ON A MULTIPLE DEFINITE INTEGEAL*. 



IN the eighteenth number of the Journal, I pointed out the 

 mode in which Fourier's theorem may be employed in the 

 evaluation of certain definite multiple integrals. The theorem 

 generally known as Liouville's, and another of the same de- 

 gree of generality, were readily deduced from the considera- 

 tions then suggested. I proceed to another application of the 

 same method. 



fadu 



ab... 



the integral being supposed to involve v variables x, y, &c., and 

 the limits being given by the inequalities 



mx + ny + . . . > h and ^ h'. f 



(Negative as well as positive values of the variables are 

 admissible.) 



DEM. Recurring to the general theorem stated at the com- 

 mencement of the paper already mentioned, we see that the 

 integral whose value is sought is equal to 



r& r * r* ... "$ 



J J-~ J-oo - (a 2 + 



Develope the cosine in a series of products of sines and cosines 

 of simple arcs. Every term involving a sine disappears on 



* Cambridge MathematicalJournal, No. XXI. Vol. iv. p. 116, May, 1844. 



