PAPER BY PROF. HELMHOLTZ. 123 



form velocity. For the surface Ih we put a. x for this velocity, for the 

 surface II, we take (— a 2 ) since we give the latter a motion in the oppo- 

 site direction to that which would be given to it in the normal cases 

 where the wind outruns the wave. 

 We have at once 



+fi=«i. A 

 — f 2 =« 2 . A 



and in the higher layers of the fluid 



$\+<P\ i=+a x ($+yi)+h 



where h x is a constant to be determined by the equation (le). 

 Similarly 



^•2+ qj 2 i=—a 2 {x+yi) + h 2 



For plane boundary surfaces when for these as above assumed 

 //•, = >/•.,=(), and also #=0. we should also have h x and 7* 2 both equal to 

 zero, and the living force in this case becomes 



i 2 1 =-' M -p 2 . f 2=l y a 2 2 . H 2 X 



When on the other hand billows have arisen, L x is smaller for a con- 

 stant value of «] and therefore also of h, since, as we have seeu then a 

 negative value of 6L\ results from an increase in the altitude of the 

 wave. We can therefore under these circumstances put 



a 



L x =^ar {B x —r x ).X 



wherein r x has a positive value that depends on the form and height 

 of the wave, but not on H x . If we imagine H x increased by the quan- 

 tity D H x and the quantity L x correspondingly increased by D L x then 

 iu the strip thus added to the field the velocity is uniformly equal to a x 

 and therefore 



DL x =%a\.DH x 



L x +DL x ="l-«\ [(H x + l)H x )-r x ~^.X. 



Therefore the same value of r x also holds good for the greater alti- 

 tude independent of the value of D H x . 

 The formula (4) gives directly 



t>i=-fi {Hi-r{) (4«) 



Compared with galvanic conditions, p x measures the total flow or the 

 intensity of the current; f, is the difference of potential between the 

 boundary surfaces. Hence (H x -r x ) is the conductivity which is pro- 



