A curious arithmetic formula deduced by Berndt from an analytic formula of Ramanujan is proved arithmetically and used to prove the formulae given by Liouville for the number of representations of a positive integer by the forms x 2 + y 2 + z 2 + t 2 + 2u 2 + 2v 2 and x 2 + y 2 + 2z 2 + 2t 2 + 2u 2 + 2v 2 .