certain Dilute Aqueous Solutions. 385 



How Joes the viscosity of the liquid affect the apparent wave- 

 lengths? Prof. Tait* has shown that viscosity does not 

 affect the true wave-length, but simply introduces a damping 

 factor of the form e _CLX , where a depends upon the wave- 

 length and upon the viscosity of the liquid. Hence the 

 apparent wave-length is the distance between the consecutive 

 corresponding points where the curve y = Ae _ou: sin mx has a 

 constant slope c. Hence it is necessary to find the roots of 

 the equation 



-~ = — Aa€~ aX sin ma + A)ne~ aX cos mx=c. 

 ax 



B cos {mx + 6) = 4- e°" , where B = a/^ + m 2 . ta n 6 — — • 



A v ' m 



Since a is very small for water and for dilute solutions, B is 

 approximately equal to m. The maximum value of c is at 



/> 

 the point x = where c = Am. Hence ^-r > 1, and the 



roots of the equation are the abscissas of the points of inter- 

 section of curves 



z = cos' (???&' + 6), and 



z = Ee a *=E(l +ax) approximately ; 



and E^-g-r > 1. By setting near a crest E can be made 



very small, and hence the slope of the line z = l£e aX will also 

 be small, practically zero, and the apparent wave-length 



must be very nearly — , which is the true wave-length. If 



we could make c exactly zero, the apparent and the real 

 wave-lengths would be exactly equal. 



To test experimentally the error that might be introduced 

 if c were not zero, the telescope was turned so as to look on 

 one side of the crests at points having as great a slope as 

 could be observed, and the apparent wave-length was 

 measured. It was then turned so as to look at corresponding- 

 points on the other side of the crests, and the wave-length 

 was again determined. Each time I measured over a distance 

 equal to twenty waves, which is the interval usually employed 

 in the rest of the work. The maximum difference of the two 

 series amounted to one-fifth of one per cent. The apparent 

 field swept over by the spider-line in turning the telescope 



* Proc. Roy. Soc. Edinburgh, xvii. pp. 110-115 (1890). 

 Phil. Mag. 8. 5. Vol. 44. No. 270. Nov. 1897. 2 F 



