TOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 541 



vious steps involve higher dimensions than the original equation. The original in- 

 dex being n, that of the equation of the difference of the roots is n X (n — l). 

 However, from the nature of differences, being taken both affirmatively and nega- 

 tively, all equations formed from them must, as observed of quantities of that sort 

 in art. 11, be universally deficient in every alternate term, which brings their equa- 

 tion to the form of equations of only half their own index, orw X -i (n — I ) : but 

 in this case their differences are 6", and their equation, with that consideration, is 

 reduced no lower than a cubic form, which is the same degree with the proposed 

 equation : therefore it does not appear that we can be enabled, a priori, to deter- 

 mine the formula of any direct resolution of this case. 



21. Let us then try to trace some relation which may convert some or all of 

 the roots, or some regular function of them, into equal quantities ; when, the 

 equation of that function having equal roots, of course those roots will be sepa- 

 rately deducible, as shown in art. 11,12. In art. 14, we may remember that 2 

 particularities were mentioned to belong to 3 quantities, viz. that their differences 

 were so related as to be every 2 of them equal to the 3d ; and that, if the quanti- 

 ties themselves have their sum equal to nothing, 2 of them also must equal the 3d, 

 and their magnitude be respectively the same, whether they are taken simply, or 

 summed in pairs. To avail ourselves of both these properties, let us suppose the 

 2d term to be expunged from the given equation, which we know may always be 

 effected, its form will then be x 3 — qx + r = O *, and the sum of its roots equal 

 to nothing. Let a and b be two of its roots, the 3d will therefore be — a — b ; 

 take their sums by 2, — a, — b, a -f- b ; take their differences, 2a -f- b 9 a -f- 2b, 

 a — b, and their negatives, which may be divided into 2 sets whose sum is nothing, 



like that of the roots, viz.| __ | ~7 £ __ ° "\ ^ "~ 2 « + by So tnat > from tne 

 given equation we derive 3 others, which make a set of 4 exactly similar. 



1st. x 3 — qx + r = O, the given equation. 



2d. x 3 — qx — r = 0, that of its roots summed in pairs. 



3d. x 3 — 3qx -j- \/(4q 3 — 27r 2 ) = 0,~> _ . ., c ,,,..,. 



4th. x 3 -3^W (4? 3 - 27?^) = o,} 2 Similar e( l uatl0ns > form ed by dividing 



aP — Qqx 4, + 9^ 2 x x — (4q 3 + 27r 2 ) = 0, the equation of the differences, into 2 

 wanting the 2d term. 



22. Now, leaving these considerations for a moment, let us speculate on the 

 further reduction of the equation. If, instead of the present form x 3 — qx-\- r = o, 

 q could be supposed to vanish as well as p, a still more powerful additional rela- 

 tion would be given to the roots ; for, the equation being then a pure cubic 

 x 3 = + r, its roots would obviously be the cube roots of r, and all cube roots are 

 alike. If r be a cube, and ^r be one of its roots, the remaining two are 

 X #r and X #r, let r be any quantity whatever, real or 



* Besides expunging p, the sign of q has been changed ; because, in cases of real roots, it will inva- 

 riably become negative on destroying the 2d term. Vide note in p. 536, — Orig. 



