DISCONTINUOUS AND INDETERMINATE FUNCTIONS. 67 
We are now to multiply various equations or terms by an 
expression of the form (A), making the exponent such a func¬ 
tion of x, y, z, etc., and constants, as will introduce the de¬ 
sired limitations. Cases will be chosen of lines, surfaces, and 
solids, and some involving physical laws. For brevity the 
coefficient 2 before qo will be dropped; that is, <x> will be 
considered to be the limit of the series of even numbers. 
But besides the monomial factor (A) there will be needed a 
complementary or binomial factor of discontinuity, 1 — (A), 
of the form 
i—(B> 
whose values are 
1 — N~° =1 —1 = 0 when x < a; 
1 — iV ~ 00 — 1 — 0=1 when x > a. 
Application to Lines .—Given y = b + ax as the equation of 
a line, between the limits x = -f- c and x = — c; outside of 
these limits y = 0 for all values of x; required an equation 
expressing all these conditions. 
We have simply to multiply the second member of the 
equation by a factor of the 
form (A) involving x and c; 
thus: 
(t)" 
y — {h 4- ax) N \ c 
Fig. 1 shows the form of -—| 
the locus. That the por¬ 
tions parallel to the axis of 
Y belong to the locus will 
be evident if one constructs 
a series of curves with num¬ 
bers 2, 4, 8, etc., as expo¬ 
nents, instead of go . The 
convenience of assuming oo 
to be even is clear; for if 
L . 
~A .| 
/*>./. 
Fig. *■ 
Ffy.J. 
Fig. 4. 
