488 PHILOSOPHICAL TKANS ACTIONS. [aNNO 1779. 



result an equation of the formula of a (luadratic, from which the other b or a 

 maybe found; whence from the equation {p = — ) may be deduced, and conse- 

 quently the resolution of the cubic required. 



In the same manner, for/) may be assumed any quantity whatever, and in the 

 equation q = a^h 4- b^p"* for b substitute its value --, or for a its value -r-, and 



there result the equations a ^ a^b 4- -^r-n and a = .,, , — b^b'^, which have 

 ^ ' "iia^p 2/b'p' ^ 



the formula of a quadratic, from which may be deduced the resolution of the 

 cubic required. 



2. Let the resolution assumed be x = a ^z p + f-'\/P' + c\/p^; exterminate 

 the irrational quantities, and there results the equation x* — {2b- + 4oc) px' — 

 4 {a'b}} + bc'p') X — a'p + b'p'' — c^ + Id'c'p- — 4ab-cp- = o; suppose /j = 1, 

 and the given equation x'' + gx'^ — rx -{- s := o; then let the correspondent terms 

 of the given and resulting equations be respectively made equal to each other, 

 and there result the 3 equations 2b- + 4ac = — q, and 4b {cr + c'^) = r, and 

 a'* — b* -\- c* — 2aV + 4a/rc=: — 5; reduce these equations into one, so that 

 the unknown quantities a and c may be exterminated, and there results the equa- 

 tion 4b'' + 2qb'' + (V ~" ^^ — ffi ~ ° ^^ ^^^^ formula of a cubic, from which 

 the unknown quantity b may be found, which being substituted for its value (b) 

 in the preceding equations, from the equations thence ensuing may be found the 

 unknown quantities a and c, and consequently the resolution of the given biqua- 

 dratic x'* -f qx' — rx -\- s = 0. 



From the same principles can be deduced different resolutions of the above- 

 mentioned biquadratic x* + qx"- — 7X -\- s = o. 



3. (1st) Let X = a^i^p -\- b'^/p'^; then will the equation free from radi- 

 cals be X-" — 2b''px" — ■^. 2 «/'""' a^jb^r"- • — liii^'-^^ x 2nb"--a'pv"~^ — 



'ix("'-'>^"''-^^^ X 2.^-3.V-.-3- "•("'- 'M"^-^-("'-9) X 2nb^-^a^ 

 1.2.3.4.5.0 ' 1.2.3.4.5.6.7.8 



_ » ■ («^ - 1 ) . («■' - 4) . («^ - 9) . (n"- - I6-) . [«- - (,i - 2)'] ,_,_ 



P^ I '2.3A.5.6.7. . . . ^2« - 2) ^ ^'"' 



hpx = a"p — b"p''. 



This equation may be deduced from the following principles. Let a, (3, y, S, i, 

 he. be the 2n roots of the equation z-" — 1 = o, then, by prop. 23 of my 

 Meditat. Algebraicae, the equation free from radicals will be the product of the 

 following quantities {x — ac^'J^/p — bx^'l^p-) . {x — afi'l/p — b(i-'l/p-) . {x — ay 



""VP - h'"^P') ■ i^' - aS'l/P - ^^'V/^') • (.^ - "^V/J - ^'f-V/*') &c. = o: 

 multiply these quantities into each other, and from the resulting product, by 

 prob. 3 of the Meditat. Algcbr. easily ci.n be deduced the equation free from 

 radicals which was to be found. 



3. (2d). Let X = a.'' '^'^ p -f- b.^"^]/ (r; then will the correspondent equa^ 



