336 PHILOSOPHICAL TRANSACTIONS. [aNNO I7O6. 



1 ± \/ 4 = 7 or 3. Or again, another cubic root of the same binomial 3 + V 



— 'Vt (^o^ ^^ has three) is 4.-)- -/ — ■-^=w+ \/n; and hence the root x =: p 

 4- 27W = 4 4- 3 = 7, and also x = p — m ± »/ — 3n = 4 — -|-+ -i/-j. = 3or 



2. Or lastly, the third cubic root of the same binomial 3 + V' — Vr is — 4. 



— a/ -~^ ^ r=. m -\- v^w; and therefore the root x •=. p •\- 1m ■=. A ~~ 1 =3, and 

 also X ■=: p — m ±_ »/ —3^ = 4 + 4-+ »/ v=7or2. 



3. In the equation af' = — • 15^* — 84a; + 100, it will bejb = — 5, 9 = — 



3, r = 135; and the cube root of the binomial 135 + ^ 18252 is 3 -f- -/ 12: 

 therefore the root is a7= — 5 + 6= 1, and the other roots a; = — 5 — 3 ± 

 i/ — 36 = — 8 + -/ — 36 are impossible. 



4. In the equation x^ = 34a;^ — 310a; + 1012, it will be/) = V, 9 = «•.•, 

 r= '■l-f*'; and the cube root of the binomial »4-f-« + -/ "'-|-f«'' is V"+ -v^ V ^ 

 therefore the root ar = y + y = 22, and the two other roots a; = y — y + 

 »/ — 10 = 6+ v^ — 10 are impossible. 



5. In the equation x^ = 28ar^ -j- 6la7 — 4048, it will be/) = y, 9 = »^r^ 

 r = — • Vr " ; and the cube root of the binomial — • Vr ° + \/ — 382347 is 

 y -|_ y/ _ •^s : therefore a? = V + V = ^3, andx = V — V ± -v/ '^» = 

 16 or — 12, the two other roots. 



6. In the equation x^ = — x^ -|- 1660? — 660, it will be /> = — 4., ^ — 4 j|.»j 

 r = — "-148 ; and the cube root of the binomial — "4-f « +-/ — '' VV°* i* 



— V + V^ — ■§■ * therefore x •=. — ^ — V = — '5, and also a; = — ^ -|- «_« + 

 V' 5 = 7 ± -/ 5, irrational. 



7. In the equation x^ = 63a:^ + 99673^7 -f- 995 1705, it will be/)= 21, q = 

 ioo^»i»6^ ^ == 6031 680; and the cube root of the binomial 603l680 + v' — 

 4T8.7 >^o*3 1 36 is 183 -|_ ^ »_ .^0; thcrcfore jT = 21 + 366 = 387, and also 

 a: = 21 — 183 + ^ 529 = — 139 or — 185. 



And thus we are to proceed in other examples. Now the theorem may be 

 investigated in the following manner. Suppose the root of any cubic equation 

 to be 2 = a + ^, the cube of which is t? =. ci" ■\- Za^h + 3a/»^ + Z?^ = a^ -|- Zab 

 y, a -\- b ■\- h^. Now instead of a + ^ substituting its value z, it gives z^ = 

 3a^z + a^ + b^i which is a cubic equation raised from the root z = a -f- ^j of 

 which the second term is wanting. But to reduce this to a better form, assume 

 the equation z^ = 39Z + 2r, instead of the former z' = 3a^z -H a^ X ^^; then 

 to change the one into the other, there is first 3^ = 3a^, or (f = c?b^\ secondly 

 2r = a* -f ^', or Ira} = a^ -j- a}b^ = a** + 9'; then resolving this quadratic 

 equation gives ci^ =z r •\- '>/ r^ — 9^, and hence b^ := 2r — a^ = r — v^ r'^ — 9*. 

 Therefore at length we obtain a = ^ r + V r" — ^^^ ^^^ b •= ^ r — \/ r^ — q^. 

 Therefore in the cubic equation z^ = 3qz + ir, we shall have the root z = 

 (a + Z> =) ^ r + V^ r' — ^» -f- ^T^TV^'^Y. 



