Prof. Helmholtz on Discontinuous Movements of Fluids. 3-A5 



If c/»log2, the whole expression as far as + AzW becomes real, 

 which latter belongs to the value of ri through its relation to yi. 



The equations (3a) and (3d) correspond, therefore, to a pouring 

 out of a boundless trough into a canal bordered by two planes, 

 the breadth of the canal being 4A7T, and the length of its walls 

 being from a?= — x to <r= — A(2 — log 2). The free divisional 

 line of the current liquid is at first bent from the edge of the 

 orifice a little towards the side of the positive value of x, where 

 for $ = 0, x=— A, and y=+A(§7r + l) it attains its greatest 

 #-value. It then enters the interior of the canal, and at last ap- 

 proaches the two lines y= + A-7T asymptotically ; so that, finally, 

 the breadth of the stream as it flows out is only equal to half the 

 breadth of the canal. 



The velocity along the divisional surface and at the straight 



end of the stream flowing out is -r. Along the rigid wall, and 



-A. 



in the interior of the liquid, the velocity is always less than 



— , so that this form of motion may take place with all values of 



velocity of efflux. 



One especially remarks in these examples how it is shown that 

 the form of the liquid current in a tube may be determined for 

 a long way by the shape of the first portion. 



Addition, relative to electrical dispersion. — If in equation (2) 

 we look upon the value ^ as the power of the electricity, we get 

 the distribution of the electricity in the neighbourhood of the 

 edge of two plane and almost touching screens under the condi- 

 tion that the distance between the two may be considered vanish- 

 ingly small compared with the radius of curvature of their curved 

 edges. This is a very simple solution of the problem which 

 Clausius* has discussed. The same distribution of electricity is 

 obtained as w T as found by Clausius, as far, at least, as it is in- 

 dependent of the curvature of the edge. 



I may add that the same method suffices to determine the 

 distribution of the electricity on two parallel infinitely long 

 plane strips whose four corners (in section) form the points of a 

 rectangle. The potential function yjr is given by an equation of 

 the form 



n^A^+^+Bg^ ... (4) 



where H(w) represents the function developed by Jacobi in the 

 Fundamenta Nova, p. 172, of the numerator of sin am u. The 



* Pogg. Ann. vol. lxxxvi. 



