6 GWYTIIEK, On the Propa<^ation of a Solitary Wave. 



rroceedinj^ with the investigation before us, we obtain 

 from the equations (7) and (8), by applying the conditions 

 proposed, 



(r"''=^/3 tan 2w/5/2w/> (9), 



and 



1 ^ + 9; 

 m/h _ A.1 I - cos 2/«/3 



sin 2W/3 / \^- 4 cos'-^2w/3 - I 



1 + 





• (lo)- 



Of these relations the first is identical with that found by 

 McCovvan, the second we shall consider later. With these 

 we combine the expression of the fact that we also suppose 

 that the defect of pressure is null at the crest of the wave. 

 This gives us 



tan 2W/3 I -7 pr- ., , . 



\ \ sin 2W/3 J ] 



It will be noted that the relations obtained from the 

 extreme part of the wave depend on the ratio X., : Ai ; while 

 those at the crest depend upon h.^ : Jix or A;, tan- ui /3 : Ai, 

 which for values of ni^ less than 7r/4 will be the smaller 

 ratio. 



Assuming Ji^ -.hi to be small, and writing 3 to stand for 

 ;;////sin 2vt^, we get 



\ + Z / I - s /aA 



cos 2;///5(l + 2Z)-\ \ + Z //i/ ' 



or, neglecting the small term 



cos 2w/5(i + 25)'''=: I + s (12). 



To test the agreement of results drawn from these ex- 

 pressions with the results of experiments on the solitary 

 wave is the only method which we can now adopt. We 

 notice that all the expressions recur with the period ir for 

 7?i^ ; also that c' has a negative value between the values 

 7r/4 and 77/2, 377/4 and tt, of mft. 



Calculating from these equations for values of wj3 



