298 PROCEEDINGS OF SECTION A. 



The most satisfying process of demonstration is certainly based 

 upon Stokes' theorem. However — strange enough to say — although 

 the importance of that theorem is everywhere recognised, a correct 

 demonstration of it I have been unable to find. 



Analytical transformations either establishing directly Stokes' 

 theorem or deducing it from Green's theorem are easily found, but 

 the difficulty, I think, consists in showing clearly what conditions 

 must be fulfilled by the functions considered to satisfy the theorem. 



I propose to give here a complete and correct demonstration of 

 Stokes' theorem, furnishing at the same time a criterion not yet 

 given for the conditions to be satisfied by the closed curve and the 

 functions X, Y, and Z. 



I will use, as far as analytical transformations are concerned, a 

 process somewhat similar to the one used by Minchin in his 

 Statics.* I will start from the following well-known theorem, 

 which is a particular case of Greerx's theorem on the transformation 

 of a triple integral. 



Theorem. — If U and V are two functions of the realf variables x 

 and .V, single valued, and continuous (and consequently finite) 

 within the plane area A (that is to say, for all values of x and y 

 corresponding to points inside A) the double integral 



taken over the whole area A is equal to the value of the simple 



integral ^ 



/Vf/y -f Vdx 



taken along the boundary curve of area A (that is to say, taken in 

 giving successively to x and y all the systems of values which 

 correspond to the different points of the boundary curve) supposed 

 described in such a manner that the area be always kept on the left 

 hand side.]: 



Now let us put u = (ii — V = ^ 



'^ dx" ^ dy 



in the expression of that theorem, being a single-valued and 

 continuous function of x., y and :; for every point on a surface 



(I) ~=/(^>y) 



z being first supposed to be a single-valued function of x and y, or, 

 in geometrical language, any parallel to oz being supposed to meet 

 the surface only at one point, and therefore the projection of the 

 surface on plane xy being then the projection of its bounding 

 edge. . 



* Minchin : A treatise on Statics, 4th edition, vol. ii., pages 2i2-245. 



t The theorem is also true for comple.'c variables. 



X By replacing V by U ^and U by - U ^1 that particular case of Green's theorem is 



obtained under the ordinary torm. 



