Bernard J. S. Cahill, Oscar S. Adams (1929 approx.)
In 1909, Cahill developed his first (compromise) butterfly map. He succeeded in having Oscar S. Adams develop the mathematics for a conformal version, which he published in 1929.
In Cahill’s original arrangement, Africa was placed in the leftmost lobe, the map shown here was centered on 20° West.
As far as I know, there’s no established name for the M-shaped variant shown here. The label Cahill Conformal M-shaped Projection was chosen by me for the use on this website.