Several well-defined cases of yellow fever outside of the Quarantine limits have occurred at Staten Island since the middle of July.

Taking the report of the Medical Society of Richmond County, in connection with the statement of Dr. Watson, the Assistant Physician of the Marine Hospital, into account, and rejecting the flood of merely controversial letters which have passed from one to the other on the subject, the following appears to be the true state of the facts: Yellow fever-cases have occurred outside of Quarantine at three localities only.

First, in a block of shanties close by the beach and a short distance below the Quarantine. Second, in the Selira House, directly on the beach, about half a mile below Townsend's Dock; and, thirdly, at New Brighton, in two dwellings on each side of the Post Office building, also very close to the beach. In the first locality there have been twelve cases and four deaths. The first case was reported on the 17th of July. In the second, two cases and two deaths, the subjects taking ill on the 11th and 13th inst. In the third, three cases and one death; they were taken ill on the 13th and 15th inst. Total, seventeen cases and seven deaths. The source of the disease in the first and second localities is distinctly traceable tot he infected shipping within the Quarantine anchorage, but the source of the malady in the third locality is wholly a matter of conjecture. Nearly two miles intervene between each case, and aside from these three points there have occurred no cases of yellow fever on Staten Island this season.

The above is substantially the statement of Dr. Walser, of the Marine Hospital, who has given his personal attention to each case. The Medical Society's report, however, makes out in all, 22 cases and 12 deaths. Mr. Kramer, who had been employed in burning infected bedding, sickened about the middle of July, at his residence in Tompkinsville, and died in the hospital. It was stated that Kramer brought to his wife articles of clothing intended for the flames which she was in the habit of washing on the premises. Mrs. Kramer was attacked on the 17th of July, of yellow fever, and died at her own residence.

A stevedore's wife named Nell, opposite and near the same house_Mrs. Nell and Mrs. Kramer were on intimate terms took the fever and died on the 19th of July. This case was not considered one of yellow fever by the Quarantine authorities. Mr. Halliday, who owned the house occupied by Nell, who also lived near by on the same lot, was seized on the 23d and died on the 27th of July. A boy in the same house took the disease but recovered. Two other cases occurred on the 3d of August, a Mr. Young and daughter, both of whom recovered. A German tailor, living immediately on the water at the end of Minthorne street, within 100 feet of Kramer's house, sickened but recovered. His son was taken ill on the 21st of July, and is still under treatment. Mrs. Finnenty living in the same block was seized with he fever on the 2d inst., but recovered. On the 18th, Mr. Miles' daughter, living on the same block sickened; on the 20th another daughter, on the 21st Mr. Miles himself, and on the 23d his wife all now under treatment, and one in a dying condition. Mr. Holland took the fever on the 23d and his death is just reported. on the 8th and 11th of August two cases occurred at Silvia's Point, one and a half miles from the above locality, in a house situated immediately on the water. The Victims were Miss Cross and a servant girl. Miss C. Undoubtedly contracted the disease by visiting the shipping on livers and sundry occasions. On the 18th of August Mrs. Quinn, living between Townsend's Dock and the gas works, took sick, she is reported at death's door. Her residence was half a mile from the house of Miss Cross. Two cases occurred at New-Brighton on the 12th inst., in a building situated near the landing. One of these was transferred to the City, at No. 16 Greenwich street. Her companion at Bridgeport has recovered. Mr. Block took the disease on the 15th inst. He lived at the corner of Jersey street and Richmond Terrace; he died on the 19th. This case was pecuniary marked. His wife sickened at the same time but has recovered.

The manner in which each person contracted the disease is speculated on by different parties according tot heir favorite theories. Some were exposed to night air, others inhaled the smoke of the burning materials from the iron scow, while others still bathed along the Quarantine shore.