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XLV. On a Fundamental Rule in the Algorithm of Continued 

 Fractions. 'By J. J. Sylvester, JF.R.S* 



a &c. be any continued fraction, and let 



the successive convergents — , , &c. be called^, *, &c, 



and let D f be denoted by (a v a 2 , . . . ^t); then the following iden- 

 tity obtains which I regard as the fundamental theorem in the 

 theory of continued fractions, but which I have never seen stated 

 in any work where this subject is treated. 



Theorem. 

 («]... a m ) x {a m +i . . . a m+n ) + (a x . . . a m -i) x (a m+2 . . . a m + n ) 



= {a 1 . . . a m a m+1 . . . a m+n ). 

 Corollary 1. 

 (ff, « 2 . . . a m )x(fl 2 « 3 ...« m +0 — (« 2 %...« m )x(^ ff 2 ...a TO+1 ) 

 = (-)-. 1. 

 This is the well-known theorem 



which, however, is only a case of a much more general theorem 

 easily deduced from the fundamental theorem given above. In 

 fact, we may derive immediately from the latter, the equation 



(a x a 2 ...a m ) . (a 2 a 3 . . . a m+i ) — (a 2 a 3 . ..a m )x(a 1 « 2 . . . a m+i ) 



= ( — ) m . (a m+i a m +i-\ ... to 2—1 terms). 

 Hence 



D m _ 1 .N m -D WJ .N w _ 1 = (-) w .l 



D OT _ 2 . N m — D OT .N TO _2 = ( — ) . a m 



D m _ 3 .N ni -D m .N w _3=(-) m K.« m - ] + l) 



Dm-4 • N 4 — D TO . N TO _4=(— ) m (a m . a m -. v .a m -.<2 + a m + a m-2)t 



&c. &c. 



Corollary 2. 

 («,...«, a p+l ...a p+f ).( ai ...a p a p+l ...a p+ J-fa...a p a p+l ...a p+g ) 

 . {a x ...a p a p + l ...a p+h ) 



=(-)^{(Vi"V/)*(Vi*'V~(Vi'"W'( ff P+r" fl /»+*)}' 



Sub -corollary. — If all the several quantities a v a 2 , a 3 . . . are 



equal to one another, the quantity Df.D k —D g .D h is constant 



* Communicated by the Author. 



t It is essential to notice that («i a 2 . . . ^.^(a^ai-i . . . a^. 



