STUDIES IN STATISTICAL REPRESENTATION. ATT 
such as the nature of the curve which represents f («) when 
x= 0or ©, etc., or when it is a maximum or minimum, 
will often decide its form. It may be evident, for example, 
from the nature of the case that y = 0 for both « = 0 and 
«= © for « = 0; or again that y has some limiting value 
or values: in other words, that certain values of y cannot 
be exceeded, no matter what the value x may be. More 
succinctly we may say that a consideration of the value of 
y for critical values of the independent variable, and of the 
possibility of the existence of straight or curved asymptotes, 
will often afford the necessary guidance in the choice of 
the type of formula which would be found appropriate. 
5. Necessity for the adoption of equations with fractional 
indices.—The unsatisfactory results arising from the use of 
inappropriate formule are, even yet, only imperfectly 
realised. Many expressions have been devised from time 
to time to meet particular cases and have had considerable 
vogue, notwithstanding that the results analysed could 
possibly have been more suitably and more accurately 
represented by a much simpler formula. This has been in 
the main owing to a somewhat remarkable habit of limit- 
ing rational algebraic expressions to forms containing only 
integral powers of the variable. 
This limitation, self-imposed by mathematicians, arises 
merely from an inadequate conception of the synthesis of 
such expressions, seems entirely unnecessary, and in some 
cases to be illegitimate. For example, the expression gio 
is usually considered as the tenth root of x°, and thus the 
idea that only integral powers of x are quite admissible is 
implicitly maintained. That «xt? may also represent, for 
example, a value through which the function <* passes, as n 
increases continuously from 0 to 1, is often not sufficiently 
kept before the mind. 
