29, 30] VARIATION OF LINE INTEGRAL, 



an integration by parts, 



85 



B 



/jrd(8x) 

 ds 



XSx 



B 



A dx j 



- I dx-j-ds. 

 J ds 



Now we have 



dX = dX dx dX dy dX dz 

 ds dx ds dy ds dz ds 



Performing similar operations on the other terms we have 



Y8y 

 dY 



dXdx ( dXdy dXdz\ 

 dx ds dy ds dz ds) 



9x 

 dZ, 



dZ 



* IdYdx dY dy dY 



~ d y(dxds"^d^"J7 



-dz ( 



^ ^ 



\dx ds dy ds dz ds. 



ds. 



Now if in the variation the en'ds of the curve A and B are 

 fixed ? dx, dy, dz vanish for A and B, and the integrated part 



Xdx + Ydy -f 



vanishes. Collecting those terms under the 



sign of integration that do not cancel 7 and removing the factor ds 

 we have, 



Now the determinant 

 8yds 8zdy 



is the area of the parallelogram in 

 the YZ- plane the projection of whose 

 sides on the T- and x?-axes are 

 dy, dz, dy, dz. That is, if we con- 

 sider the infinitesimal parallelogram 

 whose vertices are the points s, s + ds, 

 and their transformed positions, the 

 above determinant is the area of its 

 projection on the YZ- plane. If the 

 area of the parallelogram is dS and n 

 is the direction of its normal, we have 



Fig. 20. 



