On the Photography of Hippies. 33i) 



Tims for gravity waves 



^i = eoth "Itt/j,-, 

 A, 



and for capillary ripples 



S 1 ^ l 



(x = eotn 2tt/jl , 



A, 



while in the general ease 



h 



y[t P = COth 27TyU - , 

 A, 



where p is a positive number having 1 and 3 as its extreme 

 limits. 



Now suppose the minimum depth is 4 millim. and we wish 

 to set a refractive index of 2. Then 



•4 5 



2 p = coth 47r — = coth - , approx. 

 \ A. 



Let p= 1, then* 

 and 



coth - =2, 

 A. 



\ = 9 centim., roughly. 



Let /> = 3, then X = 40 centim., approximately. 



Thus, if we used a liquid whose minimum depth was 

 4 millim., no matter what its surface-tension and density were, 

 we should have to employ waves whose length was between 

 9 centim. and 40 centim., in order to get a refractive index of 

 2. Suppose the wave-length was arbitrarily taken in the 

 case of mercury to be 20 centim. in the deep liquid. In the 

 shallow portion it w r ould still be long enough for us to neglect 

 the effect of surface-tension, and the equation 



(i = coth 2tt/jl - 



would apply to this case very nearly. It gives the relation 

 between /a, h, and \ when the wave-length is taken to be over 

 20 centim. in the deep liquid, and the value of A is such as not 

 to make the refracted waves less than about 8 centim. long. 



If we neglect the difficulty caused by reflexion from the sides 

 of the trough (which would be overcome by some such device 

 as a flexible boundary or a hanging fringe), this means that 

 the trough to hold the mercury would have to be at least 



* For the approximate solution of such equations the seuii-logarithmic 

 chart of the hyperbolic functions is useful. See Vincent, Brit. Assoc. 

 Uep. 1898, ' ; On the Use of Logarithmic Coordinates,'' plate iii. 



2 B 2 



