250 
PALEONTOLOGY: H. F. OSBORN 
where the arctangents are all taken between — and -, and the integer k is 
chosen so as to make — 7r<r^x. Since the polygon is not overlapping, we 
have 
= — 2, —S^fj^S 1 (t^= 1,2, . . 
For a fixed value of m, we consider the set of all the values of ai, . . . , am, 
fjLi, . . , iJLm for which (6) maps < 1 on a simple polygon, and it is first shown 
that this set is closed. On this set, and for —t^O^t the expression (7) has 
therefore a maximum and a minimum, which are found by observing that 
f sin 
arctan :; 
1 — r cos (p 
has a maximum = arcsin r for = arccos r and a minimum = — arcsin r 
ior (p = — arccos r. In the discussion of (7), it is necessary to distinguish 
the case where all /x are negative, corresponding to a convex polygon, from 
the general case where some jjt, are positive; for this reason, the convex regions 
appear separately in the statement of the theorem. By considerations of 
continuity, and the use of elementary properties of harmonic functions, it is 
finally shown that the upper and lower bounds in (1) and (4) are reached in 
the cases (2) and (5) only, and that no closer bounds than (3) can be found 
for ^r<l in the non-convex case. 
Regarding (2) and (5), we observe that by rotating the z- and z£;-planes 
through the angle —a about their origins, we may make a — 0; in this case, 
the circle |z| < 1 is mapped by (2) on the w-plane slit along the straight line 
segment 
•w = tA—r - (0^t^+ ), 
4sm/3 ^ ^' 
and by (5) on the half plane in which the real part of we^^ is greater than — J. 
^Koehe, Gdtlln gen, Nachr. Ges. Wiss., 1909, (73). 
2Gronwall, Paris, C. R. Acad. Sci., 162, 1916, (249). 
SEVENTEEN SKELETONS OF MOROPUS; PROBABLE HABITS OF 
THIS ANIMAL 
By Henry Fairfield Osborn 
American Museum op Natural History, New York City 
Read before the Academy, April 29, 1919 
Moropus is an aberrant perissodactyl, closely related to the family of the 
Titanotheres and more remotely to that of the Horses. It occurs in the 
Lower Miocene age in France and North America, and its ancestors have been 
