From Hodge's Systematic Theology, Volume II, Chapter 7, Section "Argument from Related Doctrines," numbered paragraph 1 (pp. 520-21) (the following has been edited by being broken into smaller paragraphs):
No doctrine of the Bible, relating to the plan of salvation, is more plainly taught or more wide reaching than that which concerns the union between Christ and his people. That union in one aspect, was from eternity, we were in Him before the foundation of the world; given to Him of the Father, to redeem from the estate of sin and misery, into which it was foreseen our race would by transgression fall. It was for the accomplishment of this purpose of mercy that He assumed our nature, was born of a woman, and did and suffered all that He was called upon to do and to endure in working out our salvation.
He did not, therefore, come into the world for Himself. It was not to work out a righteousness of his own to entitle Him to the exaltation and power which in our nature He now enjoys. In virtue of the Godhead of his personality, He was of necessity infinitely exalted above all creatures. He came for us. He came as a representative. He came in the same relation to his people, which Adam, in the original covenant, bore to the whole race. He came to take their place; to be their substitute, to do for them, and in their name, what they could not do for themselves.
All He did, therefore, was vicarious; his obedience and his sufferings. The parallel between Adam and Christ, the two great representatives of man, the two federal heads, the one of all his natural descendants, the other of all given Him by the Father, is carried out into its details in Romans v. 12-21. It is assumed or implied, however, everywhere else in the sacred volume. What Adam did, in his federal capacity, was in law and justice regarded as done by all whom he represented. And so all that Christ did and suffered as a federal head, was in law and justice done or suffered by his people.
Therefore, as we were condemned for the disobedience of Adam, so we are justified for the obedience of Christ. As in Adam all died, so in Christ are all made alive. Hence Christ’s death is said to be our death, and we are said to rise with Him, to live with Him, and to be exalted, in our measure, in his exaltation. He is the head and we are the body. The acts of the head, are the acts of the whole mystical person.
The ideas, therefore, of legal substitution, of vicarious obedience and punishment, of the satisfaction of justice by one for all, underlie and pervade the whole scheme of redemption. They can no more be separated from that scheme than the warp can be separated from the woof without destroying the whole texture.
You can imagine how someone could read "the satisfaction by one for all" and think that Hodge held to a view that Christ died for the reprobate. Hopefully the remainder of the paragraph makes it clear that he and I do not think that, but instead hold that Christ came, died, and rose again for the purpose of saving his people (those for whom he is the federal head - those whom he represented on the cross) from their sins.
May God be Glorified!
P.S. Incidentally, this week has been a bit hectic. I hope to provide for the atonement debate a brief argument (my original draft got quite long and ungainly) as to why Hebrews 10 is talking about the same group Hodge is talking about in the paragraphs above.