TRACT 28.

AND INFINITE SERIES.

85

A' 3 _ x = r ({-?'■ — e 1 ), and that equation will become .r 3 —1

3 + n z _

4 ‘

,3 . Here then p — r 1 , and q — r z ,

iq

and consequently the root r = V — i /,TT. =

n+ 3 o

ex-

jl — n n— 1 p

- ./JL _

3-f?i

pressed in three different ways. The other roots, the ge-neral values of which are — ir ± e, become — \r ± \Z\nr 1— — ir ± ir </n = - \r x ( 1 ± s /n).

23. Hence then, in an easy and general manner, we canrepresent any form or case of the general equation, with allthe circumstances of the roots, by only taking, in these lastformulae, any particular number' for n, either positive ornegative, integral or fractional, &c. As, if n = 1 ; thenthe equation becomes x z — r^x = -°r 3 , or = 0, the value of

e — \r, the root r = Vp = = and the other two roots

— — ir ( 1 ± ^ 1 ) — — ir . 2 and — \r . 0 = — r and 0.

29. If n = — 1, the equation will be x z — -|r 2 x = ~r z , the

value of e = \r*J — 1, the root r —i=. V2p = ,f/2q, and

the other two roots = — ir (1 ± 1), imaginary.

30. And thus, by taking several different values of n,positive and negative, the various corresponding circum-stances and relations, of the equation and roots, will be asexhibited in the following table.