408 Reviews, and Notices respecting New Books. 
greatly facilitate the study of Sturm’s details to have Young’s general 
view of its essential parts already in the mind: to younger and less 
experienced students this course is indispensable, whilst to those who 
are but arrived at the threshold of the subject, by their previous ac- 
quirements, no inducement to pursue the obvious and natural course 
is necessary. The process itself is, we may add, in application, only 
the method of finding the greatest common measure of two alge- 
braical expressions. 
The real roots, both positive and negative, being successively evolved, 
an equation is left in which all the roots are imaginary. For all the 
purposes of actual calculation, the problem then is perfectly solved. 
Still, for many reasons, it is desirable to be able to assign the quad- 
ratic factors of which the depressed equation is composed. Is it too 
much to hope that another Horner or another Sturm may rise up in 
our own day to render the solution, in every sense, complete ? 
The process of Sturm is the same as that which leads, in the usual 
operation itself, to the detection of the equation which is composed of 
equal roots, and terminates there at once, so that we cannot but detect 
them as we proceed. This is a great advantage, in as much as we can- 
not pass over this circumstance unknowingly. We have, it is true, to 
depress the equation, and proceed anew ; but we have removed all am- 
biguity as to equal roots, and done very much towards their determi- 
nation. This advantage is peculiar to the method of Sturm. If real 
roots, in the reduced equation, lie between narrow limits, we know 
they are not equal ones, and therefore proceed to their separation 
with certainty. 
We would not, however, conceal from our readers the fact that, 
advantageous as Sturm’s rule generally is, in comparison with that 
of Fourier and Budan, still the great facility with which the derived 
polynomes are formed in the latter method, compared with the tedious 
calculations which the former often requires, is a great and decided 
advantage in this stage of the work. Wherever, from the want of 
some visible relation amongst the coefficients of the given equation 
existing, it is probable that the derivation of Sturm’s V,, V,, etc. (or 
X,, X,, etc. of Young’s notation) will give high numbers as coefficients 
of these derived polynomes, we think it better to defer the application 
of either method, till, by successive substitutions in the usual manner, 
it is rendered evident that some test will be required, by the appear- 
ance of “a doubtful interval.” Should there be but one such doubt- 
ful interval, or even a small number of them, compared with the de- 
gree of the equation, then we think Fourier’s method will be the less 
operose: but if several, then unquestionably it will be most simple 
to have recourse to Sturm’s in preference to the other. The direct- 
ness of Sturm’s process, and the less danger of interchanging the ad- 
ditive and subtractive signs of the result, is an advantage, however, 
which furnishes great relief to the attention during the operation, and 
which, to calculators who are not in the almost daily use of either the 
one or the other, will be duly valued as an important feature of this 
process. It may, moreover, often be abbreviated in practice by 
using only a few of the higher places of figures, instead of all 
