75 



Various solutions of the equation 



have been given. The following appears to be rather more 

 general than any I have seen : — 



Let (p^ -PQ + g^) (^ + I/J = ff- -rs + s^) (w - z), 

 (u + ii}V ) (r+ (osj (x + w//j = (xi + iM-v ) (p + la'-q) (w — urz). 



From these equations, observing that 



x + y+ui(x + (siy)+ to- (x-\-or>/) = 0, 

 we obtain the solution, 

 x = (r^ — rs + s^) [u^(2pr — qs — qr — psj + 2uv (2qs — qr — ps — prj 



+ v^ {2qr + 2ps — pr — qs)'\ + (u~ — uv + v^) (pP- —pq + q^)^, 

 y = (r^ — rs + s^) [ii^(pr+qs — 2qr — 2j)s)-{-2uv (qs + qr + ps 



— 2prJ + v^ (qr + ps -!- pr — 2qsJ] - ( u- — uv-\- v^J fp~ — pq 



z = (p2 —pq + q-) [«2 (^_pr— qs-^ 2qr + 2ps) + 2nv ( — qs-qr 



— ps + 2prJ + V- ( — qr — ps - pr + 2qsJ] + (u^ — uv + v^) 

 fr^ — rs-\- s^Y^ 



w = fp^ — pq + q^) [n- (2pr — qs — qr — p^ ) + 2uv {2qs - qr — ^j.s 



— prJ + v^ (2qr + 2ps — 2?r-qs)] +(u- — uv + v-) (r~ — rs 



Particular cases are 



33 + 43 + 53^53^ 13 + 63^83^36^ 20^ + 543 + 79^-87^ 

 343 + 393 + 653 _ 723^ 863 + 953 + 973 ^ 1343^ 

 93 + 103 ^ 13 + 123 26'' -4- 36^ = 173 + 39\ 



