Edgar suffers two losses in this chapter — the death of his wife and the birth of a non-heir. Because Cathy is not a male, she legally is not Edgar's heir, and complicated laws end up leaving Thrushcross Grange to Isabella, and then to her son. This is not to suggest that Edgar does not love Cathy; he adores her, and she is his world. He just hates the fact that his rival may end up with his property.
With the shock of Catherine's death, Heathcliff implores her to haunt him: "I cannot live without my life! I cannot live without my soul!" He is clearly devastated by the death of his one true love, and although Heathcliff has done dastardly deeds throughout the text, most readers tend to sympathize with him and the loss he is feeling.
Edgar is devastated too, but by burying Catherine near her beloved moors, Edgar demonstrates both the depth of his love for his wife as well as insight into understanding her character. He wants Catherine to be happy and at peace, and this is one final gesture he can give to show his love.
