A78 G. H. KNIBBs AND F. W. BARFORD. 
A simple geometrical illustration will shew even more 
clearly the inadequacy of an algebraic expression, from 
which fractional indices are rigidly excluded. Oonsider 
the family of curves y = <" where n has different values. 
Suppose 1 to be positive and to nave the values 0, 1, 2, 3. 
Then the graphs of the curves, following the wsual con- 
ventions, are :— 
n = 0, a Straght line parallel to and distant 1 from 
the axis of x, passing through 1st and 2nd 
quadrants. 
n = 1, a Straight line bisecting the angle between 
the axes and passing through Ist and 3rd 
quadrants. 
n = 2, a parabola whose axis is the axis y and vertex 
the origin: passing through ist and 2nd 
quadrants. 
nm = 3, a cubic curve witha point of inflexion at the 
origin and passing through 1st and 3rd (not 
2nd) quadrants. (The point of inflexion is 
also a minimum for one branch of the curve 
and a maximum for the other.) 
The curves so obtained are thus wholly dissimilar when n 
is even and nis odd, that is, the graph region in the former 
case is quadrants 1 and 2, and in the latter quadrants 1 
and 3. If, however, n be supposed to increase continuously 
from the value 0 and a series of curves be drawn,! which 
all pass through the origin, and also the point (1.1), a much 
clearer idea of the relationship of the curves of the family 
can be obtained. Thus in general we should expect the 
curves y = 2° 3°= a2" on. = + — etc., when 6n is very 
small, to occupy the same spatial positions approximately, 
that is to say «°° and «°” should be sensibly identical 
curves for all values of x positive or negative. 
1 See “Studies in Statistical Representation,” by G. H. Knibbs, 
Journal Royal Soc., N.S.W., Vol. xiv, p. 344, fig. 1. 
