VOL. XVIII.] PHILOSOPHICAL TRANSACTIONS. 647 



or — — \/ — — — — ; that is, in the present instance, e = 



2007 J — v' 3761406* .... . „„« , . rr,, 



— r-r -, whence the root comes out 9,880, nearly true. Ihen 



substituting this for a 2d hypothesis, there arises a + e = z = 9,8862603936495 

 ve ry accura te, scarcely exceeding the truth by 2 in the last figure,* viz. when 



^ ^ = e. And even this, if necessary, might be still further 



corrected, viz. when it is + e by subtracting '^ '^ =, or if it be — e by 



adding -7===, from or to the root before found. Which compendium is 



so much the more valuable, that sometimes from the first supposition alone, 

 but always from the second, the calculation may be continued as far as we please, 

 keeping still the same coefficients. It may be observed that the proposed 

 equation has also a negative root, which is z = — 10,26, &c. as may be found 

 on trial. 



Exam. 2. Suppose z^— 17z2 + 54z= 350. Then taking a = 10, and 

 proceeding according to the rule, 



2"=: c^ +3aae + 3aee+e^ 



— dz^ = — da^ — Idae — dee 

 -\- cz = ca -\- ce 



b X t 



That is, +10004-300e + 30ee + e* 



— 1700— 340e — I7ee 

 + 540 +54 e 



— 350 



or— 5]0+14e +13ee+e^=rO. 



Now since it comes out — 510, it appears that a was assumed less than just, 

 and consequently e is affirmative ; hence from 510= 14e+ 13ee, there arises 



= e = jg ; which gives z = 1 5,7, which is too 



much, because a was taken wide of the truth. Therefore, for a second 

 supposition, take a = 15 ; then by the like process we obtain e = 



-*"^t''"'' = '"^"af ''"S and therefore z = 14,954068. And if 

 the calculation were repeated a third time, we should find the root true to 25 



* Notwithstanding this declaration however the number b erroneous, as the true figures ought to 

 be 9,88600270, &c. 



