12 Proceedings of the Roi/al Irish Acadcmij. 



If we integrate with respect to // in the interval !/s<v<yHh "i' i" 

 that part of this interval which contains points of F, we find 



f ^^*' d>/ f /•. d.v <^ 2 M,., «,, + il/ 1 ^'^' i(,) . dii. 



Adding all such inequalities, we get 



■ f dy \ f.dx-<1M,.sars + Mc. 



J F }Gy 



Now, £ is at our disposal, 



The above proof also yields the inequality 



\dyyd.^.]j.de, 



where the y-integratiou is taken over any continuous interval containing F. 

 By changing the sign of / we may at once infer the inequalities for the 

 lower double integral. 



• 14. These inequalities are less general than in the usual case, but they 

 enable us to make the following inferences : — 



A. If the dmihle integral exists, and if either repeated integral exists, 

 the double integral is equal to the repeated integral. 



B. If each repeated integral exists, and if the doiohle inteqral exists, 

 the order of integration may be reversed. 



C. The first integration in the repeated integral is made over a seetion 

 of the domain ; the second, integration is made over the jirojeetion of the 

 domain on the axis, or over the ivhole side of the fundamental rectangle 

 unless ivhen it ceases to be integrable in tJie extended region. 



15. There is no difficulty in extending the methods of this paper to sets 

 of any number of dimensions. The inequalities for the n-ple integral in an 

 open domain of the second kind in ?i-diniensional space are 



[dxi l^dxi .... '\f.dx„ ^~Sfde, 

 in which there are n - 1 lower integrations, 

 and jdx, ]dx2 .... ifdxn $ j fde, 



in which there are n - 1 upper integrations. 



The conditions under which the n-ple integral may be determined by n 

 repeated integrations are not essentially difl'erent from those given for the 

 double integral. 



