THE TWO GREAT SOLITARY WAVES OF RUSSELL. 33.5 



Equation (21) shews that waves which give the ratio between h and k the same, have 

 their lengths exactly proportional to the spaces through which they respectively transfer a particle 

 by transit past it. 



Equation (25) shews that in waves which have the same values of It and k, those will be 

 transmitted with the greatest velocity which are the longest ; and those with the least velocity 

 which are the shortest. 



We may conclude this portion of our investigations with the determination of the exact path 

 of each particle. The materials for this purpose are supplied by equations (10) and (l.O). In 

 both of them a is constant for our present purpose. The former gives, 



y = A cos {lit - a) (26), 



in which A is the maximum value of y for that particular particle. 

 Equation (19) gives 



c - ^,ir t= — - sec {nt - a); 

 k 



ch r 



.-. X = ct — / sec {nt - a) 



k J f 



c ch 



= - (nt - a) - —- log, 5 tan (nt - a) + sec (nt - a)] + constant; 

 n nk ° '■ ' 



t being eliminated between this and (26), we shall have the equation required ; which is 

 manifestly not that of an ellipse as has been found by approximate methods ; though as far as the 

 eye can judge in an experiment, it may not be distinguishable therefrom. 



It is very easy to shew from (2+) that the surface of a wave meets the level surface of the 

 quiescent fluid in a finite angle; and that under certain conditions it may have a point of contrary 

 flexure. The actual wave surface is only a symmetrical portion of the whole curve represented 

 by the equation (24). When a wave first reaches a particle d,iP = 0, and d,y = a finite quantity ; 

 consequently the initial motion of each particle is vertically upwards with a finite velocity. When 



it has described half its path diX = c [l —7)5 and d,^ = ; consequently its motion is then 



horizontal. At the termination of its motion d,a? = 0, and d,j/ = - (the initial velocity), so that 

 the final velocity is vertically downwards, and is finite ; which indicates that the motion ceases as 

 suddenly as it began. This seems to coincide either accurately, or very nearly so, with the 

 account Mr. Russell has given {Report, p. 342) of the observations he made on the motions of 

 individual particles in his experiments. 



Before we proceed to compare the formula (20) with the results of experiment, it is necessary 

 to advert again to a circumstance which has been already alluded to. The formula of (20) 

 involves A. The value of this quantity not having been recorded in Mr. Russell's tables, I have 

 been under the necessity, as the best substitute for exact measures, of having recourse to the 

 rule, which he has given in page .S43 of his Report, for computing its approximate value. In 

 the notation of this paper, that rule may with sufficient accuracy be represented by the equation 

 X = 8/t — 2/c; for it is not necessary in computing the value of v that X should be known with 

 extreme accuracy, as the term in (2.';) into which it enters is very small, and has but little effect 

 upon the value of c. With these premises we give the following table, exhibiting a comparison 

 of theory and experiment. 



