APPENDIX A: 
NOTE ON A DOUBLE LIMIT PROCESS 
Theorem. Let f(x, y) be a function defined on asx<b, c<y<~, 
with the properties that 
(i) ae y) = y(x) uniformly for each x on asx<b 
(ii) f(x, y) and y(x) are continuous functions defined on 
asx<b. 
Then 
b 
b 
vee \ re, y)ax = [Pe y)ax 
a a 
Proof: Since f(x, y) converges uniformly to g(x) for each x 
on a<x<b, if e>0 be given, then there exists a Y (ce) (depend- 
ing on c, independent of y) such that 
| 7x, y)-9 a) | <== 
whenever y2/(c) for each x on a<x<b(b>a) 
This inequality can be written 
s— - g(x) <f lx, y)< g(x) 
EO 0 Gy b-a 
whenever y>/ (ec). 
Since g(x) and f(x, y) are integrable functions of x 
[oy 
b fo) 
|G -at)ax< [rey dx < \@or eax 
a 
Qa a 
whenever y=V(c). This reduces to the inequality 
b 
D 
[rte y)dx- | oberon <e 
a a 
39 
