8 Procfii/iiii/s (if the /loz/iil frisk Acailcmn. 



Our. 2. — If I) lie ail n\<i'n duiiKiin nl' the second kind, we can find a 

 sequence of open domains of the first kind D,, Di, . . . &c., such that D„ is 

 contained in J), and that the content of Z* = ™ of the content of D„. 

 Jordan's polygons defiumg " I'etendue interieure " may be cited as an example. 

 It is clear from Cor. 1 that 



C:"^^'''] integral in B = li^i^ f "P^^^" ) integral in 7>„. 

 Uower/ ■ '^ "^>"^ Uowery =' 



TJie Double Integral for a Plane Closed Set. 



9. Let a plane closed set S be contained in a fundamental open rectangle 

 H, and let D be the open domain complementary to S with respect to JR, then 

 I define the 



I , I inteoral in *S' = ( , ) inteo-ral in li -I , ] integral in D. 



Vlower/ \lowery ° Vlowery ° 



It follows from Cor. 1, § S, that the ( , 1 integral in a closed domain 



\lowery = 



of the first kind is equal to the ( , 1 integral in the open domain obtained 



by omitting the boundary points. This is not generally the case for domains 

 of the second kind. 



Also, by Cor. 1, we may replace the fundamental rectangle by any open 

 domain D of the first kind which contains S. Hence, if i), be the open 

 domain formed b}' the points complementai-y to ,S' with respect to B, then 



[ ^" I integral in <S' = ( , " | integral in D -i , ] integral in 2),. 



Vlower/ ° \lowery ^ Viewer; 



Piu'thermore, by Cor. 2 we are enabled to state the more general identity 



that if J) be any open domain of the first or second kind containing *S' and i), 



the complementary open domain, then 



( ^^ 1 integral in *S = ( , ^ | integral in D - 1 , '^^ ] integral in B^. 

 Viewer; "" Vlower^ ^ {lowevj 



10. The extension of the definitions and theorems in §§ 3,4, 5, and 6 now 

 presents no difficulty. It is only necessary to substitute the term " open 

 domain " for " set of open intervals," and " plane content " for " linear 

 content." 



The Bouhle Integral as a Bepeated Integrcd. 



11. Froniier of zero eontent. — If the plane measiu-able set S have a 

 frontier F, the remaining points, being internal points, form an open domain 

 B, and we may write S = F + B. The part F having zero content con- 

 tributes nothing to the integral, and the region of integration may be taken 

 to be the domain of the fir'st kind B. The evaluation of the integral by 



