According to Wintner, the study of central configurations in celestial mechanics may be reduced to an extremality problem. Wintner's Conjecture amounts to saying that the corresponding extremal zeroes for each fixed number n of different masses is finite. By the author's Finite Kernel Theorem it follows that the corresponding number of extremal values is finite for each fixed n. Thus, Wintner's Conjecture will be true or false according to whether there will be only a finite number of extremal zeroes or not. This gives thus a new way of attacking Wintner's Conjecture.