VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 539 



once complete the formula of general resolution of equations of this degree, viz. 

 x = \P + 4- a/ (^> 2 — 4^). This, as before observed in art. 6, is however the same 

 quadratic, only reverted; for, the quadratic surd it contains is frequently incapable 

 of further reduction. Therefore, generally speaking, the degree of the equation is 

 not altered ; only the place of the index, which being first affixed to the unknown 

 quantity, is now transferred to the known ones. But yet this resolution is, in all 

 cases, equally true and direct ; for, involving no other radical than belongs to the 

 degree it relates to, it faithfully exhibits the nature of the roots, and is always 

 rational or real, or not, according as they are so. 



* 18. If, instead of seeking, a priori, the formula of resolution, we attempt to find 

 the roots simply, we may instantly trace a constant connection between them, or at 

 least between their differences; which, however the quantities are varied, are 

 always related in the same manner, being a — b and — a + b, the same quantity 

 with different signs, and consequently their squares precisely the same. From 

 which it appears that the equation of those differences will always want the 2d term, 

 or be a pure quadratic ; and that of their squares will be a perfect binomial square, 

 having both roots equal; which roots may therefore, by the reasoning in art. 11, 

 be certainly found. But the inference is just the same as before : the equation is 

 not lowered in degree ; the equal relation is brought no nearer than between the 

 squares of the differences ; and, when they are found, the same quadratic surd 

 must be used to arrive at the roots themselves. This formula of resolution x = 

 -rp ± -<jV(/> 2 — 4^). is the same given for quadratics in every algebra; but it is not 

 usually remarked, or perhaps understood, that the whole operation, however varied 

 in appearance by setting about to complete the square, as it is called, or to destroy 

 the 2d term, is merely employed to obtain the difference of the roots ; that, on 

 analysing the formula, the part under the vinculum is always that difference and 

 nothing else, and why it must be so. 



]g. Next, let n = 3, and the equation bethe complete cubic x 3 — • px 2 -f- qx-—r 

 = O. If we make it our first step here, as in the last case, to inquire what is 

 necessary to construct a general representation of 3 numbers in simple equations, 

 we shall find it must consist of the same parts, the sum and the differences: but, as 

 the differences increase in number, to show the order in which they are taken, and 

 the law they observe progressively, I shall subjoin a general table of the simple re - 

 presentation of the different orders of quantities. As in every equation the sum of 

 the roots is always given, I shall, for greater simplicity in the table, suppose it 

 always to vanish. If then there be a series of general equations, beginning with a 

 quadratic, and proceeding upwards with progressive indexes, in all of which the co- 

 efficient of the 2d term p be taken = 0, and a be supposed a difference of the roots 

 of the first, a and b 2 of the differences of the roots of the 2d, a, b, c, 3 differ- 

 ences of those of the 3d, and so on ; in taking of which differences, no other cau- 

 tion is necessary than that they should be similarly situated, viz. all derived tyy com- 

 paring the same individual root with the remaining ones, as if a be taken as a root 



3z2 



