504 REPORT — 1887. 



at the end towards wHch the liquid is forced. With the same substitutions 

 as before, this becomes 



sl^-l-E .... (28) 



differing from von Helmholtz's result by the factor Ijcl, as in the previous 

 cases. ' 



This result does not involve the dimensions of the tube, and raay 

 therefore be surmised, like that contained in (10), to be of much wider 

 application than to the particular form of channel above Considered. It 

 may be shown, in fact, that if a liquid is forced by pressure through 

 any system of channels with homogeneous walls, and no external path is 

 provided for the electricity set free at various points of these, the result- 

 ing distribution of electric potential is given by 



<^= -"^1^ + const (29) 



In the first place it follows from (12) that this value of (^ satisfies 



vV = 0- 



We have next to take account of the fact that the integral amount ot 

 electricity which, in consequence of the slipping of the superficial film of 

 liquid, crosses the contour of any elementary area dS of the wall is not 

 in general accurately zero, and that each such element cZS must be re- 

 garded, in relation to the conducting mass of liquid, as a (positive or 

 negative) ' source ' of electricity. If the origin be taken in this element, 

 and the axis of z normal to it, the strength of this source is 



\dx chjl 



dz 



Now at the origin we have 



w=0 



j/du dw\ 



\dz dxj 



-.fdv .dw\ 



\dz dy J 



(30) 



and if I be small in comparison with the radii of curvature of the walls, 

 &c., we may neglect the second terms in the brackets.- Under the same 

 circumstances we shall also have, approximately, 



**=Z ] 



dx dzdx I fc-i N 



dv^^d^ f • • • • ^""-^^ 



dy dzdy ) 



' Dorn infers from a comparison of his experimental results with von Helmholtz's 

 formula that for water in contact with the glass of his tubes E/D = 3-9, about. 



^ The justification of these and the following approximations is given in the 

 Appendix 



