14 REPORT 1872. 



in faff ( — ^1=3+ — ^rr^, indicating that ^^ is nearer to VS than am/ rational 

 in lact ^^209/ (209)^ ' ° 209 "^ 



fractior whose denominator is less than 209. 



On Cube Hoots. 



It is plain, as hefore, when vi is a positive integer, that (N^— a)'", when 

 expanded by the binomial theorem, has always the form AN^-f BN* + C, since 



(N^)'=N*, (N5)'=N, (Ns)* = N.N^ (N*)'=N.N*; &c. 



Now take a, as before, by trial or guess, so that N? — a shall be a small fraction /<:5 5 



.-./'"= (N*- a)™' = AN? +BN*+C ; 

 and for another integer m' we have 



/"^'=(N*-fl)'"'=A'N^+B'N^+C'. 



Now, by eliminating N* from these two equations, we easily find 



N*=(A'C-AC'+A/"''-Ay"') -f-(AB'-A'B) exactly, 



and therefore =^C+AC_ 



AB'-A'B •' '' 



Ex. gr. Taking m=-) and m'=7, we readily find 



^i_ 7N^+105NV+120Na" + na" . ,.ery nearly • 

 ^ " N^'+eONV+UJNa^+Sg^ ' "" ^^^^^^^^^^3^' 



so, if a^=N4-D, then 



To find v/29, take a = 5; then D = a^— N, .-. here =— 2 ; and then our last for- 

 mula gives 



. . -29* 8Z(?iZ)ltia^^L_ . 12=3-0723168, 



-^-87(488)^+141x4x27+8 ' 



which is correct to its last decimal figure 8. 



1 

 Hasp Demonstration of Dr. Jlutton's elegant formula for approximating to N", 

 with an important' liemark or Estimate of the degree of accuracy attained by 



means of its use and apjjlication. 



I 



Let a be the assumed near value of N", whose exact value is =a+a; j then, as 



N must 



, , -n -,i 1 n-i . n(n—l) ,,_ o „ n(n — l')(n—2) ,,_■, , „ 

 = (a+x)"= «"+««" i»+-^-2 — -a -x''+— — 2^ \' V &c., 



• :c- N-g" ^ . 



,,_i n . n—1 .,_o „ 

 na + — o — a "*+ &c. 



■\r n 



and as x is very small, therefore it is nearly = — _ . By substituting this value 



for the first power of x in the preceding denominator and omitting the subsequent 

 terms therein containing x-, x^, &c., we now find, more nearly, 



x_ 2(N-a") 2(N-Q") 



« 2wa"+(«-l)(N-a") (»-l) N+(«+l)a'' 

 and thus the corrected root a+a? = a( 1+?) comes out 



