Everybody pursues his private interest and only his private interest; and serves thereby, without wanting or knowing it, the private interests of all, the general interests.
$$$$$$$$$$$$yup$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ The point is not that, with everyone pursuing his own private interest, the totality of private interests, that is the general interest, is attained.
$$$$$$$$$$$$$$$$That is precisely the point$$$$$$$$$$$$$$$$$$$$$$$$$$$$
Rather, from the same abstract phrase the conclusion could be drawn that everybody mutually hampers the assertion of the interest of others, and that instead of a general affermation a general negation results from the bellum omnium contra omnes.
$$Except that personal experience doesn't bear out the charge of 'bellum omnium contra omnes.' Rather we see that 'quum unum adipiscitur, omnes adipiscuntur'. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
The point is rather that the private interest is already a socially determined interest and can be attained only within the conditions set by society and with the means provided by society; that is to say, the private interest is bound to the reproduction of these conditions and means.
$$$Actually I would say that private interest and socially determined interest may merge or may not merge. Freedoms must be lost when society begins determining conditions and means of mergeing interests.$$$$$$$$$
It is the interest of private people; but its content, as well as the form and the means of realization, are given by societal conditions independent of all.
$$$$$$$$societal conditions are independent of all? People have no effect on society?hmmmmm......$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$