As Jan already said, Louis was the artist's name of Lodewijk Franciscus Hendrik, son of Johannes Cornelis Apol (also a painter) and Antonia Johanna Hage. Here are the details of the family:
Johannes Cornelis Apol, art painter Bapt. Breda 03 Feb 1806, died Velsen 04 Mar 1896 Son of Adrianus Apol, shopkeeper, and Adriana Christina Harmsen Married den Haag 31 Mar 1847 (akte 95) to: Antonia Johanna Hage Born Amsterdam 07 June 1821, died Heemstede 05 Nov 1903 Daughter of Hendrik Hage, book printer, and Johanna Paulina Prinses
Children: 1. Anna Maria Apol, born 22 Oct 1845 at 11.00 hrs Born as A. M. Hage, acknowledged by father at marriage parents, therefore after that with last name Apol. 2. Johannes Adrianus Cornelis Apol, 24 Oct 1848 at 03.00 hrs 3. Lodewijk Franciscus Hendrik Apol, born 06 Sep 1850 at 10.00 hrs
As for Henry W., this is probably his "Americanised" name... Was his wife Elizabeth and their children Bertha, Rosa, Anna and Arthur? On ancestry.com you can find their names on passenger lists. Unfortunately they traveled in 1891 (from 1892 onwards these lists are accessable free of charge), so you need a paid subscription to ancestry.com in order to view these passenger lists. Often these lists have a lot of information on birth places etc. So if you have a subscription or you know more about Henry (his Dutch name, place of birth, anything), we might be able to find the family connection. But to do that, we need to find the parents, etc of Henry...