OF THE ROOTS OF ALGEBRAIC EQUATIONS. 347 



/^ + '(0) 



fp+Uo) 



^ = /(o) +■'-1 — VV^ ' '^^os (p + i)e + 



, — —pPsm(p + 1)9 +. 



= 



The form of the curve very near the point in question will be given by taking only the terms 

 of the series above set down, and therefore we shall have 



e 



p + I' 



omitting the values and p + 1 of /c which correspond to the real branch. Hence there are p 

 imaginary branches going oiF from this point ; and to determine the nature of the curvature 

 we have 



|P + 1 "" *" 



% = /(O) + ~-^ cos fcTT . -" 



/(o) + (-i)*-'^— AJ^P... 



The second term will be alternately positive and negative as k assumes successive value 

 and when it is further observed that the equation of the real branch is given by putting k = > 

 the whole of the theorem will be seen to be true. 



In the case of simple maxima and minima, for which f{w) = and f'{x) does not vanisl 

 this proposition admits of more simple and obvious proof; for we have from equations (3), 



^ =/(.»■) -.rc^)j^ + 



o=/G^)-/>')S + 



and it is clear from these equations that when y = 0/'(.r) = 0, that is, where there is a maximum 

 or minimum there is an imaginary branch; it is also evident that the imaginary branches can 

 never cross the real plane except at points for which f{ai) = 0, that is, either at maximum or 

 minimum points or points of inflexion : this last is an important consideration, because it shews 

 that, in tracing the general form of a curve, after having traced the real branch and those imaginary 

 branches which start from jjoints at which f\a:) = we may be quite sure that all the remainder 

 of the curve lies in isolated infinite branches situated symmetrically with respect to the real plane. 



If we suppose as before the axis of z to pass through the maximum or minimum point, 

 we have, 



^=/(o)-/"(o)^, 



which shews that the form of the imaginary branch is that of a parabola curved in the opposite 

 sense to the real branch, or we may say that when there is a maximum in the real branch there 

 in a minimum in the imaginary, and vice versa. 



V Y 2 



