133, 134] STOKES'S THEOREM IN CURVILINEAR COORDINATES. 381 



35) 



' 



which as above is equal to 



Integrating the other terms in like manner we obtain the general 

 formula, 



Q*\ CCC\i*dUdV , J9 3UdV . 19 dU dV\ dq.dq.dq, 

 36) / / / {W* js -- h^lo 5 --- h --- 

 y ' 8 



= -J J Z? 



in which each integrand is found to correspond to one of those in 

 115, 20). 



134. Stokes's Theorem in Orthogonal Curvilinear Co- 

 ordinates. The proof of Stokes's theorem given in 30 can be 

 easily adapted to curvilinear coordinates. 1 ) Let P t , P 2; P 3 be the 

 projections of a vector P on the varying directions of the tangents 

 to the coordinate lines at any point. Then, the projections of the 

 arc ds being ds 19 ds%, ds s , we consider the line -integral 



37) I =p cos (P, ds) ds = i ds i + P 2 ^ 4- P 



A 

 B 



where 



1) Webster. Note on Stokes's theorem in Curvilinear Coordinates. Bull. 

 Am. Math. Soc., 2nd Ser., Vol. IV., p. 438, 1898. 



