TO THE THEORY OF EVOLUTION. 
27 
then 
SI r 
3 ' ss dx x dx. 
m+l Q ( d2 \ TT 
z~ = So( r ss ,^-r- )z 0 U, 
-o 
^ ' ss 'dx s dx s ,} 0 
where U stands for {So(u,,e i e r )}' tt . 
Hence, remembering that dz Q /dx s = — z 0 x s ,, 
| S 2 {r SS ' d ^ dr z 0 — z 0 S 2 (^xa,)U + z 0 S a (V, 
^ 2 U 
dx s dx s , 
c / ( dU . cl U 
- z 0 S 2 (r ss ,[x s ^ 
= z 0 S 2 (r w ,2^)11 + z 0 So{r^(a;, — e,)(av — e,,)}U 
^0^ 2 {Uk' £«') -h X s t(x$ Ci) } U 
= *o{S a (r,^^)}* +1 , 
which had to be proved. 
But it is easy to show by simple differentiation that 
®2 {^ ss 'cJr sl ^ Z ° = Z ° ,<y l F^l) ~ Z 0 S 2 ( r ss' e J e «') 
Zq — Zq S 2 (7' W ' s v 2 s'Vq -j - ss’ pV\ qV i ~h 2 Vp s r ss i s v 2 pV x ^i) 
= z 0 {S 2 (r^e,e y )} ? .(lxxvii.). 
Hence the theorem is generally true. 
Thus we conclude that 
Z = Zn 
1 
I “h “1” p-)] So(^ SS'^S^S’') 
+ • • • + lvl j S 2 (n y e J e y ) }> + 
1 
(lxxviii.). 
It is quite straightforward, if laborious, to write down the expansion for any number 
of variables. 
Now let Q be the total frequency of complices of variables with x x lying between 
h x and co , x. 2 between h 2 and oo , . . . x s between h s and oo , . . . x n between h n and oo ; 
and let Q 0 be the frequency of such complices if there were no correlations. 
Then 
/.oo /.oo /.oo /.co 
Q = ...... \ z dx l dx 2 . . . dx s . . . dx n 
• hi J ho J h 3 • hn 
• hi J h 2 
/» CO -cc 
Now let 
r co -co -oo -oo 
Q 0 = ...... z 0 dx x dx 2 . . . dx s . . . dx H 
j hi j h-2 J hs j hn 
1 r" 
