31, 32] 



DEFINITE INTEGRALS. 



57 



occurrence in mathematical physics. The curl may be derived 

 from the primitive vector by the application of Hamilton's vector 

 differential operator V ( 16) to a vector point-function R, 



dy 



/ax dY dz- 



dY dZ\ ./dZ 

 j j _i_ ^ j 



dy dzj \]cjy 



= 



dz 



d_z 



to 



dz 



dx 



dz 



-o- 



dy 



curl R = WE = i 



So that the vector part resulting from the application of the 

 operation V to a vector point-function gives its curl. The scalar 

 part 



\ dx dy dz 

 has an important interpretation to be given shortly. 



[The significance of the geometrical term curl can be seen from 

 the physical example in which the y 

 vector R represents the velocity of a 

 point instantaneously occupying the 

 position x, y, z in a rigid body turn- 

 ing about the ^/-axis with an angular 

 velocity o>. Then the vector R = wp 

 is perpendicular to the radius p and o 

 its components are 



FIG. 12. 



X = R cos (Eos) = - R sin (px) = -R- = - 



= xco, 

 P 



Y = R cos (Ry) R cos (px) = 



where &> is constant, and 



dY dX_ 



dx dy ~ 



So that the ^-component of the curl of the linear velocity is twice 

 the angular velocity about the ^T-axis.] 



32. Lamellar Vectors. In finding the variation of the 

 integral / in the previous section, since the variations &e, Sy, 8z 

 are perfectly arbitrary functions of s, if the integral is to be inde- 



