Throughout his poems, Arnold frequently alludes to figures of Greek and Roman mythology in order to further his ideas. Here is a quick explanation of the stories he is alluding to.
Cadmus and Harmonia
Cadmus was a Phoenician prince and the founder of the city of Thebes. He carried off Harmonia, the immortal goddess of harmony and concord, into marriage. Also in love, Harmonia then accompanied Cadmus when he was forced to leave Thebes, and the two successfully helped a group called the Encheleans wage a war against the Illyrians. But because Cadmus had killed a sacred dragon in his former quest to protect Thebes, the gods turned him into a serpent. Harmonia begged to be changed into one along with him, and her wish was granted.
Bacchus was the Roman counterpart to Dionysus, the Greek god of wine. He invented wine, and spread the art of tending grapes, bringing joy and ecstasy to all who encountered him. He was the only god with one mortal parent, as the son of Zeus, king of the gods, and Semele, a mortal woman. The Bacchanalia were a group of his loyal followers; he and these followers could never be tied up or fettered. Finally, Bacchus (and Dionysus) is often considered an expression of man's purely emotional, visceral capability.
Apollo was the Greek and Roman god of the sun and of music, and twin brother to Artemis (Diana in Roman mythology). He was known for harnessing his regal chariot with four horses, and for flying across the sky to move the sun. Apollo was the leader of the Muses, a group of nine goddesses who served as inspiration for literature, poetry, and the arts. Finally, Apollo is often considered an expression of man's intellectual capacity.
Sophocles is not exactly a mythological figure; he was a living playwright in ancient Greece who composed some of the most famous plays about mythology. His most celebrated work is Oedipus the King, which tells the story of the Theban man who unwittingly killed his father and married his mother. Overall, this still-famous play explores the power of fate, a theme quite resonant with Arnold.
Though Arnold never specifically mentions the Fates by this term, he alludes to the goddesses of destiny in "Consolation," and often talks about forces which control us. The Fates were three goddesses of classical mythology who were entirely in control of destiny. They controlled the metaphorical "thread of life" of every mortal from birth to death, and after chose the manner of each person's death.