ROOT. 



2. x ' + px = a. thefe branches can be found in other binomial furds, v. 



, when poffible, is done by the following formula : 

 Put — ./ n = tan. z; then 3 . . , 



^ A ' (1) / (« ± </*) = * + */.T- 



f +- */? x c °f' z » or ( 2 ) *' — i * */ ( a ' — ^) — i« = o. 



I - V? x 



tan. 2. 



(3) y = *■ - V («' - *)■ 



3. X -f- px — ~ 1' By Sines anil Tangents. 



p ut _L /7 = fin. %■ then VI. In this cafe, it is neceilary to feparate the abov. 



p general formula into the following particular ones, ac- 



?— ^' q X tan. \ z, or cording as a and £ are pofitive or negative, as follows; vi-z. 



~ </? X cof - h z - Form 1. * + ax - b = O. 



4. Jc' - p x — — q. 



2. .r ! + fl.v f £ = O. 



3. ,v : — ax — b = O. 



Put — */y == fin. z; then 4- *' - «* 4- * = o. 



* I. Solution of Form T : x' -(- a .r — b = O. 



* = I + V 7 X tan. i ss, or , , 



l + A /y X cof. i e. p ut _/i| = tan. z ; and p tan. (45 s - £ - 



Cubic Equations. tan. u. 



By Cardan's Rule. Then x _ 2 / JL x cot . 2 . 



V. Let*' + ax — b = o, or*' 4- ax = b, reprefent V 3 



any cubic equation wanting its fecond term, and in which 2. Solution of Form 2 : !•' |jx + i = o. 



<j and b may be either pofitive or negative ; then will 1 / . -. • 



3 /n //*> 8 K1 Put T (X/ = tan. 8 ;and^tan.( 45 o-i z ) = 



, -* aB Vit + v(j + 2 - 7 )\ + 



(/IT ~ vAtO}' «—.-.- .^/f x co, 2 , 



3 / f b / / ^ a '\ 7 3' Sot" 1 ' " °f Form 3 : r - ux - i = o. 



\/ l"a" v \X 27/ J _ This form refolves itfelf into two cafes, according as 



4 a T \ — ' )' ' 6 ^ s or g reat:er 'han I. 



\/ 1 2 + \/ \ 4 + 27/" In the firft cafe, put j (— Y = cof. z, and ^ tan. 



Which latter form is frequently the moft convenient, as it / 1 „\ , an 



requires only one extraction of the cube root, whereas the ^ 2 ' 



former requires two. Thcn wju v _ 2 / JL x co f cc . 2 u. 



The fecond term of a cubic equation may be taken away \/ 3 



by the following formula? : b / 1 \- 



In the fecond cafe, put — ( — V = cof. z, then has * 



Let/ + py* + qy + r = o. r 2 \ a / 



x — p the three following values, viz. 



Afiume y = ' ; a rr 9 q — $p' ; and b = gp q — 



2A> ,„• 1. ' * = 2, /- x cof.-. 



zp - zjr, y 3 3 



fo (hall x'-\- ax — b = o be the transformed equation 1 a / % s. 



required, which will, under this form, have integral co- 2 - « = — z \/ ~T x cof. ( 6o Q -| \. 



efficients. V 3 V 3 / 



When a is negative, that is, when the equation is .x 1 — _ _ /_f_ f /fi o s ^ 



j x = ^, the preceding formula becomes *" ' ~ \/ 3 " \ 3 /' 



3 / f b /(^ a \ X 4" Solution of Form 4: x~' — ax + b = o. 



1 his has alfo two cafes, according as — I — 1 is lefs 



VIA A^l-flM * V3/ 



V 1 2 V V 4 27/ J " or grater than 1. 



2 / a \ ,- t 



Aii l a ' ^ *' u .u i v. r .u . u In the firft cafe, put — I — V — cof. z, and ^/ tan. 



And here, when — > — , both branches or the root be- b \* / 



27 4 . 



come imaginary, and the equation is faid to be of the ir- *45 ~ 2 ~) 



reducible cafe ; no folution being then obtainable by this Then will x = — 2 / — X cof. 2 a. 



rule, except in thofe cafes in which the cube root of each of y 3 



In 



