NOTES. 243 



i.e. by (4) and similar equations 



.du du du 



dv dv dv 

 dx-^dy-^T. 



dw dw d 



This being true for any small circuit whatever, the coefficients of 

 -4, B y C on the left-hand side must separately vanish. The equations 

 thus obtained are equivalent to (3). 



If we introduce the Lagrangian notation of Arts. 16, &c., and apply 

 (5) to the circuit which initially bounded the rectangle dbdc, we have 



and therefore 



where is the initial value of at (a, b, c). This and two similar 

 equations which may be written down from symmetry are Cauchy s 

 integrals of the Lagrangian equations. These integrals contain, as 

 Stokes pointed out, the first rigorous proof of Lagrange s theorem. See 

 Art. 23. The remark that they follow in the above manner from 

 Thomson s circulation-theorem is due to Prof. Nanson. 



If we eliminate p between the equations of motion of a viscous 

 liquid [(15) of Art. 180], we obtain three equations of the form 



9 .du du du , , 



The first three terms on the right-hand side express, as we have seen, 

 the rate at which varies for a particular particle when the vortex-lines 

 move with the fluid and the strengths of all the vortices remain constant. 

 The additional variation of due to viscosity is given by the last term 

 of (4), and follows the same law as the variation of temperature due to 

 conduction of heat in a medium of uniform conductivity. It appears 

 from this analogy that vortex-motion cannot originate in the interior of 

 a viscous fluid, but must be diffused inwards from the boundary. 



162 



