66 ON THE ALGEBRAICAL SOLUTION OF 



number, 7, which is of the form 2 + 36 2 , since 7 4 =49 2 = ( 1 2 + 3-4 2 ) 2 , 

 we have 



i.e. j*+if+(x+y)*=(-~ y ) +(^p2 } +(^pM (3) 

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



and "JI ^respectively (or by x ~ ^ and - x ~ & respectively, 



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

 the next repetition gives 



I 



+ 



72 / y 72 ) y 72 



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 



55|/+39a;\ 4 , /39y- 16x\ 4 ,, 



72; +( v 72; 



Thus for example if we put in (3), (4), (5), (6), x=l, 2/=3 we 

 obtain 



49 4 + 147M- 196 4 =84 4 + 119 4 + 203 4 =9 4 + 172 4 + 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 



