Mr. A. J. Robertson on the Positive Wave of Translation. 523 

 through — — , if the distance traversed by it in the half-period 



were L. But this distance is Lh ; hence the velocity is 



increased in the proportion of L H : L ; 



L + V 



,\ V = 



- </(a + k)g=\/?±t.(a + 2k)g* (14.; 



It was determined before, from the oscillation of a single 

 column, that 



from (13.); 



/. c= ^.(a + *) 2 L= X= (a + k) 

 J -' v c 



= 7r(a + ^), if c=l and /*,= -° , . 



There is no proof that c=l ; but it is remarkable, that, if 

 it be true, then the half length of the wave =7r(a + &), which 

 agrees very exactly with the observations of Mr. Scott Rus- 

 sell, who has assigned from them a value L = 7ra. The 

 velocity, however, is independent of the length, and con- 

 sequently remains very nearly the same, although the wave 

 becomes lower and longer-}-. Is not the degradation of the 

 wave the result of imperfect fluidity ? 



The cause of the increase of velocity consequent upon an 

 increase of depth is evident; the volume of the generator 2V 

 being constant, the amount of horizontal movement of the 

 particles is inversely as the depth. 



The first five columns in the following table are copied 

 from Mr. Scott Russell's Second Report on Waves J. 



The column F gives the velocities calculated by the above 

 formula, and G gives the difference between observation and 

 theory. 



* The empirical formula given by Mr. Scott Russell is v= y (a-\-2Jc)g. 



f It is remarkable, that, if these principles be applied to oscillating waves* 

 as in the note, pp. 518, 519, where the middle line QN is the level of repose* 

 and where the particles having no positive motion of translation, the distance 

 travelled by the crest of the wave in the time T is only L, the velocity is 

 that acquired in falling through half the simple depth of the water j or since 

 the length varies as the depth, the velocity is as the square root of the 

 length. 



I Report of the British Association for 1844, page 336. 



