PAPER BY PROF. HELMHOLTZ. 119 



"We now indicate by a a proper fraction and imagine that we have ex- 

 ecuted a variation of Li to the amount expressed by ar, such as would 

 correspond to the variation a. d^. On the other hand we perform the 

 variation dLi^ to the amount (1— «). Then the total variation for ^ is 



6^= [«+(!-«)] 6^^ 



6L = a. 6Li + (1 — ^1'). 6L2. 



If now SLi > 6L2 we obtain the maximum variation of dL when we 

 make a = 1; but for the opposite case we shoukl have to make « = 0. 

 Tlius 6L attains the greatest value that it can have for the given value 

 of (J^ and the adopted form of wave. 



When the greatest positive value of SL is smaller than 6<^ then a 

 value for pi^ can be found that in any case will make 



and therefore, for at least one method of change of form, which need not 

 necessarily be a minimal form, will make the variation d {0 — L) nega- 

 tive. 



Since ^ always remains finite one can always execute finite varia- 

 tions in its magnitude that shall be of the same order of magnitude as 

 tbe displacement 6N of the elementary line ds, and which latter give 

 always finite variations of Li and L-z, at least for finite velocities of fiow 

 along the surface. 



Infinite velocities can only occur at the projecting cusps of the wave- 

 lines and, when there is a current there, give infinite negative pressures, 

 that is to say, the phenomena of breaking or frothing. Oulj' when there 

 exists no relative motion of the wave with respect to the medium into 

 which the sharp edges of the waves project, namely, when the wind has 

 precisely the same speed as that of the wave, can such cusp points 

 long endure. 



Except these latter cases, that lie on the boundary of breaking and 

 frothing, we shall therefore for all continuously curved forms ot waves 

 have for every S^ a maximum of 6L of the same order of magnitude. 



And when we seek for the smallest value of the ratio -. — and seek for 



d(p 



a value of p^ which shall be greater than the greatest of the values of 



1 



6L thus obtained, then for the corresponding strength of current the 



60 



possibility of stationary wave-formation for the prescribed wave-length 



A is entirely excluded. 



Therefore stationary waves of a prescribed wave-length are only possible 

 for such values of the velocities of flotv \\- and p2^ as are less than cer- 

 tain definite extreme limits. 



On the other hand, these same considerations further show that the 



