F R A 



in very large numbers which are prime to one another, as by their means 

 we may find an approximate value in less terms. 



To represent y in a continued fraction. 



Divide as in the rule for finding the greatest common measure, thus 

 ft) a (p a I 



__ 



d)c(r 9+ r 



e)d(s T+-J&C. 



&c. &c. 



The first approximation is p, which is too small, the next p 4. - , 



1 9 



which is too large, the next p + - p which is too small ; and thus 



t + 7 



\ve may form a series of fractions, each succeeding one being nearer the 

 true value of the proposed fraction than the one which preceded it. 



This series of fractions requires some trouble in their formation after 

 the first two or three ; but the 3d, 4th, &c. may be expeditiously found 

 thus. Arrange the figures of the quotients in a line, as 



P) q, r, s, t, &c. let the successive fractions be -~, y, , , , &c. then 

 to find any of them after the 2d, as -^, we have - = ^ ! 1^ J j = 



*g+._ ; ~ = - !t-, &c. where the law of formation is evident. 

 s h ~\-f n tl -f- n 



Ex. To approximate to J,'^ , proceeding as if finding the greatest 

 common measure we have for the quotients 

 3, C, 1, 1, 2, 1, &c. 



1 1 19 



Now first approximation p 3 ; 2d. = p -f = 3 -f ~ = , ,*. 



vve have by the rule the following series of fractions 



3, W f , I ^L, * where 3 is too small, too large, &c. 



FRACTIONS vanishing. 



If u , where P and Q are functions of x, which are both = o, 



Q 

 when x a, then the value of u, in this case, is the same as the values 



. d P ds P d* P . 

 in this case of ^-, ^, ^, &c. 



125 G4 



