S 



1821.] Finite Values of Circulating Decimals. 17 



» = I D + (5F + &T + K7 * R 



Generally 



= vt; tt; • (cor. prop. 1). 



10" — iQ»- m v t i > 



Where 



S represents a given multiple circulate, 



n the distance of the last figure of the repetend from unity, 



m the number of figures that repeat, 



R the common numerator. 



It is plain from inspection, that m cannot exceed n, but it may 

 be equal or less ; therefore, when m is equal to n, then will S 

 it 



— io» - r 



Problem^. — To find the finite value of any mixed circulate. 

 0,34 = i 3 o + S =7b^ + S > 

 °- 3 ° 4 « = To°o + S = i£i + S > 



1 • 4fi 4fi 



°-° 4672 = l£> + S = 1^ + S > 

 46-3 = i 6 + S = T iL i + S, 



238-004 =r^ + S -^+S 



100 T ° — 1003-> ^ ' 



374-2358 = ^ + S = ^ + S. 



Generally, A = lQn _- + S. Where A represents any mixed 



circulate, and N its finite part. By Problem 2, S = — — ~ 



r j > 10"— 10"-'" 



*U f A N R N(10-» - 1) + R 



therefore A = — — -+- — — = — —— (a) 



10"-"' 10" — 10*- m 10 n — 10"-'" v ' 



Since circulates that begin in the integral part may be reduced 

 to pure ones by dividing by 10, 100, 1000, &c. according to the 

 situation of the decimal point, the quotients thus produced may 

 be found by equation (a), which being multiplied by 10, 100, 

 1000, &c. (the multipliers agreeing with the clivisers) will give 

 the finite value of the mixed circulate proposed. See the fol- 

 lowing examples : 



New Series, vol. n. c 



