$(this+"p").slideDown("slow");
$(this)+$("p").slideDown("slow");
$("this+p").slideDown("slow");
does not work.
$(this+"p").slideDown("slow");
$(this)+$("p").slideDown("slow");
$("this+p").slideDown("slow");
does not work.
Next.
$(this).next("p").slideDown("slow")
Make sure that the "p" element is directly adjacent, though. Otherwise you'll want to use nextAll.
Yeah, your syntax is bad. You should use the jQuery Sibling function:
$(this).siblings().find("p").slideDown("slow");
The jQuery API site is awesome for looking stuff like this up, I rely on it nearly daily. I'd keep an eye on it.