REPORT ON CERTAIN BRANCHES OF ANALYSIS. 299 



Renting real magnitudes affected by the recognised and known 

 signs of affection only, are competent, under all circumstances, 

 to answer the required conditions of the problem. 



If the value of one root can be ascertained, and that root be 

 real, the problem can be simplified, and the dimensions of the 

 equation depressed by unity; for the coefficients of the reduced 

 equation Pj . Pg . P„-i, which are also real, can be successively 

 determined. If more real roots than one can be found, the 

 dimensions of the equation can be depressed by as many unities 

 as there are real roots. If the root determined be not real, and if 

 a similar process for depressing the dimensions of the equation 

 be adopted, the coefficients of the new equation would not be 

 real, and the conditions of the problem with respect to the re- 

 maining roots would be changed. But if we could ascertain a 

 pair of such roots, such that their sum = x + x^ and their pro- 

 duct = xxy should be real, then the dimensions of the equation 

 might be depressed by two unities, without changing the con- 

 ditions of the problem with respect to the remaining roots; for 

 if we supposed Qj, Qa, Qz, &c., to represent the coefficients of 

 the reduced equation, we should find, 



J^ + ^1 + Qi = Pv 



X x-^ -{■ {x -\- X-^ Qi + Q2 = p<i, 



X x^Q^ + {x + X^} Q2 + Q3 = ^3, 



X X^ Q„_4 -\- {X + X-^ Qn-3 + Qb-2 = Pn-U 

 X X^ Q„_2 = Pn, 



from which equations we can determine successively rational 

 values of Qj, Q2, . • • Qb-2- It remains to show, therefore, 

 that in all cases we can find pairs of roots which will answer 

 these conditions. 



If the number of quantities x, x^, . . . .r„, be odd, it is very 

 easy to prove that there is always a real value of one of them, x, 

 which will satisfy the conditions of the general equation (1.) *, 

 and that consequently the dimensions of the equation may be 

 depressed by unity, and our attention confined therefore to 

 the case where the dimensions of the equation are even. If m, 

 therefore, be any odd number, the form of n may be either 2 m, 

 2^ m, 2^ m, 2" m, and so on. Let us consider, in the first place, 

 the first of these cases. 



The number of combinations of 2 m, things taken two and two 

 together, is m (2 m — 1,) and therefore an odd number : these 



* This may be easily proved without the necessity of making any hypothesis 

 respecting the composition of the equation. See the Article ' Equations' in 

 the Supplement to the Encyclopedia Britannica, written by Mr. Ivory. 



