certain Dilute At/neons Solutions. 373 



where p is the density of the air in contact with the surface. 

 For the accuracy obtained in measuring A,, the first and 

 simpler formula is sufficiently exact. 



Of the terms on the right of these equations, the first is 

 much the larger for small waves. Hence the value calculated 

 for T will be proportional to the cube of the wave-length and 

 to the square of the frequency. Here is the first objection to 

 this method. The value found for T will be only one-third as 

 accurate as the measurement of the wave-length, and one-half 

 as accurate as the determination of the frequency. Again, 

 since the formulas are deduced for waves of infinitely small 

 amplitudes, we must use very moderate vibrations, and these 

 are difficult to measure. 



Several, among whom we need mention only Matthiessen, 

 TJiess, and Ahrendt, have attempted to test experimentally the 

 extreme accuracy of Lord Kelvin's formula. 



When a needle is placed so that its point dips slightly into 

 a jet of water, a series of stationary waves is formed behind 

 the needle, and their length is equal to that of waves whose 

 velocity is equal to the velocity of the surface of the jet at that 

 point. This is the method employed by Ahrendt *. He found 

 that the value of the surface-tension calculated from the ob- 

 served wave-length is too great, but that it decreases as the 

 needle is removed from the orifice. This proved that the fault 

 lay in assuming that the surface of the jet moves with the 

 velocity calculated for the ideal case. In reality, the surface 

 has a smaller velocity, probably owing to friction with the edge 

 of the orifice and with the air. Hence he proved nothing, so 

 far as the formula is concerned. 



Both Riess | and Matthiessen $ failed to satisfy two of 

 the conditions which were assumed by Lord Kelvin in 

 obtaining his formula. One of these is that the amplitude of 

 the waves shall be very small in comparison with the wave- 

 length ; and the other is that the surface of the liquid may be 

 represented by an equation of the form 



y = a sin {mx + nt) ; 



i. e., the formula is deduced for the case of a single series of 

 very small plane waves. The waves employed by these ob- 

 servers were of such an amplitude that they could be easily 

 seen with the unaided eye : this alone might very probably 

 introduce a discrepancy between the observed and the cal- 



* Exner's Pep. der Phys. xxiv. p. 318 (1888). 

 t Exner's Pep. der Phys. xxvi. p. 102 (1890). 



\ Pogg. Ann. cxxxiv. p. 107, cxli. p. 375 ; Wied. Ann. xxxii. p. 626, 

 xxxviii. p. 118. 



