MySQL best index for static table - mysql

Question

I have the following table (T) in Mysql:
+--------+-------------+------+-----+---------+-------+
| Field | Type | Null | Key | Default | Extra |
+--------+-------------+------+-----+---------+-------+
| first | varchar(50) | NO | PRI | NULL | |
| second | varchar(50) | NO | PRI | NULL | |
| third | varchar(50) | NO | PRI | NULL | |
| count | bigint(20) | NO | | NULL | |
+--------+-------------+------+-----+---------+-------+
This table contains several million rows. I have created the following indices:
+-------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| Table | Non_unique | Key_name | Seq_in_index | Column_name | Collation | Cardinality | Sub_part | Packed | Null | Index_type | Comment | Index_comment |
+-------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
| T | 0 | PRIMARY | 1 | first | A | 591956 | NULL | NULL | | BTREE | | |
| T | 0 | PRIMARY | 2 | second | A | 67927032 | NULL | NULL | | BTREE | | |
| T | 0 | PRIMARY | 3 | third | A | 271708128 | NULL | NULL | | BTREE | | |
| T | 1 | SECONDARY | 1 | second | A | 398399 | NULL | NULL | | BTREE | | |
| T | 1 | SECONDARY | 2 | third | A | 45284688 | NULL | NULL | | BTREE | | |
| T | 1 | SEC | 1 | second | A | 4382389 | NULL | NULL | | BTREE | | |
+-------+------------+-----------+--------------+-------------+-----------+-------------+----------+--------+------+------------+---------+---------------+
Searches of the type:
SELECT * FROM T WHERE first = "WHAT" AND third = "EVER";
and
SELECT * FROM T WHERE first = "WHAT" AND second = "EVER";
also usually fast (results are always obtained under 1 second). However the searches like:
SELECT * FROM T WHERE second = "WHAT" AND third = "EVER";
are very slow (usually more than 1 minute). I created the index SEC (see indices table), but that doesn't improve the results.
What index should I use to make these searches faster? (I haven't kept experimenting because the creation of one index takes around 5 hours)
MORE INFO: The table is static (i.e. I won't be adding any more rows - I am only interested in search speed), and disk space is not an issue.

Answer 1

Use additional indexes comprising of fields which match your queries. If the row combinations are unique then use primary indexes. These give quicker access than secondary indexes.
As the table is static - the number of indexes will not affect performance (any updates, deletions and insertions require updates to each index of a table).
So for quicker retrieval from this query create an index of second and third columns:
ALTER TABLE T ADD PRIMARY KEY (second, third);