66 ON THE ALGEBRAICAL SOLUTION OF 



mimber,7, which isof the forma 2 +36 2 , since 7 4 =49 2 =(l 8 +3-4 2 ) 2 , 

 we have 



\ 

 (3) 



Hence it is clear that we may replace x and y by -^-- 



and x ~t y respectively (or by x ~ y and - x ~ y respectively, 



etc.) without altering the value of the left-hand side. Thus 

 the next repetition gives 



/65a;+39y\* /39z-l%\ . 



~7 2 ~~7 + ( V~) ' 



and the process may be continued ad infinitum. Also we 

 observe that we may interchange x and y without altering the 

 left-hand side so that we have also 



72 



Thus for example if we put in (3), (4), (5), (6), x=l, y=3 we 

 obtain 



49 4 + 147 4 + 196*=84 4 + 119 4 + 203 4 =9 4 + 172*+ 181 4 



=28 4 + 161 4 + 189 4 =101 4 + 103 4 + 204 4 . 

 Cor. 2. Changing the sign of y in equation (3) we get 



