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XXIX. New Criteria for the Imaginary Roots of Equations. 
By J. R. Youne, Esq., Professor of Mathematics in Belfast 
College*. 
ie is shown in my recently published work on Equations of 
the higher orders, that Sturm’s function, X,, derived from 
the general equation 
Ay 2” + Age 2"1+ Ano v2? 4+ .. Ag2?+A,+N=0 
is XK, = {(n—1) A®,_. —2n A, A,_.} 2*-?-+ &e. 3 
and it is known that if the leading coefficient here exhibited, 
that is the expression within the braces, be negative, the pro- 
posed equation must have one pair of imaginary roots, at least. 
Hence we have the criterion 
Bn Ay Aj 1) AMR sion dis eo. | 
the satisfying of which will always imply the entrance of a pair 
of imaginary roots. 
If the order of the coefficients of the proposed equation be 
reversed, we shall have a new equation, whose roots will be 
the reciprocals of the roots of the former equation. Hence 
the condition 
QaYINWA, ON Ly AS ter sa > es pane 
will also imply the existence of a pair of imaginary roots. 
And it is plain, from the nature of Sturm’s theorem, that in 
either of these criteria > may be changed into =, whenever 
the proposed equation is above the second degree‘ ; it is also 
obvious that the common criterion of imaginary roots, in qua- 
dratic equations, is only a particular case of the more general 
conditions [1.] and [2.]. 
Itis worthy of notice, that when the condition [1.] has place, 
the imaginary roots, thus implied, can never be converted into 
real roots by means of any changes among the coefficients 
after the third: nor, when the condition [2.] has place, can 
any alteration in the coefficients which precede the last three 
terms convert the imaginary roots, thus implied, into real 
roots. 
The criteria just exhibited, being very easily applied, will 
often save a good deal of labour in the analysis of equations. 
For example, the equation 
@ — 3627 + 722° —377+72=0 
is immediately seen to satisfy the second criterion, and there- 
fore to have a pair of imaginary roots. The partial analysis 
* Communicated by the Author. 
+ As in the equation of the second degree, the roots in this case may all 
be equal : in order to which however al/ the coefficients of X: must be zero. 
