and infinite Series. 397 
24. When e is a real quantity, the general equation 
is x 3 zft 'x = r . — <r, and all the roots are real. 
25. When e is imaginary, the general equation is 
x fifz'x = r . \r z + e 2 , and two of the roots are imaginary. 
26. When e is between thefe two hates, or = o, the 
equation becomes x z - |r 2 a; = ^r 3 , and the root r — s/\p 
— 21; for in this cafe p — |r 2 , and q = |r 3 . 
Alfo the other two roots —\r±.e are each = - ~r. 
27. Affume now any general relation between the 
foot r and the fupplemental part e of the other two 
roots, as fuppofe r* : ** : : 4 : n, or <?* = ” or e = jry/n f 
where n reprefents either nothing or any quantity whether 
politive or negative, that is, pofitive when e and all the 
three roots are real, or negative when e and two of the 
roots are imaginary. Subftitute now inftead of in 
the general equation x 3 1 fix - r . ~r x ~ e % , and that equa- 
tion will become x 3 - r T x — l —r $ . Here then p — 
44 r 
and q — bll/* 3 , and confequently the root r = 
sf ^r n — \/ = n p~i ’ \ ex P re ff e d in three different ways. 
The other roots, the general values of which are -\r±e> 
becom z-\r±s/\r , '--\r±\r-Vn-~\r x 1 n. 
G g g z 28. Hence 
