540 VISCOSITY. [CHAP, xi 



The admissible values of m, 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 = - <x> to y = + QO , 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 



j / ,-w 



u = - I dm I f(\)cosm(y \)e~ vm d\ (iv), 



n J J -oo 



if =/(y) (v) 



be the arbitrary initial distribution of velocity. 



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



We thus find 



f e- (y ~ x ^ vt f(\)d\.. ...(vii). 



_ 00 



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 



TT C<B 



U " \ \& ~~~ & } ff\ .... ....... ( Vlll ). 



O f ./N* / 

 ^ \TTVl) J o 



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

 form 



where in Glaisher's (revised) notation f 



Erf #= -*<fc .............................. (x). 



9 



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

 equal \U when y/2v*t* ='4769. For water, this gives, in seconds and centi- 

 metres, 



* Lord Rayleigh, "On the Stability, or Instability, of certain Fluid Motions," 



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



t See Phil Mag., Dec. 1871, and Encyc. Britann., Art. " Tables." 



J Berl. Ast. Jahrbuch, 1834. The table has been reprinted by De Morgan, 



Encyc. Metrop., Art. "Probabilities," and Lord Kelvin, Math, and Phys. Papers^ 



t. iii., p. 434. 



