PAPER BY PROF. HELMHOLTZ. 119 



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

 ecuted a variation of L y to the amount expressed by rv, such as would 

 correspond to the variation a. 60. On the other haud we perform the 

 variation 6L 2 , to the amount (1— a). Then the total variation for is 



d&=[a+(l-a)] 60, 



6L = a. 6L X -f (1— a). 6L 2 . 



If now 6L\ > 6L 2 we obtain the maximum variation of 6L when we 

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

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

 of 60 and the adopted form of wave. 



When the greatest positive value of 6L is smaller than 60 then a 

 value for pf can be found that in any case will make 



\\ 2 6L>60 



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

 necessarily be a minimal form, will make the variation 6 (<P — 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 

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

 always finite variations of\Li and L 2 , at least for finite velocities of flow 

 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. Only 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 of waves 



have for every 60 a maximum of 6L of the same order of magnitude. 



^ r 

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



a value of f which shall be greater than the greatest of the values of 



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



6~0 



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



A is entirely excluded. 



Therefore stationary leaves of a prescribed icave-lencjth are only possible 

 for such values of the velocities of flow fr 2 and \h 2 as are less than cer- 

 tain definite extreme limits. 



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



