134 



THE ROYAL SOCIETY OF CANADA 



The result arrived at is 



/ = v^ 



P A 



(3) 



where r is the radius of the cylinder and lo {x), Ii {x) are Besel's 

 functions.^ 



lo (x) = <^ 1 + 



X' 



T2 



+ 



2-4 • 2-4 



+ 



Il (x) = y 



1 + 



+ 



+ 



2-4 ' 2-4 • 2-2 • 2-4 

 As will be seen later, the values obtained for r and X are such 



2 Tcr 



that I] does not differ much from Iq and 



is large compared 



with unity. Under this condition formula (3) tends to coincide with 

 formula (2). 



Before discussing the experimental details of determining the 

 three unknowns AS r, and v of formula (3) , it may be interesting to notice 

 the photographs of a falling stream of water when the latter is illumin- 

 ated indirectly. (Figs. 1 and 2). 



The stream is allowed to flow into a basin of water, waves travel- 

 ling upwards from the point where the jet strikes the surface. In 

 each case the source of light illuminating the stream was placed in a 

 position such that the stream was illuminated from the side. The 

 hollows of the waves are clearly shown at the side of the stream farthest 

 away from the source of illum.ination. There are also curious light 

 effects shown along the centre of each jet. These are due to the waves, 

 each hollow and each crest acting as a lens. This effect enables one 

 to measure the wave-lengths very accurately by placing the source 

 of light directly behind the stream ; the light passing through each wave 

 hollow and each wave crest is thus focussed to a point, the rest of the 

 stream appearing to be dark. (Fig. 3.) 



The hollows focus the light to very fine points (see also Fig. 5.), 

 and the wave-length can be very accurately determined by measuring 

 the distance between two such points. It is plainly shown in Fig. 3 

 how the surface of the water rises to meet the stream. The dark 

 band above the water line is due to the meniscus against the side of 

 the glass box containing the water into which the stream is falling. 



With a view to utilizing the wave lenses, described above, the 

 apparatus was designed as shown in Fig. 4. 



^Numerical tables of these functions may be found in Brit. Assoc. Reports 1889, 

 1893, 1896. 



