MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



747 



If in this we suppose « = a, J = /3, we obtain 



A'D=G = f3^+2a^ eosS + a^ 



GD=A=2a (13+ a cos 6), 



Wherefore, if in a trigon A7/3, the two sides /3, a, be denoted by integers, 

 while the cosine of the angle A7/3 is rational, the three sides of the trigon AGD 

 having an angle GAD double of 7A/3, will be commensurable. 



The subject may be presented in another light, thus — 



Let the cosine of the angle 6 be denoted by the rational fraction -, then c, h, a, 

 being the sides of any trigon having 180°—^ for the angle opposite c, we have 



V 



2s §2 ^2 _ g2 



= b'^ + —ah+ -g-x^ + —To — a^ 

 t V r 



= {h+-^a? 



f-s" 



or, 

 whence 



(tc + tb + so) {tc — ib — so) = {t + s){t — s) OD^y"^, 



if we put ocy (either of which may be unit) for a. Decomposing the latter 

 member into factors prime to each other, we may put 



tc + tb + sa =(t + s)o(P 



tc, — ih — sn, 



which give 



tc — tb — sa ={i- 



-s)2/^ 





C=2tc=(t + s)x^ 





+ {t-s)y^; 



B = 2tb=(t + s)a;^ 



—2sxy 



-(t-s)y^; 



A = 2ta = 



2txy; 





in which a; and p may be any two numbers whatever ; it is needless, however, to 

 assume them with a common divisor. 



When the three sides BC, CA, AB, of a trigon, having the angle ACB equal 



