VOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 337 



V • ■ , ' ' 



But this root is really threefold, from the triple value which ^ r + v r^ — ^» 



and ^ r — -/ r^— q^ can assume ; for the cube root of any quantity whatever is 

 threefold, and the cube root of even unity itself is either 1 , or — 4- + -r V' — 3, 

 ^ y^ -_ 3 ; as will appear by cubing each of these expressions, which 



or — 



T -r 



gives unity. Therefore if J X (r -^ \^ r' — q^) or r -f ^ r^ - q^ be considered 

 as a cube, its cubic root or roots will be ^ 1 X r -\- y/ r^ — q^ -, that is, first 

 \ X^ r-\- ^ r^ — 9% which we have above called m -\- V Uy ov I X w + / n. 

 Secondly "^+/"^ X ^7+~7^^^' = ^ ' V ^ ^ ^T+^T^. Thirdly 

 _i- y.-3 ^ ^ ^ ^ /7nr^3 __ - 1 - ^/— 3 ^ ^ _|_ ^ „, In like manner the re- 



sidual r — V r'^ — <^' or 1 X (r — '^ r* — ^^), being considered as a cube, will 

 have three cubic roots, viz. first 1 Xr— v^r^ — y'= 1 X m— \/n=^ m — Vn: 

 secondly, ''+/~^ - X ^T^TVp^^ = "'+/"" ■■ X m- ^n: thirdly 



r^~ ^~^ xr— '^r* — ^^ = ^^^ — ~ Xm^ -v/ w. And by a due connec- 



tion of thcvse roots, we have the three roots, viz. z=r-}-Vr'^-- q^-^ 

 ^V_V^7^^rp==77^+'/y^ + ^^--'V^n==2m;andz = ~^^^^^~^ ^r4-^7^^^ 



-j -^1 ^ r — V 7-' — / = ^^^ Xm-\- i/n-\ ^ Xm— ^/n 



^ 1 — / — 3 



= — 1 Xw + / — 3X'v/n= — m+>/ — 3n; and z = -^ X 



-1 + V-3 



2 ^m--v^7<=— IXw— V' — 3X'/n= — ?w— v' — 3/1, 



These therefore will be the three roots of the cubic equation ^ = Zq% -\- lr. 

 And that the parts are duly connected in the foregoing manner, by multiplying 

 them together in the usual way. Finally make z •=. x — ^, and the equation 

 z^ = 3^z -j- 2r will become x^ — Spa?' + Zp^x — ^^ = ?,qx — ^fq + 2r, which 

 by transposition will be the same as the general equation first proposed, and its 

 roots are the same as there exhibited. 



Here it will be worthy of notice, that all the roots of any cubic equation are 

 real and possible whenever the irrational member i/ r^ — (f of the binominal in- 

 cludes an impossibility ; that is, when y is a positive quantity, and (f greater 

 than r^. But if that member be possible, that is if $- be a negative quantity, 

 as also if it be positive, and q^ less than r^, then the equation has only one real 



VOL. V. X X 



