DISCONTINUOUS AND INDETERMINATE FUNCTIONS. 69 
/3C 2 -f- ?/2\ 
by which is meant that either x or y — go ON v V ) . 
For values of x or y that make the value of the fraction 
less than unity, the value of “ the factor ” is unity, so the 
second member is indeterminate. But if the value of “ the 
factor ” becomes 0, the second member reduces to zero, and 
the equation cannot be satisfied except by making x or y— 0. 
Equation (5') therefore represents the surface within the 
circle and the portions of the axes outside of it. Since these 
outside portions are only lines, they must be neglected when 
a surface is under consideration. 
Similarly, to represent the portion of a plane outside of a 
circle of radius r, write 
x, y = co 0 
( 6 ) 
The surface of a ring bounded by concentric circles whose 
radii are R and r, R being the greater, is represented by 
/ x~ 4- y -\ 00 I - _ / x 2 + .v 2 \ 00 ~] 
x, y = co 0 N v is 2 / 1 — N V r 2 / I. (7) 
If the breadth of the ring is r (]/ 3 — 1), the equation may 
be written 
/2 r 2 — x 2 — 1 / 2 \ 00 
x, y — co 0 N v r 2 /. (8) 
Here the equation is indeterminate when x 2 + y 2 > r 2 and 
< 3 r 2 . This is an example of exponents in which the vari¬ 
able has the negative sign. Sometimes these prove quite 
puzzling, and so they should generally be avoided. 
Recalling the equation for a rectangle, we get for the sur¬ 
face within it and outside of it, respectively, 
a,?/= 00 0iV _ [(f)” + (1)"] ; (9) 
x,y = ao 0 (l — N~ C(l)“ + (£)"]). 
(10) 
