138 
Proceedings of the Royal Society of Edinburgh. [Sess. 
a sequence with a as limiting value, the value a being expressly excluded 
from being a possible value for any x n , and further, the series 
f(x i) , f(x 2 ) , . f(x n ) , . . . . 
has only one limiting value u (which may be + oo or — oo ), then we say 
that n is one of the single limits of f(x) at a , and write 
u = Lt fix n ) = one of the L It fix). 
n=oo x=a 
The function f(x) may or may not be defined for the value x = a m , 
but, if defined, the value fia) must be disregarded in considering the 
limits at a. 
If f(x , y) is a function of two independent variables x and y, it becomes 
a function of x alone when we keep y constant, and has a corresponding 
set of simple limits, 
L It fix, y) . 
x=a 
If there is only one such limit for each value of y* this limit defines a 
function of y , and has, as such, a set of limits for y — b\ these are called 
the repeated limits of f(x ,y) first with respect to x and then with respect to 
y , and written 
Lit L t fix,y). 
y—b *=« 
Similarly, if there is only one limit when y is kept constant, 
Lit L t fix , ?/) 
x=a y=b 
denotes the repeated limits of f(x , y) first with respect to y and then with 
respect to x. 
The idea of double limits of fix , y) is different. 
Let x lJ x 2 , . . . . be a sequence of values of x having a as limit, but not 
including a, and y x , y 2 , . . . .a sequence of values of y having b as limit, 
but not including b, then f(x,y) has at the points (x 1 ,y 1 ), (x 2 ,y 2 ), . . . . 
* The same is true if a law is given by means of which one of the limits is defined 
for each value of y ; e.g. the maximum limit. In this case a number of quantities can 
be identified as limits of limits of fix , y), or repeated limits ; e.g. the upper upper limit, 
the upper lower limit, etc. The whole set of such limits, which may theoretically be 
denoted by 
Lit Lit fix , y), 
y=b x=a 
can only be regarded as perfectly defined when all possible laws which can be used are 
in some manner specified. 
