348 On Linearity of Eegression 
Accepting the simple formula (xxiii) for we get 
logJl +^ 
iV^' 1 1 1 
Assuming - < 1, we can expand this expression in a series of ascending powers 
of ^, and, neglecting cubes and higher powers, we get 
Proceeding in exactly the same way from equations (xxvii) and (xxix), we get 
= Y + 1' + I • (1 - r') {xxxvii), 
''^ + ?+?•(! + 3r=) (1 - r"-) (xxxviii). 
When - is small these approximate formulae will be very nearly accurate, and 
they exhibit the arithmetical reason why > > e^t" work out nearly equal 
and, moreover, we see in what manner they will differ when they are not equal. 
^ . 
In our statistical examples - is small and we should expect from these second 
order approximations that , would be greater than ^r?— , which will be 
seen to be the case on referring to the table of values. Again, = according 
> \ ^ t 
as 1 + 3?'- = 8r'-, i.e. as 1 = 57-'-; if r- is nearly -j^r , will be nearly equal, but 
we cannot say which will be the greater as the third order terms ai'e then likely to 
become important in determining the difference. These conclusions are seen to be 
borne out by the table of values, and other peculiarities in the values might be 
explained in the same way. 
When - is small, equations (xxxvi), (xxxvii), and (xxxviii), obviously suggest 
the arithmetical approximation 
~ ■67449 ^ {xxxix) 
smce + 
~= ' {2r + v-r]=-L-^. 
