Miscellanea 
299 
Accordingly: a, = -i=^^^f^\ 
We now pass to the fourtli order regression line function : 
and we find : 
-C43A3 
= _^ _ ft/3 2-ft)/34 _^ (/33-_^2'' + fe-ft)fe ^ (fe-2 ft^, + /3i^) 3o 
Vft V/3i(/32-/3i-l) V/3i02-ft-l) V/3i02-^i-l) 
- = {^5 (|92 - ^1 - 1 ) - ^4 (ft - ^2 - ^1 ) + /33 (ft -~ /32- + 2ft - ^1 ) + ^1 /32 Ol " 2^2)} Vft (ft - ^1 ^ 1 )• 
ft (S2 - ^1 - 1 ) - ft (ft - ft ft - ^i) +ft ^ (ft - ft' + 2ft - /3i) + ft ft (/3i - 2fe) 
Vft 
^6 
Let <^5 = ' 
Thus : C43 = 
Again: —^^f ='^42X2) 
JV 
ft-ft-ft = C42(ft.-ft-l), 
or •■ • <^4-<^>3Vft + <#>2ft' = ^4'> aay, =f42'^'2- 
Thus: C42 = ^. 
92 
ft 
Further : -p= = C41 , 
Vft 
and /32 = C4o- 
Thus : ^^4= A'^ - . . „ ft . 
</>4<^2 - <^3- <^'2 Vft 
To find a4 we must determine K^ and A4 : 
X4 = - 
-(ft-ft-ft')- 
33 
(^4</>2-<^3')<^>2 (/la " ft 
</>2 {<^i<p2-4>3^] 
where 
Til 
<t>2 
ft^ 
ft 
K4— _^„-^^^^,^.,3-^-.2j-^(^:2 Wft) 
Let us write ?^'' = fi4, then 
Vft 
a.=: ■ 
^6 
04'/>2 - 03^ 
1^3 
We have thus obtained the regression orthogonal functions up to the fourth order. Higher 
order terms can also be found, but their expressions become very complicated and such ex2;)ressions 
involving fifth product-moments and eighth marginal total moments will be subject to very large 
probable errors. 
