66 
WEAD. 
locus more and more nearly approaches the circumscribing 
rectangle, 2a X 26. If n = 2 go , the locus coincides with the 
rectangle; so it is discontinuous at four points. It is here 
assumed that what is true up to the limit is true at the limit. 
2 
Similarly, if the exponent become —, the limiting curve will 
coincide with the axes, and have eight points of discontinuity. 
It is to be particularly noted that the discontinuity does not 
arise in these cases, because one of the variables—for ex¬ 
ample, y —becomes imaginary, as in the usual cases in the 
text-books; nor because of an infinite series, as in some har¬ 
monic curves; nor because of an integration, as with Diricli- 
let’s so-called factor of discontinuity used in the reduction 
of multiple integrals. 
We see, then, that an expression affected with an infinite 
exponent may become discontinuous at a desired point. 
Consider, now, more carefulty the values of either term in 
the first member of equation (1), or write 
*=©"• 
Whatever value x may have between 0 and ±<x> ,u has only 
three values, viz: 0 , 1, and qo , according as x is less than, 
equal to, or greater than ± a; the second of these holds only for 
an infinitesimal range in the value of X, and will be disre¬ 
garded in what follows. 
To get in place of the function having an infinite value 
when x exceeds a, one that is finite, a further step is needed. 
Take any positive number, N, greater than unity (10 is con¬ 
venient, so that if one desires to plot the series of curves with 
exponents increasing up to infinity he may use common 
logarithms) and give to it — u as an exponent, thus: 
N~ (tt) ’ ( A ) 
This clearly has only two values, viz: 
X ~ 0 = 1 when x < a, 
