ON THE EVALUATION OF DEFINITE 

 MULTIPLE INTEGKALS*. 



THE following pages contain some general results obtained 

 by means of Fourier's theorem. A few words will be sufficient 

 to explain the manner in which it has been applied. 



A definite multiple integral, where the limits are given by 

 the inequality 



may be treated as if the limits of the different variables were 

 independent of one another, provided the function under the 

 signs of integration be considered discontinuous, and equal to 

 zero whenever f(xy ...... ) transgresses the assigned limits \ 



and h. This idea has been made use of by M. Lejeune Dirichlet. 

 It had, however, occurred to me before I was acquainted with 

 his paper on multiple integrals, and the way in which I have 

 applied it is, I believe, new. 



Suppose the function to be integrated were of the form 

 <f) (xy ...... }^{f( x y ...... )) : the limits being given by the in- 



equality already mentioned. Then, by Fourier's theorem, writing 



1 /"* [ k 



i|r/. = - I doil -^ru . cos a (/. - u) 



7T J o J HI 



du, 



for all values of f. which lie between \ and h ; moreover for 

 the purposes of integration the formula may be applied, except 

 in particular cases, so as to include these limiting values 

 (see Poisson, Theorie de la Chaleur) ; while for all values of/. 

 which lie without these limits, the second side of the equation is 

 equal to zero. Consequently the integral 



* Cambridge Mathematical Journal, No. XIX. Vol. iv. p. i, November, 1843. 



