33] DEFINITE INTEGRALS. 



or, transposing, and denoting the symmetrical integral by J, 



mtv ,dUW-\, 



" 



~" I / l a" cos ^ nx ^ + 7T cos ( n y) + T~ cos ^"^ I 



This result is known as GREEN'S THEOREM*. 



By the definition of differentiation in any direction the paren- 

 thesis in the surface integral on the right is 



if P v is the parameter of V. 



Since the integral on the left is symmetrical in U and F, we 

 may interchange them on the right, so that 



Writing this equal to the former value, and transposing, we obtain 



which will be referred to as Green's theorem in its second form. 



It will be noticed that the integrand on the left in the first form 

 is the geometric product of the parameters of the functions 7 and F, 



We shall, unless the contrary is stated, always mean by n the 

 internal normal to a closed surface, but if necessary shall dis- 

 tinguish the normals drawn internally and externally as ^ and n e . 

 If we do not care to distinguish the inside from the outside we 

 shall denote the normals toward opposite sides by n^ and n 2 . 



* An Essay on the Application of Mathematical Analysis to the theories of 

 Electricity and Magnetism. Nottingham, 1828. Geo. Green, Reprint of papers, 

 p. 25. 



