4!fl# Mr green, on THE DETERMINATION OF THE 



% + %^%+ +% z 1 and u = .(3) 



are satisfied simultaneously, considering as we shall in what follows 

 the limits of the multiple integral (1) to be determined by the conr 

 dition (a)*. 



In like manner it is clear that when 



Z^2+ Jl+ + 77-2>^ (4)» 



a?' 



the expansion of V in powers of u will contain none but the even 

 powers of this variable. 



Again, it is quite evident from the form of the function f^ that 

 when any one of the * + 1 independent variables therein contained be- 

 comes infinite, this function will vanish of itself. 



3. The three foregoing properties of F combined with the equa- 

 tion (2) will furnish some useful results. In fact, let us consider the 

 quantity 



fd.,d^,...d..duu-'.[[^)\ [^)\ + (g)\ (^^)] (5) 



where the multiple integral comprises all the real values whether posi- 

 tive or negative of x^, x^, x,, with all the real and positive values 



of u which satisfy the condition 



/!« 2 A< 2 « 2 /|/2 



^■^^^ + -^^^^F^^ ^^^ 



* The necessity of this first property does not explicitly appear in what follows, but 

 it must be understood in order to place the application of the method of integration by 

 parts, in Nos. 3, 4, and 5, beyond the reach of objection. In fact, when V possesses this 

 property, the theorems demonstrated in these Nos. are certainly correct: but they are not 

 necessarily so for every form of the function V, as will be evident from what has been 

 shewn in the third article of my Essay on the Application of Mathematical Analysis to 

 the Theories of Electricity and Magnetism. 



