SOME DISCONTINUOUS AND INDETERMINATE 
FUNCTIONS.* 
BY 
Charles Kasson Wead. 
[Read before the Society, March 31, 1900.] 
Introduction .—In many physical problems we have to deal 
with discontinuous quantities. Up to a certain point one 
law holds; beyond it, another. For example, in Helmholtz’s 
discussion of the motion of a violin string, separate equations 
are needed to express the conditions of the two parts of the 
string. Again, we may desire to express the fact that an 
equation is indeterminate within certain limits, but cannot 
be satisfied outside of them. To express these two kinds of. 
peculiarities it is usually, though not always, necessary to 
abandon mathematical expressions and use words. Some¬ 
times, however, imaginaries may indicate the change of the 
law. Thus, the inequality x 2 + y 2 < r 2 is satisfied for any 
point within the circle whose radius is r, but has imaginary 
roots for points outside. 
It is the purpose of this paper to indicate a simple method, 
or an extension of familiar methods, of expressing these lim¬ 
itations and peculiarities. 
Description of the Functions .—The principle on which the 
following functions are based was suggested by a considera¬ 
tion of the equation 
( 1 ) 
When n = 2 this is the familiar equation of an ellipse 
having semiaxes a and b, and as n becomes 4, 6, 8, etc., the 
* A part of this matter was presented at the Buffalo meeting of the 
American Association for the Advancement of Science, 1886, but not 
published. 
10—Bull. Phil. Soc., Wash., Vol. 14. (65) 
