338 PHILOSOPHICAL TRANSACTIONS. [aNNO 1707. 



and possible root, the two others being impossible ones. In this theorem, if /> 

 be = O, that is, if the second term of the equation be wanting, then it be- 

 comes the case of Cardan's rules, the solution of which is contained in what is 

 done above. 



§ II. Of the General ) ** = 4px^ -f Iqx^ + Qrx + As 

 Biquadratic Equation ) — 4/)"^ — 4pq — q^ 



the 4 roots are * = /> — a + \/ p'^ -\- q ■— a^ ^ , 



where a^ is the root of this cubic equation a^ = p^a* — ipra} -J- '"*• 



+ 7 — -s 



Now when any biquadratic equation is given, a comparison must be made 

 between its terms, and the corresponding terms of this general equation, by 

 which means the quantities />, ^, r, s will soon be found. Then the value of a 

 will be discovered by the theorem in the former section ; and lastly the four 

 roots of the proposed biquadratic equation by the theorem just given. 



An example or two may suffice to illustrate this solution. 



Example 1 . — ^To extract the roots of the biquadratic equation, x* = Sx^ -f- 

 83.t* — l62x — 936. First, by comparing the terms, 4/) = 8, or p = 2. 

 Secondly, 2q — 4/)» = 83, or ^ = V- Thirdly, 8r — 4pq =. — J 62, or 

 r = >4-'. Fourthly, As — q* — — 936, or * = «-H-'. Hence p^ -\- q =. 

 »4', and Ipr + 5 = 'ff", and r* = "-yV " 5 and therefore a® = 'f' a* — 

 1^9 a« _|. »3j^8 9^ Now that this equation, which is in effect a cubical one, 

 may be resolved into its roots, we must have recourse to the first theorem, in 

 which there will be /> = »^', q = '-H-S-", r = 's^^yS and r* — ^' = — 

 110^075^ But the cubic root of the binomial « Vrw » + ^ — > ' »^° ' » i* 

 — -fl-H- -/ — *-§-"; and therefore a^ = •-§-' — V =9? and also a^ = '|^' -f- 

 44 + i/ 400 = '4» or 4. Or, which comes to the same, the six roots of the 

 same equation, which is really cubo-cubic, are a = + 3, and a = + Vj and 

 a = + 4-, any one of which may be taken indifferently for the root, and 

 will answer the purpose. As suppose in the present case we take a = 3 : then 

 by the theorem 



x = p^ a ± \/ p-" + ^-a''~| = 2-3±\/4 + f-9-y = 



/ 2r 



— 1 + 5=4 or — 6; ?xA x ^ f -\- a ± v/+7— a' + - = 2 + 3 



- V4 +^""9+^=^ ±8= 13 or — 3; which are the four roots 

 of the given equation. 



