The method of solution which I have the honour to lay before the Society is founded upon an old and well-known arithmetical process called “ casting out the nines.” It consists in substituting for any number the remainder which that number leaves when divided by nine. As this remainder contains only a single figure, whereas the original number may contain any number of figures, it is obvious that if the remainder can be employed instead of the number itself, a great saving of time and trouble must follow.