28 Proceedings of the Royal Irish Academy. 



(2) I have shown in a formercommnnication to the Academy " Pro- 

 ceedings," vol. X., p. 373), that if any cubic equation he represented by 

 P = 0, and the first and second functions derived from P, by Q and 

 P' respectively, the equation Q^ - ZPP' = v/ill always be a quadratic 

 equation. Let one of the roots of this equation — either root indifferently 

 — be represented by r\ and let each root of the proposed cubic equation 



P - A^,x^ + A,x'' + A,x + Ao= 



he diminished by r. The resulting transformed equation will be 



A3x'''-^{3Asr + A2)x''+{3A2r'+2Aor+A{)x + Air^+A^r^+A,r + Ao=0, 



the second triad of coefficients of which fulfils the second of the two 

 conditions [1] ; and, consequently, by the formula [3], we have for 

 x the expression 



^,_ 3A,r' + 2A,r+A, _ ( _ ^ A,' - 8A,A ^] 

 3A,r + A, • } (3A,r+A,fy 



Therefore, multiplying numerator and denominator of the fraction 

 under the radical sign by 3^.3^ + ^,) and remembering thatx = r+a;', 

 we shall have for x, the expression, 



SA,7-^ + 2Aor + A^ 



•r 



3A,r + A,-^{ 'yA^- - 8AAz){^A^r + A^:) J ■ 



(4) 



And this is a general symbolical formula for the three roots of the 

 cubic equation P- 0. 



(3). Whenever r, determined as already explained, is real, we may 

 be certain that only one of the three roots of the cubic is a real root, 

 since a cube root has but one real value ;* and this real root the for- 

 mula (4) will enable us to determine, by introducing into that formula 

 the real cube root only of the number under the sign v ; and the two 

 imaginary roots of the equation will be expressed by introducing into 

 the formula the two imaginary cube roots of the number under the 

 sign v, after the real cube root of that number has been determined. 

 But whenever r is imaginary, then, although we know, in that case, 



* And in this manner is the truth of the second of the general properties, noticed 

 at Article 5 following, otherwise proved : \h& first of those properties showing that, 

 if r be imaginary, the above expression for a;, notwithstanding its being then so 

 encumbered with imaginary quantities, must have all its values real values. More- 

 over, \)ins, first of the properties at (0) may also be deduced from the general for- 

 mula above ; for since, as just seen, two roots of the equation must be imaginary 

 whenever r is real, the roots must all be real when, and only when, r is imaginary. 



