If all the values of x and y given by $$ ax^{2}+2hxy+by^{2}=1, a'x^{2}+2h'xy+b'y^2=1 $$ (where $a, h, b, a', h', b'$ are rational) are rational, then $$ (h-h')^2 - (a-a')(b-b'), (ab'-a'b)^2+4(ah'-a'h)(bh'-b'h) $$ are both squares of rational numbers. (Math. Trip. 1899)
Note: the first case is trivial, the second is more difficult, ill be granted if someone could tell me from where Hardy and collaborators get this miscellaneous problems, to be specific the problems with the reference (Math. Trip. yyyy).