PKOF. K. PEARSOIT ON THE MATHEMATICAL THEORY OF EVOLUTION. 73 
Between the six constants on either side of this equation an infinite variety of 
relations can be reached by giving x an infinite variety of values, and it seems 
impossible to satisfy this series by the same set of values of the constants. For 
example, let x be very great, and suppose cr^ to be the largest of all the quantities 
I (x - 
^ e~ -“■i" and putting x very great we have 
r r / I 1 \ „ / 1 1 \ „ X? c \ 1 
— L 3 g 2 fo- 2 - — —1 g 2 V 1^32 (T^) _j-i g 2 V<r*2 “ 
^1 *^2 ^3 i 
whence, proceeding to the limit, 
^ 1/^1 “ 
unless o-i = 0-3 or cr^. 
The first is impossible by hypothesis, therefore the latter must be true, say 
cTi = 0 - 3 . This gives us at once = C 3 . 
Beturning to the original equation, and making x large in it, we see that the ferst 
two terms become equal on either side. Hence, the second two terms must become 
equal as x approaches infinity, or 
C:y _ iii c, — 
— e 2(7-22 — _A g 2 (r,-‘ ^ 
^2 ^ i 
cTp cTg, 0 - 3 , and cr^. Dividing by 
Dividing again by this leads in the same manner as before to — 0 - 4 , and, 
ultimately, to = c^. 
Our original equation may now be written 
U\/ (-7^) 
(X - 6 i )2 
2(7,2 
e 
(.r-ftjVi-j 
} 
CTov/ (^tt) 
(x - 6^)2 
2(T2^ 
e 
■ 
(’?)• 
Put a? = A ( 6 j + 63 ), then the left-hand side vanishes and, accordingly, the right 
must vanish, but this involves either 
h = K 
or 
+ ^3 = ^2 + ^4* 
Similarly, putting x = ^ (fy -j- fy), we find that either 
— ^3) 
^1 + ^3 — ^2 + ^4.(“)• 
Thus, either the two sets of components are identical, or (a) is true, 
MDCCCXGIV.—A. L 
