31] 



DEFINITE INTEGRALS. 



55 



Y$y + Zbz I vanishes. Collecting those terms under the 

 sign of integration that do not cancel, we have, 



Now the determinant Bydz Szdy is the area of a parallelogram 

 in the F^-plane the projections of whose 

 sides on the Y- and ^-axes are dy, dz, y> 

 Bz. That is, if we consider the infinitesimal 

 parallelogram whose vertices are the points 

 s, s + ds and their transformed positions, the 

 above determinant is the area of its pro- 

 jection on the F^-plane. If the area of the 

 parallelogram is dS and n is the direction of its normal, we 

 as in 30 



Sydz &zdy= dS cos (nx), 



Sz dx Sxdz = dS cos (ny), 



Sxdy ydx = dS cos (nz\ 



and 



dY\ 



w cos (nx} 



dZ 



-- cos 



+(E-!?)<n 



which is in the form of a surface integral over the strip of in- 

 finitesimal width. 



If we again make an infinitesimal transformation, and so 

 continue until the path has swept out any finite portion of a 



FIG. 11. 



