8 GwVTIIER, O71 the Propagation of a Solitary Wave. 



The values of A, : Ji-^ are given to show the rate of 

 increase for the larger values of ;;//3, and the values of the 

 other quantities for values of /;//3 less than tt; 18 have been 

 omitted because /a, : h\ being exceedingly small, and h\ : |3 

 being also small, there is little interest attaching to them. 

 It will be noticed that about the value 1377/72 of wj3, 

 the ratio //., : h^ being about one-twentieth, the values of 

 Ji^ : ft and tan 2inftJ2iuft shew signs of inordinate increase, 

 and it is about this point that I should place the limit by 

 which the validit}' of this investigation is bounded. It 

 also appears to be somewhere about the limit of the height 

 of the wave in e.xperimental investigations. 



For the purpose of comparison with the investigations 

 of Scott Russell on the velocity of the wave, on writing 



//'- for ^i,7) I I + - ), V- for ,t,'/-] tan 2w/5/2w/3, 



and Kfor Scott Russell's experimental value, we see from 

 these number.? that // and z' become about equal when /r : j3 is 

 about "04 and that for higher values of this ratio 7' is slightly 

 in excess of Ji, so that when Ii : j3 is about "4, v : u is about 

 ro3, and the ratio for higher values increases more rapidly. 

 In discussing a formula proposed by Airy, Scott 

 Russell* in his Report gives two tables of small and large 

 waves respectively, from which we can very conveniently 

 compare the values of F and ?( for values of /i : (5. It is 

 then seen that V is smaller than // for the smaller values 

 of// : ft and greater than u for the greater values of// : ft. 

 The values calculated from these tables (retaining the 

 order in the tables) run as follows: — 



Sm.\ll \V.\ves. 



* Brit. Assoc. Report, Vol. Ixiii. (1844) pp. 335—337. 



