PAPER BY PKOF IIELMIIOLTZ. 101 



The equations (2) and (2a) reniain trne when we increase either the 

 values of the two coordinates x and y or those of y-j or tf-2 in any given 

 ratio. Since the densities Si and Sz do not occlir in these two equations, 

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

 quires that the quantities 



^^r* Y I and -j^rm i 



shall remain unchanged. When therefore Si and s-z vary and we ])ut 

 their ratio 



S-z 



-and when further the coordinates increase by the factor n, but ipi by 

 the factor «i and </'2 by the factor (tz, then the quantities 



0- «i' „„^ 1 «2' 



and 



l — ff'n^ l — <j'n^ 



must both remain unchanged. 



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



6, = ^iand&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 "'' 



l — ff n 1 — <J n 



remain unchanged. 



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

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

 the tico 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 j)roposition 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 g is varied, the quantities 



bi^ Si bi^ 



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

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

 this collection of Translations. ] 



