ON stokes' theorem. 299' 



Thus U and Y are single-valued and continuous functions within 

 the plane area A, bounded by the projection of the bounding edge 

 of the surface on plane x y, and we obtain 



* -^ \3y dx dx dy J ' ■' \9y '^ dx J 



But denoting by I, vi, n the direction cosines of the normal at x^ 

 y, z to surface (1) reckoned in positive direction we have — 

 9z dz 



and 



dx dy =: ndS = 



V/Z 



Where f/S is the element of area at x, y, z of the surface con- 

 sidered. We have besides by differentiation of (1) — 



— dy -\ dx = dz 



dy dx 



And now noticing that a passage from one point to another point 



on the area A corresponds to a passage from a point to another 



point on surface z-=f{x,y), and similarly that a passage from 



one point to another point on bovmding curve of area A corresponds 



to a passage from one jjoint to another point on the bounduig edge 



of our surface, we can write from (2) — 



90 d(h 



m — 



dy dx , 



where the first integral is to be taken over surface z ■=. f {x, y) 

 and the second one over its bounding edge. 



Let us now suppose that z is a many- valued function of x and y^ 

 or, in geometrical language, that a ])arallel to oz meets the surface 

 at several points ; then the projection of surface z =.f(x, y) on 

 plane xy is no more the projection of its bounding edge, but 

 an other curve, B. 



Now let us consider the narrow path determined on the surface 

 by two planes, parallels to xoz, and infinitely near. On the corres- 

 ponding projection we have an infinitely narrow strip parallel to 

 ox. If we go over the surface along that path always in the same 

 direction — for instance, the direction of the arrow — the corres- 

 ponding motion on plane xy will consist in : — Starlmg from A to B,. 



.//•( 



r-^- m^^) dS =f(fidz 



