134 Royal Society. 



force of gravity. Hence it appears that the velocity is greater for a 

 positive than for a negative wave in the same channel. 



Comparing this result with Mr. Scott Russell's experiments, it 

 appears that on fourteen experiments of positive waves, the total 

 error is 3J per cent, on the sum of all the velocities ; and that on 

 sixteen experiments of negative waves, the error is scarcely 2 per cent, 

 on the sum of all the velocities. The author infers that it may 

 therefore be safely considered that the experiments bear out the 

 theory, which shows that the positive and negative waves are phe- 

 nomena of the same class, and not distinct, as maintained by Mr. 

 Scott Russell and Mr. Earnshaw. 



The next point considered is the horizontal motion of the indi- 

 vidual particles occasioned by the passage of a wave. It is shown 

 that the velocity of any particle is shown by the equation 



2k 

 "^-^A + 2^ sin«g(c/-j^), 



• v u ^ K 9 ~ h^-1k\ 

 in which 9=-A / I ^,^ ,x • — =^ — )• 



""■ <2k 



At the crest of the wave this becomes + c- r ; and when the 



-' h^2k 



height of the wave is equal to the depth of the water, the velocity is 

 -, or half the velocity of the wave itself. In a negative wave the 



motion of the particles is in the direction opposite to that in which 

 the wave is moving. 



According to Mr. Scott Russell's observations, a wave breaks 

 when its height is equal to the depth of water, but the author con- 

 siders that Mr. Russell did not succeed in producing a wave of a 

 cusped form at once, and that it assumed that form and broke only 

 when it travelled up a channel with a rising bottom. He further 

 observes, that at the commencement and termination of the motion, 

 the direction of the particle is vertical ; under the crest of the wave 

 it is horizontal ; and that the path has an oval form, but evidently 

 not an ellipse. He next deduces X, the length of a wave, in terras 

 of T, the period of the wave, or the time which elapses from the 

 commencement to the termination of the motion of a particle, viz. 



on which equation he remarks, that setting aside the variation caused 

 by a change in the value of k, both in that of c and in the radical, 

 the length of the wave varies directly as T ; and that we have thus 

 an explanation of the observed fact, that a wave becomes gradually 

 diminished in height and increased in length. 



The author then enters on the subject of Oscillating Waves, He 

 remarks that, in the wave of translation, it appears from the preceding 

 investigation, that the horizontal and vertical motions of the particles 

 commence and terminate together ; and that consequently the par- 

 ticles left at rest when the wave has passed must continue in the 

 same state, unless some fresh disturbance again set them in motion. 



