540 VISCOSITY. [CHAP, xi 



The admissible values of w, and the ratios A : B are as a rule determined 

 by the boundary conditions. The arbitrary constants which remain are then 

 to be found in terms of the initial conditions, by Fourier s methods. 



In the case of a fluid extending from y= oo to y = + GO , all real values of 

 m are admissible. The solution, in terms of the initial conditions, can in 

 this case be immediately written down by Fourier s Theorem (Art. 227 (15)). 

 Thus 



u = - { dm( /(X)cosw(y-X)e-&quot; m2 ^X ............... (iv), 



&quot;&quot; J J -oo 



if t*=/(y) ....................................... (v) 



be the arbitrary initial distribution of velocity. 



The integration with respect to m can be effected by the known formula 



w/todfc-jf-) 1 *-^ ..................... (vi). 



\ a / 



We thus find u= i- f e ^^ 1 ** f(\} d\ .. ... (vii). 



2 &amp;lt;w_. 



As a particular case, let us suppose that f(y}=U, where the upper or 

 lower sign is to be taken according as y is positive or negative. This will 

 represent the case of an initial surface of discontinuity coincident with the 

 plane y = 0. After the first instant, the velocity at this surface will be zero 

 on both sides. We find 



u= U r 

 2 (**)*&amp;lt;/ 



(viii). 



By a change of variables, and easy reductions, this can be brought to the 

 form 



where in Glaisher s (revised) notation f 



Erf x = e~ x &quot;dx .............................. (x). 



The function 27r~*Erf x was tabulated by EnckeJ. It appears that u will 

 equal \U when y/2j/M = 4769. For water, this gives, in seconds and centi 

 metres, 



* Lord Rayleigh, &quot;On the Stability, or Instability, of certain Fluid Motions,&quot; 



Proc. Lond. Math. Soc., t. xi., p. 57 (1880). 



t See Phil Mag., Dec. 1871, and Encyc. Britann., Art. &quot; Tables.&quot; 



J BerL Ast. Jahrbtich, 1834. The table has been reprinted by De Morgan, 



Encyc. Metrop., Art. &quot;Probabilities,&quot; and Lord Kelvin, Math, and Phys. Papers, 



t. iii., p. 434. 



