•"Tract 28.

AND INFINITE SERIES:

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gative, with q on the right-hand side of the equation, thesebeing either botli positive or both negative together; and theother two roots of the contrary sign. For if r be thegreatest root, then is \T greater than e, and \r z greater thane 2 , and ~r 2 — e 2 always positive, and consequently the pro-duct r (^r 2 —e 2 ), or q, will have the same sign with r. Butif r be one of the less roots, the contrary of this will hap-pen ; for then 4 r is less than e, and consequently Ar 2 lessthan e 2 , and so ~ : r z — e z a negative quantity, and thereforethe product r (|-r 2 — e 1 ), or q, will have the sign contrary tothat of r; that is, q and the less roots have different signs,and consequently q and the greatest root the same sign,since the sign of the greatest root is always contrary to thatof the other two roots.

14. Again, when q or r {\t z — e z ) is positive, then r de-

notes the greatest root; for then ^i’ 2 is greater than e 2 , orgreater than e, and r greater than either or — \r— e.

But when q or r (^-r 2 — e z ) is negative, then r represents oneof the other two roots in the equation ; since then e is greaterthan -Jr, and — -|r — e greater than r. Lastly, when q isbetween the positive and negative states, or q — 0, then rought to be neither the greatest nor one of the less roots, ifwe may so speak, that is, two of the roots are equal, andthe third root = 0, since then \r z must be = e z , or Ar = e.

15. Hence it appears, then in general, that the sign of pdetermines the nature of the roots as to real and imaginary,and the sign of q determines the affection of the roots as topositive and negative. Let us illustrate these rules by a fewexamples.

16. The equation x* — 9r = 10 has all its three rootsreal, because p = — 9 is negative, and (-fp) 3 = 3 3 = 27 isgreater than ( \qf, which is = 5 2 = 25 ; and the greatest ofthe roots is positive, because ^ = 10 is positive; and thetwo less roots negative.

17. The equation x z — 9x = — 10 has- the same three realroots as the former, but with the contrary signs, the sign of

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