242 VORTEX MOTION. [CHAP. VII 



ing system of electric current-sheets. This proves a statement 

 made by anticipation in Art. 141. 



153. Under the circumstances stated at the beginning of 

 Art. 152, we have another useful expression for T\ viz. 



~P\\\\ 



To verify this, we take the right-hand member, and transform it 

 by the process already so often employed, omitting the surface- 

 integrals for the same reason as in the preceding Art. The first 

 of the three terms gives 



(dv du\ fdu dw\\ , , , 

 y \-j --- 7- }z I-, --- s-U dxdydz 

 U dyj \dz dx)\ 



~3 



Transforming the remaining terms in the same way, adding, and 

 making use of the equation of continuity, we obtain 



^ / / 1 V + ^ + W * + XU cT + ^ IT + ZW d) dvdydz, 

 or, finally, on again transforming the last three terms, 

 ip fff(u* + v 2 + w*) dxdydz. 



In the case of a finite region the surface-integrals must be retained. This 

 involves the addition to the right-hand side of (4) of the term 



P JJ (Q u +mv+nw)(xu+yv + zw')-%(lx+my+ nz) q 2 } dS, 

 where q 2 = u?+v 2 + iv 2 . This simplifies in the case of & fixed boundary*. 



The value of the expression (4) must be unaltered by any displacement of 

 the origin of coordinates. Hence we must have 



JJJ Otf- i0j;) dxdydz = Qj\ 



SHM-vQ dxdyds=0, [ ........................... (i). 



JJJ (^17 - *y|) dxdydz=Q } 



These equations, which may easily be verified by partial integration, follow 



also from the consideration that the components of the impulse parallel to the 



coordinate axes must be constant. Thus, taking first the case of a fluid 



enclosed in a fixed envelope of finite size, we have, in the notation of Art. 150, 



P = pMudxdydz-ptil<j>dS ........................ (ii), 



whence * = dxddz - 



Cf. J. J. Thomson, I.e. 



