Why does Holmes believe that Moriarty is responsible for doughlas murder?

Sometime after Douglas flees England, Holmes receives a cryptic message which reads only “Dear me, Mr. Holmes, dear me!” (319). From this, Holmes deduces that Moriarty, the sender of the note, has succeeded in killing Douglas. Watson and McDonald speculate as to whether or not Moriarty will ever be brought to justice. Holmes assures them that he will capture the Professor, but he needs time in order to do so.