72 



(in which 0^—0) by putting tan J B^ = q^!p^. We thus obtain 



X^=2^-'p^q^P^q^ . . . P^-,qn-APn-2-9n~^){Pn-l + 9n-l) 



a-n = (P,' - Ql') (P2' + ^/) (p.' + ^3') . • • ^Pn^l + ql-x) 



u = (p^^ + q^^) (p^^ + q^') . , . ip^U + gn-^) 

 A particular case when n< = S is 



5^+6^ = 652-482. 



An evident solution of 



x^ + ^^ = z^ + w^, 

 is x = pr — qs, z = ps — qr, 



y = qr-\-ps^ w = qs-\-pr. 



And a similar solution of 



a?,2 + ^^2 _j_ ^^2 ^ ^^2 ^ _y^2 ^ y^2 ^ ^^2 ^ y ^2 ^ 



derived from Euler's expression of the product of two sums 

 each of four squares as the sum of four squares, is 



Xx~PP + qi' +^^' + ^^' 1 Vx—PP' + Q.q' +rr' —ss', 

 ^2 — q^' — q.'''' + ps' —p's, y^= -qr' -{. q' r -^ ps' +p's, 

 x^=rp' —r'p+qs' —q's, y^= —rp' + r'^+ qs' + q' s, 

 x^ = pq' —p'q + rs' —r's, y ^— —pq'-\-p'q + rs'-^r's. 



When we pass to equations of degjfee higher than the 

 second, there are very few cases which admit an easy solution : 

 perhaps the simplest is the equation 



x^ -\-y^ + z^ — ?>xyz = w^. 



If w^=l, o) + w2=— 1, a solution is obtained by putting 

 w = p (r^ — rs + s^), 

 x + y + z=p^^, 

 x+(x)y+ (x)^z = (r + o)sJ^. 



In particular if n = 2, 



x = p {3p + 2r+2sJ + r^, y = p (3p + 2r +2s) + 2rs, 

 z = p (3p + 2r + 2s J + s^ , tr-^Sp (r^ — rs + s-) -\-r^ + s^, 

 satisfy the equation 



x^ + y^ + z^ — ?>xyz = w"^ . 

 And when n — Z^ 



x^q^r^A^s^^ y = qj^2>r^s, z=^q + 2>rs^, 

 w = 3p fr^ — rs + s^) + r^ + s^, 

 where Q = ^P ^ ^P^ + ^P ('>' + s) + (r + sp [-, 

 satisfy 



x^ -^y^ + z^ — Zxyz = iv^ . 



An interesting particular case is 



233 + 243 + 253 - 3.23.24.25 = 6\ 



