1892.] Conflict Between Organisms. 925 
its four vertical sides agij, acil, celk, gejk. Then also, would 
the point or organism o be accessible only from a’+-4 (3a)=3a? 
points or from only half as many as in the preceding case. 
That 3a? must represent the possible number of paths along 
which o may now be approached, must be self-evident from 
the fact that the plane through v and z divides the ideal envel- 
oping cube supposed in the first case into similar and equal 
halves, since one-half of 6a?=3a. If an organism is supposed 
to lie or move at o on the plane determined by whl, on the 
ground for instance, as a reptile, or at the bottom of the sea as 
a flounder, then will the possibilities of attack by enemies, with 
the factor a still equal to 100, be only 30,000. 
A third case may be supposed where the point o is placed 
in the centre of a square plane with four equal sides (fig. 2) 
ab, be, cd and da and with axes x and y across its two dimen- 
sions. Here ifthe number of points in any side of the square 
are a as before the number of points of approach will obviously 
be 4a, since there are as many pencils of lines converging at 
o as there are sides, namely, four. If, as in the case of heavy 
terrestrial organisms, attack by equally heavy or formidable 
enemies is only possible from every direction on a plane and 
not from every point on the surface of the whole or upper half 
of an enveloping cube, the possibilities of attack now sink to 
400 or to onlyxisth of the number in the first supposed condi- 
tion and sth in the second. 
A fourth case may be supposed where the point o may lie 
in the centre of a plane surface, which is perforated at the 
same point more or less deeply, so that o may, if it be a sensi- 
tive organism, retreat more or less into such perforation or 
cavity, now supposed to be excavated in a solid substratum. 
The small opening, as indicated in Fig. 2, into which o may 
retreat obviously represents only a very small part of the plane 
abcd and o is now accessible to an enemy only through some 
fraction of the number of points represented by a’. This is 
still better shown in Fig. 1 where m is the circular periphery 
of the opening in one of the faces of the now solid cube envel- 
oping o, on all sides except one, o now lying at the bottom of 
a cavity with parallel or converging sides nn. The accessi- 
