In a previous post (link), I identified what Charles Hodge and I believe about the atonement, as taken from Hodge's Systematic Theology. There's a great deal more that Hodge said that is relevant to the issue: the relevant chapters of his Systematic Theology stretch over 100 pages.
I thought I'd switch over, for a post, to his son, Archibald Alexander Hodge. Neither Hodge requires any introduction in Reformed circles, so I'll get right to the meat, with these quotations from his commentary on the Westminster Confession of Faith:
Chapter 8, Section 6, Numbered Paragraph 2:
Christ thus has, in strict rigor, fully satisfied all the demands of divine justice upon those whom he represents. Jesus Christ has met the divine demand that the original covenant of works be fulfilled through the sorrow of His life and death, and he has met the divine demand for essential justice in the punishment of sin through the obedience unto death. Christ suffered as the representative of sinners. Our sins were laid upon him. He, "hath redeemed us from the curse of the law by being made a curse for us." He died, "the just for the unjust." "He is the propitiation (expiation) for our sins." He "gave his life a ransom for many." We are "bought with a price." (Gal. 3:13; 1 Pet. 3:18; 1 John 2:2; Matt. 20:28; 1 Cor. 6:20.) Christ suffered only in his single human soul and body, and only for a time. Nevertheless, his person was the infinite and transcendently glorious person of the eternal Son of God. Consequently his sufferings were precisely, both in kind and in degree, what the infinitely righteous wisdom of God saw to be in strict rigor a full equivalent, in respect to the demands of legal justice, for the eternal sufferings of all for whom he suffered. This is the doctrine of the whole Christian Church. The Thirty-nine Articles of the Church of England, say, Art. 31: "The offering of Christ, once made, is that perfect redemption, propitiation, and satisfaction, for all the sins of the whole world, both original and actual."
Chapter 8, Section 6, Numbered Paragraph 3:
This proves, therefore-(1) That Christ did not die simply to make the salvation of those for whom he died possible-i.e., to remove legal obstructions to their salvation-but that he died with the design and effect of actually securing their salvation and of endowing them gratuitously with an inalienable title to heaven. (2) It proves, in the second place, that the vicarious sufferings of Christ must have been, in design and effect, personal and definite as to their object. Salvation must be applied to all those for whom it was purchased. Since not the possibility or opportunity for reconciliation, but actual reconciliation itself was purchased; since not only reconciliation, but a title to an eternal inheritance was purchased, it follows (a) That "to all those for whom Christ hath purchased redemption, he doth certainly and effectually apply and communicate the same." (Conf. Faith, ch. 8. s. 8.) And (b) That he who never receives the inheritance, and to whom the purchased grace is never applied, is not one of the persons for whom it was purchased.
Chapter 8, Section 8, Numbered Paragraph 3:
Our Standards, it will be observed, very explicitly teach that Christ, as mediatorial Priest, made expiation and purchased salvation for certain definite persons. Thus, in chap. 3. s. 6, it is said: "As God has appointed the elect unto glory, so has he by the eternal and most free purpose of his will, foreordained all the means thereunto. Wherefore they who are elected being fallen in Adam, are redeemed by Christ. . . . Neither are any other redeemed by Christ, . . . but the elect only." Here it is expressly affirmed (1) That Christ died upon the cross on purpose to carry out the eternal purpose of God in the election of certain individuals to eternal life. (2) That Christ died for the purpose of saving no other than the elect.
(source) (for all quotations above)
This is, of course, the same thing being said by Hodge's father, and by Shedd and many other Reformed authors as well. Compare, for example, the similar remarks by Robert Shaw (link).
May God be praised,