380 



is not indicated in X y = 0, real roots of X 0 = 0 (if such exist) may 

 separate. 



4-10 



2 



+ 9 

 - 4 



- 3 

 2-5 



+ -25(-5 

 - -25 



~S 

 2 



~~ 5 

 - 3 



- -5 

 1- 



0 



2 



~~2 

 - 2 



•5 





~± 

 2 



0 







~2 









Here we see that *5 is a root, not only of X 2 = 0, but also of X 0 = 0; 

 and that there is no indication of imaginarity in X x = 0. A real root 

 of this equation is, however, passed over; and it further appears that, 

 for the transformation '5 - S, one variation, in the entire series of signs, 

 would be lost — the signs for this transformation being + - + + -, as is 

 obvious. Hence, one real root of the proposed equation X 0 = 0, lies 

 between 0 and *5 ; a second real root, as just seen, being b itself. 

 Again, diminish the roots by - 7. 



4-10 +9 -3 + -25 (-7 



- -1596 



2-8 



- 5-04 



2-772 



- 7-2 



396 



- -228 



2-8 



- 3-08 



•616 



- 4-4 



~~88 





2-8 



- 112 





- 1-6 



~24 





2-8 







~p2 







0904 



As before, one root, and one root only, of X x = 0, is overstepped ; but 

 there is indication sufficient that the other two roots of this equation — 

 and, therefore, two roots of X 0 = 0 — are imaginary. Hence, the equa- 

 tion has two real roots, -5 and [0, *5], and two imaginary roots. The 

 approximate value of the latter of these two real roots, found in the 

 usual way, is -12256 : it is somewhat more expeditiously found by em- 

 ploying for the purpose the depressed equation of the third degree, 

 4X 3 - 8x 2 + 5x - '5 = 0,* as given by the first row of coefficients in the 

 former of the two operations above. 



* We need scarcely remind the reader that the first member of this equation is the 

 quotient arising from dividing the first member of the given biquadratic equation by 



