452 Notices respecting New Books. 



Now assuming, according to custom, that A n is always plus, 

 it is plain, that if any one of the right-hand conditions have 

 place, without regarding those on the left, a pair of imaginary 

 roots will necessarily be indicated ; because, although the ac- 

 companying left-hand condition should not have place, yet, 

 by ascending to the preceding pair of conditions, and thence 

 to that next in order, and so on, we should evidently at length 

 arrive at a co-existent pair. It is equally evident that no two 

 consecutive pairs can exist simultaneously, since the second 

 condition in any one pair is opposed to the first in the pair 

 next following. Hence, although all the right-hand condi- 

 tions marked [I.], [2.], [3.], &c. were found to have place, yet 

 only one pair of imaginary roots could be inferred : these con- 

 ditions, therefore, can be regarded only as so many concurrent 

 indications of the same thing. Whenever this concurrence 

 ceases, by a failure of one of the right-hand conditions referred 

 to, or which is the same thing, by a fulfilment of the left-hand 

 condition next in order, then preparation is made for a new 

 indication, totally distinct from those that have preceded ; and 

 if such new indication offer itself, a distinct pair of imaginary 

 roots may be inferred. 



Hence the series of conditions [1.], [2.], [3.],&c. furnish us 

 with criteria of imaginary roots somewhat analogous to those 

 of Newton ; much simpler, however, in form, but to be em- 

 ployed exactly in the same manner. I shall give two examples 

 of this application : — 

 1st. 5x 8 — 2x 7 + Sx 6 — 24.r 5 — 16x 4 + a 3 — 4# 2 — 2d? — 60 = 0. 



Applying the criteria to this equation, we discover the exist- 

 ence of six imaginary roots ; the same as by the rule of New- 

 ton (Researches, &c, p. 47). 

 2nd. 9 xr> — 5 x 4 + 4 x 3 — 3 x" 2 + 6 x + A = 0. 



In this example Newton's rule detects only a single imaginary 

 pair; the criteria above discover two pair, so that the equa- 

 tion has but one real root. 



It may be added that, as in the rule of Newton, the sign > 

 may be changed into ==. 



Belfast, November 7, 1843. J, R. YoUNG. 



LVI. Notices respecting New Books. 



1 . Philosophical Theories and Philosophical Experience. 



2. Connection between Physiology and Intellectual Philosophy. 



3. On Man's power over himself to prevent or control Insanity. Pick- 



ering. 



AT a time like the present, when the general spread of education 

 has put the keys of knowledge into the hands of many, to 



