438 
Miscellanea 
and proofs of the formulae concerned have been given by me for many years in college lectures 
without making this assumption. I have, however, stated the fact once or twice in print that 
the formulae are general, and it seems desirable to reproduce my proofs (from lecture notes) 
at the present time as some doubt seems to have been cast on the generality of the formulae. 
(2) I begin with a preliminary proposition, which is fairly familiar. Let x u x 2 ...x n be 
n variables, continuous or discrete but quantitatively measurable ; their means, 
u\ , cr 2 , . . . u n their standard deviations, r m the correlation coefficient of the pth and qth. variables 
supposed found by the product moment formula. Then : What is the best linear function u of 
a?!, x 2 ...x n such that an (ft + l)th variable x n+ i will have a maximum correlation r nx n+1 —'Pi sa J> 
with u ? 
Let 
Clearly 
U = C\X\ + C 2 X 2 + ... + c n x n 
= S(cx). 
u = <S Y (iix), 
Np<r u ar n + , = 2 ( u - u) (x n + i-x n + 1 ) = S2c p (x p - x p ) (x n + i-x ll + i) 
= o2 {c p a-p cr n + 1 r Pt n+ {) 
P = — S(c p o- p r Ptn + 1 ) 
.(i). 
JVo-u 2 = 2(u-u) 2 = S2 {c 2 {x p - x p f) + 2S2c p C q (x p - x p ) (x q - x q ) 
= NS (c p 2 a-p 2 ) + 2A T S (c p c q r vq tr p o- g ), 
o» 2 = S (c 2 a-p 2 ) +2S(c p c q (T p <r q r pq ). 
But if p be a maximum, we must have : 
for all values of p. 
Or, we reach type-equations of form : 
dc„ 
= 0, 
U n (T u Why, 
>'p, n + 1 = {°p Vp + $ ( C 8 °> °1 r pQ>} 
Where : 
= — (cj o"! r lp + c 2 <r. 2 r 2 p + ...+Cpap+ ...+ c n <r n r n p) 
= bi r lp + b 2 r 2p + ... + b p + ... + b„ r np 
.(ii). 
Now equations of type (ii) are easily solved by aid of the determinant 
R = 
? 'l,2 
^1,3 
J '2.3 
'»+l,l> ' n +1,2) Ml + l,3l ••• 
We have b q = - R g> n + JR n + hn+l 
where R q , n + l is the minor of qth row and «+lth column. 
Hence it follows that : 
1 
