VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 545 



real, and the other 2 imaginary : the cube produced by them will therefore be 

 real. When all the roots are real, if 2 be equal, one difference necessarily vanishes; 

 therefore the imaginary factor will only appear about the 2 that remain ; and here 

 again the cube produced will be real. But if all the roots are real, and unequal, 

 their sums and differences will all be real : whence all the binomials will involve the 

 imaginary surd ; which constitutes the irreducible case. 



To give examples of this, let, 1st. x 3 — 1x ■+• 4 = O, a cubic equation, whose 

 roots are 2, and — 1 4- V — 1, and — 1 — a/ — i ; the binomials constructed by 

 taking their sums and differences as before, will be 

 2 -(l~^_i) = 1 + ^-1^ 



2 + (1 - •— 1) = 3 - </- 1 J 

 2 — (1 + •— 1) = 1 — </— 1 



2 + (1 + •- J)= 3+ • 



13 



' ~*~ , . . , , k which last binomial 



— 1 -f- a/ — 1 + 1 + •— 1 = 2-/ — \y 



— 2 ■+ 2 /— 1, when the latter quantity 2 y' — 1 is drawn into the imaginary 



2 



surd is/ — 4-, becomes — 2 — , a real quantity. 



2dly. Let x 3 — 3x 4 2 = be proposed, whose roots have been, in art. 23, 

 shown to be 2, — 1 , — 1 . Here 



2 — 1 = + n . This latter binomial must evidently remain real, since the 

 2-f 1 = + 3 J difference into which the imaginary factor was to have been 

 2 — 1 = + I "1 drawn vanishes. 3dly. Let x 3 — Tx 4 6 be given, whose roots 

 2 + 1 = + 3 J are l, 2, — 3. The binomials derived from these have been 



— 1 — 1 = — 21 before given, in the 2d example to the first section of this ar- 



— 14-1= / tide ; and the cube they produce shown to be 24 — V V — i» 

 the cube root of which cannot be extracted; it being from the quadratic surd, it 

 involves, in truth, not a cube, but a truncate 6th power in a cubic shape: and 

 when, to remove its equivocal state, it is made rational, shows itself to be pro- 

 perly the 6th power equation x 6 — 6x 3 4 Vt = 0, as before demonstrated. 



26. This is the common reduction of a cubic equation, to one of the 6th degree, 

 but in form a quadratic, obtained, by clearing of its quadratic surd, the pure 

 cubic formed by either of the 2 sets of binomials before described; and this is the 

 only reduction of it yet discovered. Perhaps the method called Cardan's rule is 

 the shortest mode of effecting this reduction; but I am not aware, that the real 

 principle on which it is founded has been any where fully analysed and explained, 

 except in the foregoing investigation of it. The ordinary expositions of it certainly 

 disclose nothing of the principle, and are even in many respects faulty; for they 

 treat it as the effect of a supposition or lucky conjecture, when, in fact, there is 

 no supposition or lucky conjecture made; a regular cine, furnished by certain de- 

 monstrable peculiarities in some functions of this order of quantities, being pur- 

 sued, till such a relationship among the roots may be inferred, as may be converted 



vol. xviii. 4 A 



