28 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
where 
H, = 
\/27r 
e ih,i .(lxxix.). 
We have Q 0 = N . . /3 S . . (3 U H x H 8 . . H, . . H„ 
But by (xviii.) 
where 
and as above, 
Thus 
yA 
7) ( s^p & ' clx £ — H, s Vjj_] — hb/3, 
m Jh. 
1 
ft ’ 
• v P- 1 = L> v P-i]x. = h . .(lxxx.), 
ft = [ e~ w dx s fe~ y “'\ .(lxxxi.). 
J h s / 
* 00 00 /* 00 /.SO 
yw— ...... s v ps .v p , s „v r . . . e -Pa 2 + + • • ■ +*?+ • ■ ■ + ^dx 1 clx 3 . . . <Ar, . . . <£c„ 
(V 2 177 ") J ki k., h. hi,, 1 1 
= H.H, . . . H, . . . . . . A • • . &'-%* 
r^7?—1 srV/nt__ i ■J//27 
-1 1 
& ft' ft. 
or 
«CO /%00 1*00 /»00 / — 
...... 2 0 n(^)daydan . . . dx s . dx n — Q 0 n (%u 
J 7ij J 7i 3 J/i, J/t» \ Ps 
(lxxxii.). 
where II denotes a product of s v p for any number of v’s with any s and p. The rule, 
therefore, is very simple. We must expand the value of z in vs as given by (lxxviii.) 
above, then the multiple integral of this will be obtained by lowering every vs right- 
hand subscript by unity (remembering that s v 0 =1), and further dividing by the /3 of 
the left-hand subscript. The general expression up to terms of the fourth order has 
been written down ; it involves thirty-four sums, each represented by a type term 
All these would only occur in the case of the correlation of eight organs, or when we 
have to deal with twenty-eight coefficients of correlation. Such a number seems 
beyond our present power of arithmetical manipulation, so that I have not printed the 
general expressions. At the same time, the theory of multiple correlation is of such 
great importance for problems of evolution, in which over and over again we have 
three or four correlated characters to deal with,* that it seems desirable to place 
on record the expansion for these cases. I give four variables up to the fourth and 
three variables up to the fifth order terms. Afterwards I will consider special cases. 
* In my memoir on Prehistoric Stature I have dealt with five correlated organs, i.e., ten coefficients. In 
some barometric investigations now in hand we propose to deal with at least fifteen coefficients, while 
Mr. Bramley-Moore, in the correlation of parts of the skeleton, has, in a memoir not yet published, dealt 
with between forty and fifty cases of four variables or six coefficients. 
