NOTE ON STOKES'S THEOREM IN CURVILINEAR 



CO-ORDINATES. 



By Arthur Gordon Webster. 



Presented April 13, 1898. 



The expressious for the components of the curl of a vector point- 

 function, when required in terms of orthogonal curvilinear co-ordinates, 

 are usually obtained by direct transformation of their values in rectan- 

 gular co-ordinates. 



The proof of Stokes's Theorem, given in my Lectures on Electricity 

 and Magnetism, due to Ilelmholtz, may be easily adapted to curvilinear 

 co-ordinates so as to prove the theorem independently of rectangular 

 co-ordinates. 



Let Pi, Po, Pi, be the projections of a vector P on the varying direc- 

 tions of the co-ordinate axes at any point. Let the projections on the 

 same axes of the arc ds of a curve connecting the points A and B be 

 d Si, ds.2, ds^. The theorem concerns the line integral of the resolved 

 component of the vector along the given curve. 



fPcos (P, ds) ds 



= I Pi dsi + P., ds.2 + Ps dss. 

 But in terms of the curvilinear co-ordinates pi, p^, ps, we have 



ho 



s=l,2, 3. 



Let us now make an infinitesmal transformation of the curve, so that 

 the transformed curve shall lie on a given surface containing vl and B, 



