PAPER BY PROF HELMHOLTZ. 1Q1 



The equations (2) and (2a) remain true when we increase either the* 

 values of the two coordinates x and y or those of i\, x or rp 2 in any given 

 ratio. Since the densities s x and s 2 do not occur in these two equations, 

 therefore also these can change to any amount. But equation (3) re- 

 quires that the quantities 



jV&Y 1 and -*-(.* f ! 



shall remain unchanged. When therefore s, and s 2 vary and we put 

 their ratio 



»2 



and when further the coordinates increase by the factor w, but tp x , by 

 the factor a, and y> 2 by the factor « 2 , then the quantities 



1 — a n' l — ff n 3 



must both remain unchanged. 



Or when we, in the expressions for these quantities, put 



b l= ai and b 2 =?2 

 n n 



as the ratios by which the velocities are altered, then the above propo- 

 sition becomes equivalent to saying that the geometrically similar wave- 

 forms can occur when 



and 



1—ff n 1—6 n 



remaiu unchanged. 



(1) If the ratios of the densities are not changed then in geometrically sim- 

 ilar waves, the linear dimensions increase as the squares of the velocities of 

 the two media ; the velocities therefore will increase in equal ratios. 



Therefore for a doubled velocity of the wind we shall have waves of 

 four times the linear dimensions. 



This proposition is not limited to stationary movements, but is quite 

 general.* The following propositions however will hold good only for 

 stationary waves. 



(2) When the ratio of the density a is varied, the quantities 



A 2 _«i &i 2 

 > b 2 2 ~s 2 b 2 



Gin— — rpconst. 



*See my paper "On a Theorem relative to geometrically similar movements of Fluid 

 Bodies," in the Monats b. tier Akad. Berlin, 1873, pages 501 to 514 ; [or see No. IV of 

 this collection of Translations.] 



