756 MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



BC=2A, CA=2&, AF=^ + g; ¥C=p-q. 



It is evident that we may introduce powers of the primes «, /5, 7^ and decom- 

 pose such a product as a^ /3^ 7 8* into two squares in two different ways. 



For example, if we take a = 5, /S = 13, we obtain 65 = 7^ + 4^ — 8'^ + V, whence 



BC=8, CA=14, AF = FB = 9, FC=7. 



If CF were produced to an equal distance, and the extremity of the produced 

 part joined with A and B, a rhomboid would be formed having its sides 8 and 14 

 with the diagonals 18 and 14 respectively; or, halving, the sides are 4 and 7, 

 with the diagonals 7 and 9. As 5 and 13 are the smallest primes of the class 

 4:71+ 1, we may infer that the above are the smallest dimensions of a rhomboid 

 having its sides and diagonals all integers. 



The subject may be viewed in another light thus : let the cosine of the angle 



st 



at F be denoted by the fraction — , in which one or both of the factors of the 



numerator may be 1 or zero ; while one of the factors of the denominator may be 

 unit ; then denoting AF by ^ and FC by / as before, we have 



a' = k^' + 2- kl+P = u^a;'' + 2st,v\ + v''X^ 

 uv 



uv 

 if we put 



k = ux, l = v\. 



Hence 



a--b'^ = (a + b) {a-b)=4sta;\ = isx . t\, 



which is satisfied on making 



a = 2sa; + ^ (\; b = 2sx — J i\ 

 which gives 



so that we must have 



M V + ^^2^2 = 4g2^2 ^ 1 ^2X2 . 



or 



(4s2-M2)^2^(y2_1^2')X2; 



that is 



{2s + u) . (2s-u) x^=v''\^-ie\\ 

 whence 



{2s-^u) + {2s-u)x'^=2v\=2l; 2ux=2k 



{2s + u)-{2s-u)x^ = t\; 

 SO that ultimately, multiplying by 2 to remove fractions, 



2a = A=4:sx + (2s + u)—{2s — u) x"^ 

 2& = B=4s^-(2s + M) + (2s-M)a;2 

 / 2k^^ = 2ux 



2Z=L (2s + m) + (2s-m)«2 



in which any values, positive or negative, may be assigned to s, m, and x. 



