46 DEFINITE INTEGRALS. [INT. III. 



where D sr is the oscillation in the interval # s _ 1} x s , y r -\, y r , 

 approaches the limit zero. 



In forming the double sum we may proceed with the summa- 

 tion according to x first, in which case 



n rb 



(y r - y r -i) Km 2 (x - x s -^f(% s , r) r ) = (y r - y^l f(x, rj r ) dx, 



=oo 1 Ja 



and the double limit is 



rh / rb \ 



( f(a>,y)dx\dy. 



J g \J a / 



Or we may sum first with respect to y, in which case 



ra f rh 



(x s - a? M ) lim S (y r - yr-i)/(&, Vr) = (#, - a?*-i) 



?Jl=00 1 J gf 



and the double limit is 



/*(/*/<.?)%)<** 



But we have always 



w I'n 



2 (yr - 2/r-i) ^ (a?, - ^ 

 1 U 



n fm 



= 2 (X 8 - X s ^) \ S (yr-2/ 



however small # s ^-x and y r y r -\> Accordingly, 



rh f rb 1 rb f rh ) rb rh 



j I J /(#> y) ^4 ^ = J a | J /fa ^ ^} ^ = J a J 



(In writing a double or multiple integral we shall write the 

 integral signs with their limits in the same order as the differ- 

 entials.) 



We might now deduce theorems for the double integral similar 

 to those that we have already deduced for the single integral. In 

 particular, the independence of the limit on the mode of sub- 

 division, and the theorem of the mean may be demonstrated, and 

 the extension of the definition made when the integrand or the 

 limits become infinite. The definition of an integral may be 

 extended to triple and multiple integrals in an obvious manner. 



28. General Definition of Definite Integral. We have 

 in the preceding definition of a double integral assumed that the 

 limits of integration with respect to x and y were independent. 



