328 Mr. J. W. L. Glaisher on a 



merators alone, the equations given by the lemma are, from (i), 



31 = 12 + 19, 



19=12 + 7; 



whence 



Similarly, from (ii). 



31=2.12 + 7. 



29=12 + 17, 

 17 = 12 + 5; 



29 = 2.12 + 5, 



whence 



and therefore 



31 + 29=4. 12 + (7 + 5)=4. 12 + 12 from (iii), =5.12. 

 And similarly, for the denominators, from (i), 



974 = 377 + 597, 597 = 377 + 220; 

 from (ii), 



911=377 + 534, 534 = 377 + 157; 

 from (iii,), 



220 + 157=377; 

 whence 



974 + 911 = 5.377. 



12 



Also, the numerators 7, 5 of the convergents of ^7 are ^ ne 



remainders when 31 and 29 are divided by 12 ; and the deno- 

 minators 220, 157 are the remainders when 974 and 911 are 

 divided by 377. 



The following example, also taken from the same series, 

 affords another illustration of the relations connecting the frac- 

 tions : — 



20 47 27 61 34 41 48 55 62 7 

 303' 712' 409' 924' 515' 62l' 727' 833' 939' 106* 



§ 6. The reasoning of § 3 shows that the two properties 

 are still true in the more general case in which the series con- 

 sists of fractions in their lowest terms, having numerators not 

 exceeding any given number m and denominators not exceed- 

 ing any given number n, m being of course not greater than n. 

 For consider only the second property (which includes the 

 first), and suppose this property to be true for the series of 

 fractions having numerators not exceeding m and denomina- 

 tors not exceeding m + r. Introduce the fractions having nu- 

 merators not exceeding m and denominators equal to m + r + 1 ; 

 then, as in § 3, the property is true for all the fractions up to 



