/On extracting the Roots of Equations. 1 77 



in x—i the equation in i' — 4 has stUl the same number of 

 changes of signs as that m ^— 1, viz. one change ; but in pass- 

 ing from the equation in :r— 4 to that in x— 5, that change of 

 signs has disappeared ; we therefore conclude that one or three 

 roots of the original equation in x lie between and 1, and 

 that another of the roots lies between 4 and 5. 



Therefore, if all the three first roots are real, zero will oc- 

 cupy the unit's place of each of these roots, and the first figure 

 of the other root is 4. 



As an exercise of making a research of the roots in the place 

 of tenths, the following example is given. 



• Exaynple 1 . — Let it be required to find the initial values of 

 the three roots of the equation a* — 5a^ + 3x- — 9jr + 4 = 0, which 

 lie between zero and unity. 



X, 1-5+ 3- 94- 4(0-1 



— 49 



— 48+251 

 -47+203-8749 



jr-0'1, 1— 46+156-8546+3I251{0-l 



—45 



-44+111 

 -43+ 67-8435 

 x-0-2, 1—42+ 24-8368+22811(0-1 



X— 0-3, 



"-37 

 -36-133 



— 35 — 169 — 8575 



X — 0-4, 1—34 — 204 — 8744+5856(0-1 



— 33 

 -32-237 



— 31-269-8981 



X— 0-5, 1-30 — 300 — 9250—3125. 



In passing fi-om the original equation in x to the equation 

 in X— 0-2 four changes of signs still remain; but in passing 

 from the equation in x—0'2 to that in x—0-3 there are only 

 two changes of signs ; therefore two roots, real or imaginary, 

 are contained between 0-2 and 0-3 : therefore if these two roots 

 are real, the initial figure of each of them is 2. Again, in 

 passing from the e(|uation in x—0-3 to that in j: — 0-4, the two 

 changes of signs still remain ; but in j)assing from the equation 

 in JT- 0-4 to that in jt- 0*5 one of these clmnges has disap- 



Vol.60. No.293. 6^. 1822. Z peared; 



