MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 733 



This proposition may also be put in the form " to find two numbers which 

 may diflfer by unit, and the sum of whose squares may be a square," 



Putting B = A + 1 in the equation C^ = B^ + A^ we have C^ = 2A^ + 2A + 1, or 

 2C^=(2A + 1)^+ 1 ; that is to say, we must resolve the indeterminate equation 

 D = 'n/(2C^— 1) in integers. For this purpose we put ^2 in the form of a con- 

 tinued fraction, and obtain the progression,— 





1 



2 



2 



2 



2 



2 



2 



2 



2 









1 



1 







1 

 1 



3 

 2 



7 

 5 



17 

 12 



41 

 29 



99 

 70 



239 

 169 



577 

 408 



985'^"- 



1 7 41 

 of which the alternate terms ^, ^, oq belonging to our equation, the interme- 



diates ^, ^, kc, belonging to the twin equation D = v'2C^ -I- 1 . The former 



follow the law (—1, 6), that is to say, if the progression of the numerators be 

 Di, Da, D3, . . . D„, D„+i . . we have D„+2 = 6D„+i— D„, and so also of the denomi- 

 nators ; the progression thus formed being, — 



-1 1 7 41 239 1393 8219 

 1 ' 1' 5' 29' 169' 985 ' 5741' ^' 



in which the denominator of each fraction gives a hypotenuse, while the integers 

 immediately above and below the half of the numerator represent the two sides. 



Eocample 2. 

 31. To construct a right-angled trigon in integers, such that its lesser angle 



may be the tenth part of a revolution. 



The tangent of 18°= 20^ is a/1 — |V5; or 3249 1969 6232 907, which, when 

 resolved into a chain fraction, gives the quotients, — 



3, 12, 1, 6, 1, 6, 25, 4, 1, 1, 15, 1, 1, 1, &c., 



whence the approximations, — 



3 12 1 6 1 1 



10 1 12 13 90 103 708 



&c. 



13 37 40 277 317 2179 

 which give the trigons, — 



10, 8, 6, or 5, 4, 3; 1513, 1225, 888; 1769, 1431, 1040; 84829, 68629, 49860, &c.; 

 the third of which gives an angle 36° 00' 30 ". 



Example 3. 



32. To construct a right-angled trigon in integers of which the lesser angle 

 may be nearly 23° 27' 26", the obliquity of the ecliptic. 



VOL. XXIII. PART III. y L 



