70 
WKAD. 
Combining (10) and (5'), assuming a — b < r, gives the 
equation of the surface within the circle and outside the 
square whose side is 2 a , like a Chinese coin (Fig. 6), thus : 
The equation 
represents the 32 black squares in an ordinary chessboard. 
The first factor limits the indetermi- 
nateness to four squares in any direc¬ 
tion from the origin, a being the side 
of the square. Without this factor 
Fi ^' 6 the board would be of indefinite ex¬ 
tent. 
This is an example of periodic 
indeterminateness. 
Application to Solids .—These will probably require no ex¬ 
planation. Their difference from the former equations con¬ 
sists essentially in that three variables are required. Take 
three illustrations: 
The equation 
represents any point outside of an ellipsoid— e. g., any point 
in space outside of the earth. 
The equation 
x,y,z = <*> 
represents any point within the parallelopiped whose edges 
are 2 a, 2 b, 2 c; or in the same sense in which equation (5') 
represents a surface, equation (14) represents a solid. 
