2C/ + 2D/, &C. 



The fluent of this feries ,s a = j, + — -^ - — -^ + _-- + A^ / 4- 2 A B jr« + 2 A C;.^ &c 



+ B'/, Stc. ;> = o. 

 I - -;7 X y 4- -^ xy- -7 x/ 4- — X /, &c. 



^ 4- ^. &c. 

 a' a 



la' s» T^ 

 — , &c. = the length of the curve A M required. 



y I ' y 



Other wife : the above x = — X (a" + 4J'^)- is = - — 



a a 



4J' 



a'yy + 4yy 



\a'yy 4- ^y'y 



■^ayy^'^yy Hence A= —j B= -- 



')5 «x(^V + 4/r ^'^ * 





2-^'= ~rr6~aT»' ^ 





('j' 4- ^yf a X («•/ + 4^-'; 



- X (a^ j^' 4- ^y'Yh X (i <2' jyi + = 7;» - -^^ = ^' + 4:^ + 767^' ^'- ^^- Therefore i 



,« X (<jM 4J'')' 

 4/-') 4 ^'J X n-T 



two terms 



But the fluent ofthefirft of thefe = i V (' + -^. &c.) =i x (i 4- A/ + By.&c.) 



3 Y 2 «* -f 



(i'''4-y')^ 



aa 



