Umm, SphinxClient class is for SphinxAPI . You pass the query() function a full text search directly. NOT a SQL statement.
When you run
$sql = "SELECT * FROM essays,posts LIMIT 1, 15;";
var_dump($s->query( $sql, "essays,posts" ));
You are looking for documents that contain the words 'select', 'from', 'essays', 'posts', 'limit', '1' and '15'. Only documents containing ALL those words will be returned - you must have 5 documents with all those words.
To get all documents, sort of your example $sql, you would use
$s->setLimits(1,15);
var_dump($s->query( "", "essays,posts" ));
If you want to query sphinx via SphinxQL syntax, which sort of mimics mysql, then you need to use a MySQL client in your code - NOT Sphinx
Client. ie use mysqli, pdo etc, to run queries.