56 J. B. Luce on the Theory of Numbers. 



2a 



£- = m, is 



1 



mdti 1 



2a±l 



m± 1 



2a±JL 



m± &c, ad infinitum, where the 

 square root of any number n, either integral or fractional, is trans- 

 formed into a continued fraction, and may be converted into a 

 series of converging fractions as follows, 



l. 2. 3. 4. 6. 



a am±:l 2a 2 m±3a 2a 2 m 2 ±:Aam + l 4a 3 m 2 ±10a 2 m-|-5a 



-,&c. 



1' m ' 2am±i\ ' 2am 2 ±2?n ' 4a*m 2 ±:Gam + l 



2a 

 If instead of m we supply its value -7-, we shall have the fol- 

 lowing converging fractions : 

 l. 2. 3. 4. 5. 



a 2a 2 ±:b 4a 3 db3a& 8a*±8a 2 b + b 2 16a 5 ±20a 



V 2a » 4a 2 ±6 ' 8a 3 ±4a6 * 16a 4 ±12a 2 6 + 6 



— , &c. 



Now, if in the equation x 2 — ny 2 —z\ we assume #— the 

 numerator, and y=the denominator of successively the first, 

 second, third, &c. fraction, according as i = 0, 1, 2, 3, we shall 



have a 2 -rc==F&; (2a 2 ±6) 2 - n x4a 2 = b 2 ; (4a 3 db 3a&) 



n(4a 2 ±6) 2 = =t 6 3 , &c. Thus when i is an odd number, every 

 odd power of z shall be positive if n~a 2 - b ; and negative if 

 n = a 2 -\-b. 



Let us apply this theorem to some examples ; and in the first 

 place let it be proposed to find the expression of a right angled 

 triangle, or solve the equation x 2 +y 2 =-z 2 . 



a 2 a 2 -l-8 2 



Here n= — 1. We may assume -1 = „ 2 ^j — . which 



gives a =^, and & = — p— . By placing these values instead of 



2a 2 —b - 

 a and b in our second converging fraction — ^— , we obtain 



2~p, and the numerator « 2 - r ?2 =^, and the de 



2a 



=#, and the denominator 2«^=y» 



where « and /? may be any number assumed at pleasure. If we 

 assume m* - 1, we find ar=2« -t t p --2«(« - 1), and taking suc- 

 cessively « = 1, 2, 3, 4, &c, we obtain the following series, 



x = 3, 5, 7, 9, 11, 13, &c. 



y = 4, 12, 24, 40, 60, 84, &c. 



