318 Review of the Cambridge Course of Mathematics. 



a denominator too great, the fraction joined to 4 will con- 

 sequently be too small, and the last denominator being too 

 small, will render the last fraction too great. 



By reducing, at first, 9| to a fraction, we have — ; ~ 



will be then y\, and the approximate value will become 

 1— L_ ; now — !— gives |f, which joined to unity becomes 



m, or f I for a fourth approximate value of VW' 

 Resuming the expression, 1 — - — 



2 I, we divide the two 

 terms of the last fraction f by 5, and obtain l-L, and 



l-J— H 



4-1- 

 9— i_ 



2— !— 

 1 I ; neglecting the fraction |, there will remain 



1— !— 



4-i- 

 9 1 



2 T ; and we see as above, that this value is smaller 

 than the true value. 



The fraction -!- reduces itself to \ ; and since the pre- 

 2| 

 ceding — gives /g' ^^^ next preceding becomes — L_ , equal 



9i '*28 



to yYs ; so that the fifth approximate value is lyVs or 

 ill. 



Dividing, finally, by 4 the two terms of the fraction | 

 which was neglected above, we have for a quotient -L; and 



by suppressing the fraction \, we obtain the new value 



1-1- 



4^L_ 



9— !— 



1 i, greater than the true value. If we reduce, suc- 

 cessively, all the denominators to a fraction, to obtain the 

 simple fraction which it represents, we shall find 1 yVg or 

 f ||. By restoring the fraction | to the side of the last 

 denominator, we form the expression 1— !— 



2-i— 

 1— L_ 

 1 J, which be- 

 ing reduced as the preceding, reproduces the fractional 

 number VW» 



