ROOT. 



(n — l)r" + (n - 2)ar"~' + (b — 3) b r"~' + 



(a — 4) cr"-'+ &c 



'-■ + (n — 2) b r" 

 c r"~*+ &c. 



+ 



n r" ' 4- (b — 1) a r 



(■-$) 



nearly, which general form refolves itfelf into the following 

 particular ones ; w'z. 



Equations of the third Degree. 



2. At 1 + a x"- + bx + c = o; 



2 r ! 4- a r~ — e 



x = -. 



3 r" A- 2ar -r I 



Equations of the fourth Degree. 



3. x* 4 a jc 1 4 b x* + c x + d = o; 



3r' -f 2ar' + br" — d 



x = J ■ . 



4.1-' 4 3 a r -\- 2 b r + c 



Equations of the fifth Degree. 



4. .*•"' + a x^ + bx~ 4- f x 1 4- </ *■ f f = o ; 



4 r' + 3 a r' 4- 2 £ r ; ttr*-( 

 5c 1 f 4ar' + 3<r : + 2tr +</ 

 &C. &c. 



By the fecond Method. 



XII. Let .r"+«-4h*- : + cx"-> + dx n -*+ Sec. 

 = 10, be any general equation, as before, and ;■ an approxi- 

 mate value of x ; then making 



r"+ ar"-' 4 b r"~> 4 cr"~> 4 «fr' , -*+ &c. = <u 

 we (hall have 



(to — 1)) 2 r 



- 3- 



1 . j=r + 



Or, 



2. jr = r ■' 



(b — 1) w 4 {>' 4 1) r"+ (n — 1) a f~ l + 



(b — 3) br"~- + &C. 



(< 



v) 2 r 



(b - l)w 4 (» + I) »•"+ (»- i)«f""' + 

 (» — 3) 4 r"- ; + &c. 



The firft formula being applicable to the cafe in which 

 r is greater than unity, and the fecond to thofe in which 

 it is lefs. 



Thefe general formula? refolve themfelves into the follow- 

 ing particular ones ; vh. 



Equations of the third Degree. 

 4 a \ ! -j- b x ~ to ; 

 (to — -v) r 



x = r + 



to or v -f- 2 r 1 4 a r 



Equations of the fourth Degree. 

 4. .\- v + a x'> 4 hx* 4 c x = to ; 



(to — i») 2 r 



r + 



3worjD+ 5r' + 3ar' + Jr- c r 

 Equations of the fifth Degree. 

 5. .r s 4- <i *' 4 b x~ 4 c x z + dx — . to ; 

 (to — i') r 

 2w or 2v+ 3 r'+2ar'-{-br — d r 



The latter formula?, which are by far the mod converg. 

 ing, were firft pubiilhed by Mr. Barlow, in No. 12. of Ley- 

 bourn's Mathematical Repofitory ; with reference to which 

 we propofe giving one example by way of illultration. 



Example. — Given *' — ,2 x = 5. 



Aftume r = 2, then (by formula 1), 



r ■•■= 8 



— 2 r =n — 4 



1) = 4 



TO = 5 



TO — f = I 



r = 2 



2 r = 16 



TO = 5 



2 1 divifor. 



vhence 



1x2 



21 



= .094. 



Therefore x — 2.094 nearly. 



Affume, therefore, r = 2.094, then, 



r' = 9.181846584 2 r' = 18.363693168 

 — 2r— 4.188 to= 5 



•y = 4'993 8 4 6 5 8 + 

 if = ^5j 



TO — 01 = .OO6153416 



r = 2.094 



r(w-v)= .012885253 



.012885253 

 whence 



*3-3 6 3 6 93 I<58 



.0005515. 



23-3 <5 3 6 93 I 



Therefore x = -09455 15 nearly ; which is true to the ne 

 figure in the eighth place, by only two fubftitution*. 



TA8I.& 



