68 
WEAD. 
it were odd, y would become infinite for negative values 
of x. 
The other forms of discontinuous straight lines need no 
explanation, whether they involve factor (A) or factor (B). 
y=(b + ax) |jl — N (t) J ; see Fig. 2. 
y = b + axN \c) ; see Fig. 3. 
y = b -b ax [ 1 — N~ (f ) 1 ; see Fig. 4. 
( 2 ) 
(3) 
W 
Application to Surfaces. —In the cases next to be considered 
a function will be rendered indeterminate over a given sur¬ 
face, but equal to zero outside of it, by the use of a monomial 
factor of the type (A), or vice versa by the binomial factor, 
type (B). 
Take first a circle (Fig. 5) whose equation is z 2 + y 1 = r 2 , 
r f and draw through it the straight 
line EF f the general equation of 
which is y — y l = a (x — x l ). Let 
\ y x = 0 , x x — x, and a— oo , the line be- 
J 1 F9 J ing parallel to the axis of Y; then 
the equation reduces to y — oo 0. 
This value of y obviously applies 
to all points in the line EF t and to 
E 
all points in any line parallel to EF. It is equally obvious 
that the value of y is indeterminate in the very nature of 
the case assumed, and so a determinate value of y cannot be 
found by any applications of the calculus. But this inde¬ 
terminateness may be limited to that part of EF within the 
circle by using factor (A) with a suitable exponent; thus 
( 5 ) 
Precisely the same expression may be found for x; so 
we may combine the two and write: 
