MATHEMATICS: G. A. BLISS 
173 
A NOTE ON FUNCTIONS OF LINES 
By Gilbert Ames Bliss 
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO 
Presented to the Academy, December 2, 1914 
A function of a line 
b 
F[y{x)] 
(1) 
a 
may be regarded as a generalization of a function F (yi, y2, . . . , yn) of 
a finite number of variables (i = 1, 2, . , n). Instead of having a 
well defined value when a point (yi, y^, . . . , yr) is given, the value of 
the function (1) is determined only when an infinitude of 3^- values 
belonging to an arc of the form 
is prescribed. The index i ranging over the integers 1, 2, . . ., in the 
function of a finite set of variables, is replaced in (1) by the index x 
ranging over the interval a ^ x ^ 6. Examples of functions of this 
sort are the length of the arc (2) , the time required by a heavy particle 
to fall from one end of the arc to the other, the area of the surface gen- 
erated by revolving the arc about the x-axis, and many others. 
For such functions Volterra has defined continuity and a derivative 
function. 1 The function (1) is said to be continuous at the arc (2) if 
for any given e there always exists a 5 such that 
Let {x) be further restricted not to change sign and to vanish identi- 
cally except on an interval of length less than h containing a fixed 
value X = Then the derivative of F at the value ^ is defined by the 
equation 
y = ^{x) 
{a<x<b) 
(2) 
F'y{x), ^] = Urn 
AF 
5=0 0- 
h =0 
where 
