This is the ship that transported August and Rika (Knaack) Heiden and their two sons, Heinrich and Ernst and one daughter, Meta, to America on June 4th of 1873.  The voyage probably went from Hamburg, Germany to Southhampton, England to New York but we don't know for sure.

It was part of the Hamburg American Line of ships which was prospering at the time due to the heavy volume of Germans emigrating to America.

The steamship Saxonia was what they called an iron screw steamer. She was 312 feet long and 42 feet wide at the foot beams. As you can see by the picture, it had one smoke funnel and three masts. The ship was retired from trans-Atlantic trips later that year.

So far we have been unable to track this specific trip of the Saxonia, but earlier trips that year also went from Hamburg to Le Havre, France and then to New York. Those trips took about 21 days and were under the command of ship's master, I. J. Meyers.
 

S.S. most often refers to a Steamship. Occasionally, it was attached to a Ship Under Sail.
 

District of New York,  Port of New York 

I, C.D. Brandt, Master of the German Steamer Saxonia do solemnly, sincerely and truly swear that the following List or Manifest, subscribed by me, and now delivered by me to the Collector of the Customs of the Collection District of New York, is a full and perfect list of all the passengers taken on board of the said, steamer Saxonia at Hamburg, Germany from which port said _________ has now arrived; and that on said list is truly designated the age, the sex, and the occupation of each of said passengers, the part of the vessel occupied by each during the passage, the country to which each belongs, and also the country of which it is intended by each to become an inhabitant; and that said List or Manifest truly sets forth the number of said passengers who have died on said voyage, and the names and ages of those who died.

Sworn to this 4 June 1873

List or Manifest OF ALL THE PASSENGERS taken on board the German Steamer Saxonia whereof C.G. Brandt is Master

 No.        Names                Age        Sex        Occupation

303     August Heiden       36         Male       Mason

304       Rica                   31         Female    his wife

305       Heinrich               6         Male       his children

306       Ernst                   4         Male

307       Meta                   9 mos.   Female    his baby

[Information from: Noel Reginald Pixell Bonsor, North Atlantic Seaway; An Illustrated History of the Passenger Services Linking the Old World with the New (2nd ed.; Jersey, Channel Islands: Brookside Publications), vol. 1 (1975), p. 388.]

The SS Saxonia was built by Caird & Co, Greenock, Scotland, for the Hamburg American Line, and launched on 21 August 1857. It was the first of three ships of this name owned by the Hamburg America Line. She was rigged for sail and was one of six sister ships, the others being "Hammonia", "Borussia", "Austria", "Bavaria" and "Teutonia".

• 2,684 tons, 95 x 13 meters (311.7 x 42.6 feet, length x beam); clipper bow, 1 funnel, 3 masts; iron construction, screw propulsion, service speed 10 knots; Passenger accommodation: 60 in 1st Class, 120 in 2nd Class, and 450 in Steerage-Class.

 It was chartered by the British government as an Indian Mutiny transport.

• 1 April 1858, first voyage, Hamburg-Southampton-New York.

• 5 October 1873, last voyage, Hamburg-New York
  (subsequently ran Hamburg-West Indies).

• 1879, sold to the Russian Volunteer Fleet and renamed the NIJNI NOVGOROD (Russian Volunteer Fleet).

• 1895, scrapped

[Pictured in Michael J. Anuta, Ships of Our Ancestors (Menominee, MI: Ships of Our Ancestors, 1983), p. 300, courtesy of the Peabody Essex Museum, East India Square, Salem, MA 01970.]