143-144] VON HELMHOLTZ' THEOREM. 227 



the rate at which da is increasing is equal to the difference of the values of u 

 at the two ends, whence 



It follows, by (iii), that 



du r] du du 

 jFr~ f -- 7 he 7 h c - -j- 

 Dt /j dx p dy p dz 



Von Helmholtz concludes that if the relations (viii) hold at time t, they will 

 hold at time t + 8t, and so on, continually. The inference is, however, not quite 

 rigorous ; it is in fact open to the criticisms which Sir G. Stokes* has directed 

 against various defective proofs of Lagrange's velocity-potential theorem f. 



By way of establishing a connection with Lord Kelvin's investigation we 

 may notice that the equations (i) express that the circulation is constant in 

 each of three infinitely small circuits initially perpendicular, respectively, to 

 the three coordinate axes. Taking, for example, the circuit which initially 

 bounded the rectangle 86 Sc, and denoting by A, S, C the areas of its pro- 

 jections at time t on the coordinate planes, we have 



d(b,c) d(b,c) 



so that the first of the equations referred to is equivalent to 



144. It is easily seen by the same kind of argument as in 

 Art. 41 that no irrotational motion is possible in an incompressible 

 fluid filling infinite space, and subject to the condition that the 

 velocity vanishes at infinity. This leads at once to the following 

 theorem : 



The motion of a fluid which fills infinite space, and is at 

 rest at infinity, is determinate when we know the values of the 



* I. c. ante p. 18. 



t It may be mentioned that, in the case of an incompressible fluid, equations some- 

 what similar to (iii) had been established by Lagrange, Miscell. Taw:, t. ii. (1760), 

 Oeuvres, t. i., p. 442. The author is indebted for this reference, and for the above 

 criticism of von Helmholtz' investigation, to Mr Larmor. Equations equivalent to 

 those given by Lagrange were obtained independently by Stokes, I c., and made the 

 basis of a rigorous proof of the velocity-potential theorem. 



t Nanson, Mess, of Math., t. vii., p. 182 (1878). A similar interpretation of 

 von Helmholtz' equations was given by the author of this work in the Mess, of 

 Math., t. vii., p. 41 (1877). 



Finally we may note that another proof of Lagrange's theorem, based on ele- 

 mentary dynamical principles, without special reference to the hydrokinetic equa- 

 tions, was indicated by Stokes (Camb. Trans., t. viii. ; Math, and Phys. Papers, t. i., 

 p. 113), and carried out by Lord Kelvin, in his paper on Vortex Motion. 



152 



