96 On the Importance of Quaternions in Physics. 



This is the result we should have obtained if we had considered 

 the change of contents of & fixed unit volume in space. Next 

 consider the rate at which the element gains momentum 

 as it proceeds. We write at once, since momentum cannot 

 originate or be destroyed by processes inside the element, 



where P is the potential energy of unit mass at p, and $Uv 

 is the stress-function due to pressure and viscosity. We 

 have already had the form of this function ; so that the 

 equation transforms at once into 



e^ = -ey'P--VP- n (V 2 <r + ivSV -)', 



which contains the three ordinarily given equations. Here 

 n is the coefficient of viscosity, and the pressure p enters the 

 equation in the form 



To obtain v. Helmholtz''s result as to vortex-motion, put n = 0, 

 and we deduce for the rate of change of vector-rotation of an 

 element, as it swims along, 



|-V#«r=V.vV..«rVvi«i. 



ot 



If the fluid be incompressible, this becomes 



From either it is obvious that the rate of change of the 

 vector-rotation vanishes where there is no rotation. But time 

 forbids any further discussion of formulas. 



Hydrokinetics, as presented by Lagrange and Cauchy, 

 was rather a triumph of mathematical skill than an inviting 

 or instructive subject for the student. The higher parts of 

 it were wrapped up in equations of great elegance, but of 

 almost impenetrable meaning. They were first interpreted, 

 within the memory of some of us, by Stokes and v. Helmholtz, 

 after we know not what amount of intellectual toil. The 

 magnificent artificers of the earlier part of the century were, 

 in many cases, blinded by the exquisite products of their own 

 art. To Fourier, and more especially to Poinsot, we are 

 indebted for the practical teaching that a mathematical for- 

 mula, however brief and elegant, is merely a step towards 

 knowledge, and an all but useless one, until we can thoroughly 

 read its meaning. It may in fact be said with truth that we 



