The Analyst

Sandbox heating was used also in Sweden, but later than in Finland. Instead of boxes they used cyliders. The history of Swedish State Railways: http://runeberg.org/sj50/3/0217.html "Already in 1864, attempts were made to heat up some passenger cars in the express trains, and the following year, heating of all 1st class compartments was introduced in these trains. Passengers in the other carriage classes as well as in trains other than express trains still got for quite a long time time to renounce this advantage. The restriction to only the most expensive wagon class was based on the cumbersome and costly manner in which the heating was effected at that time. The first system for train heating used by the state railways consisted of loose heating cylinders made of sheet metal, which were filled with heated sand and which were inserted through a hatch in the carriage wall under the sofas in the compartments. It is clear that this system would suffer from a lot of shortcomings. So, of course, it was impossible to maintain a reasonably even temperature in the compartments, for from being high, as long as the sand cylinders were still hot, it sank as far as they cooled. Admittedly, a control device consisting of a pair of openings and sliding dampers is used sheet metal sheaths, in which the hot sand cylinders were inserted, but the possibility of regulation was in any case very incomplete. The heating also depended on whether hot sand cylinders could be obtained, which was impossible in many cases when heat had been more desirable than usual, such as when the trains were delayed or even stuck in snow drifts. Furnaces were required to heat the sand, and as early as 1865 there were such built at the stations in Stockholm, Katrineholm, Örebro, Falköping, Gothenburg, Jönköping, Alfvesta and Malmö." Later on the heating system was introduced even in lower classes. At least on the Nora-Karlskoga railway: A 3rd class coach of the Nora-Karlskoga railway in 1872, with heating cylinder hatches. https://www.jarnvag.net/vagnguide/historia-vagnar

I wonder if this charcoal burning under-the-seats heater system was the one developed by Alwin Nieske of Dresden. He developed a chracoal brick burning stove, which didn't produce smoke, but only carbon dioxide, as he stated. Actually the stove produced also carbon monoxide, which lead even to fatalities. Nieske's firm produced i.a. heaters for railway coaches.

Also charcoal bricks have been used as heating material for heating boxes under the seats. Hammer G. (1908) Heizung und Lüftung der Wagen. In: von Stockert L.R. (eds) Fahrbetriebsmittel. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-34001-1_14: https://link.springer.com/chapter/10.1007/978-3-662-34001-1_14?noAccess=true With compressed coal heating, individual compartments were heated with heater boxes, located under the seats and operated from the outside through small doors in the long sides. As fuel pressed charcoal bricks were used, made of powdered charcoal, nitrate of potash and starch as binder. The bricks were burned in boxes of perforated sheet metal or wire mesh. The heater boxes were usually under the seats. In compartment cars they usually ran the full length of the compartment and could be served from either side; in through cars they were about the length of the longest bench seat.

The hot-water-box heating was used already in 1858 in the Royal Train of the Kaiserin Elisabeth-Bahn, KEB (Empress Elisabeth Railway) in the Austro-Hungarian monarchy. The heating boxes were located under the seats. They had to be changed every 4 hours. The system was used till 1871. This information is found in the booklet 'Der Hofsalonwagen der Kaiserin Elisabeth', 2002. As the Finnish railway pioneers were activly searching for solutions for the future rolling stock, they most probably were aware of the heating box systen, and included it in the specifications for the composite 1st & 2nd class coaches. - For now this is speculation, as I have not found the specifications. But other known specifications show very detailed instructions for rolling stock; determining even what type of screws had to be used.

To clarify the original question, I made this picture. The base drawing represents a composite 2nd & 3rd class Brown, Marshalls & Co coach, which was pretty similar to a composite 1st & 2nd class coach, but without heating. The red squares repreprent the locations where the heatingbox hatches were in a composite 1st & 2nd class coach.

The coach in the photo is not in its original form. The body is here moved on a longer, iron frame, and lengthened with the 2 rightmost windows. Originally it was a regular side door arrangement with 4 compartments. - A typical Brown, Marshalls & Co coach. The 1st class compartments were in the middle of the coach, with heatingbox hatches opening to under the seats.

The heating boxes of the first Finnish coaches were not compartment wide. They were inserted thru hatches on both sides of the coach. The actual measures of the boxes were 105.70 * 21.00 * 10.30 cm. The boxes were inserted under the seats.