Not only have I tried to determine exactly how the name "Lachance" is to be spelled (as it was originally), with or without an uppercase "C", but I've always been curious about the "space" between the "La" and the "Chance" that is often found in the US spelling of the name. Sometimes it is there, sometimes it is not. I have always written my name as one complete word, but I haven't always used a lowercase "c".
Some of my research has led me to various original documents. For instance, my grandfather Antoine (Tony) signed his immigration application "La"c"hance", no uppercase "C". In Sept 2001, I obtained a copy of my father’s birth certificate. On here I found the name spelled "La"c"hance". In his case, I suspect it somehow got changed in Military records upon his enlistment in the Marine Corps.
Perhaps a response I received from Larry Coderre of Canada may help to further explain the dilemna.
"In reply to your question about spelling: In the early parish registers I have seen a large number of the people could not sign their names. By the 1880's and 90's some, but not all, could. As a result, the spelling of a name was left up to the priest, census taker or immigration officer. Often the same priest in the same parish would spell the same name differently from one entry to the next. In the Ottawa-Hull phone book (what's a phone book?) there are just under two hundred (200) Lachances listed. Only those which are spelled entirely in capitals have a capital C. Hull is in the province of Québec and Ottawa (in Ontario) has a large French population. Incidently there are no Desjardins spelled with a capital J either although my wife, Lucille, says that she has seen the name spelled Des Jardins. I AM ONLY GUESSING but perhaps the capital C and the capital J was first used by an American immigration officer".