Vernam found a simple mathematical function, V, that operates on two pieces of data to compute a third piece: Z = V(X,Y), such that X, Y, Z are totally interchangeble.
By applying his function to the plaintext, P; ciphertext, C; and cryptographic key, K -- Vernam secured:
C = V(P,K)
P = V(C,K)
K = V(C,P)
This was sufficient to secure full equivocation because for any captured ciphertext C, there is a key K that decrypts it to any desired plaintext P.
This means that a cryptanalyst holding C cannot pinpoint which plaintext was actually sent off. Only the intended recipient, holding the right key can resolve this equivocation.
We shall further study the Vernam function, and its inherent weaknesses, which prevent the Vernam discovery from becoming the end event in cryptography.