510 



XL HYDRODYNAMICS. 



instant, the function always vanishes. Stokes 1 ) objects to this method 

 of proof as not rigorous inasmuch as it is not evident that the func- 

 tions I, rj f can be developed by Taylor's theorem, and replaces it 

 by the following demonstration. 



Let L be a superior limit to the numerical values of the coef- 

 ficients of 9 i in the second member of equations 57). Then 

 Q Q Q 



evidently , 17, cannot increase faster than if their numerical or 

 absolute values satisfied the equations 



58) 



A 



dt 

 _d 

 dt 

 d 



dt \ Q 



instead of 57), |, ??, g vanishing in this case also when # = 

 these three equations and writing 



Adding 



we obtain 



59) ^ 



The integral of this equation is 



and since 52 = when t = 0, c must be zero, and 1 is always zero. 

 Since the sum of the absolute values cannot vanish unless the separate 

 values vanish, the theorem is proved. 



Let us now consider two points A and B lying on the same 



vortex line at a distance apart ds = s > where s is a small constant. 

 Since the particles lie on a vortex -line we have 

 xvx dx dy dz ds e 



We have for the difference of velocity at A and B 



1\ ^ U J % U 7 V U J 



bl ) Un MA == o u>% ~~r ^r~ dy ~r ~^~ u>% 



or by equations 57), 

 62) 



J_ 



Q dx^ Q 



1) Stokes, Math, and Phys. Papers, Vol. II, p. 36. 



