WEBSTER. — NOTE ON STOKES's THEOREM. 385 



is equal to the area of the projection on the surface pi of the infinitesi- 

 mal parallelogram swept over by the arc d s during the transformation. 

 Calling this area d S, and its normal n, we have 



(Spo d p3 — 8 pg d P2) ^ cos (n«i) d S, 



8 p2 d ps — 8 ps d po, = k.2 hs cos (nui) d S. 



Now, repeating the transformation so that the original curve 1 passes 

 into a second given curve 2, the total change is represented by the sur- 

 face integral over the surface lying between the curves, 



/./=/,-/. =//[/M. 11 g) -4 (Jf)\ CO. („».) 



+ Kh < ^— ( ; M — ?— ( M COS {nn^) d S. 



Spi \ ^^2 / ^P2 V ^^1 



j COS {nus) 



But the difference of the line integrals /, — I^ is the line integral 

 around the closed contour 12, so that we have the line integral of the 

 tangential component of the vector P ai'ound the closed contour proved 

 equal to the surface integral over a surface bounded by the contour 

 ot the normal component of a vector f2 whose components are 



VOL. XXXIII. — 25 



