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HOLLOW SPHERES AND HOLLOW CYLIND 





213 



If the refractive index of the substance of the cylinder rises 

 to 1'5, the bright line is moved somewhat further inwards. Its 

 distance is reduced therefore to '5458 R, \i r = -8 R ; the difference 

 rises in this case to '0054. The influence of the cylinder-wall 

 upon the distances in question is therefore in all cases limited to 

 the third decimal, as the preceding examples prove ; consequently 

 the displacement of the line produced by it amounts to less than 

 1 mic., when R is less than 100 mic., and may reasonably be 

 neglected. In this case we can determine, from the position of 

 the line in the enclosed air, the radius r of the lumen, and there- 

 fore also the thickness of the wall (R r). We therefore get, in 

 accordance with what has preceded for the air-bubble, F = '68836 . r, 

 and consequently for the hollow cylinder, taking as mean dis- 

 placement -00636 .r, F = '683 . r, whence r = . = 1464 F. 



The quantity Fis most accurately determined by direct measurement 

 of 2 F t that is, of the distance of the bright lines on either side. 



The inner bright line lies nearer to the centre than any other 

 point which is illuminated in the same way, precisely as the 

 inner ring in the air-bubble. Consequently there appears within 

 it (with reference to this particular path of rays only) a perfectly 

 dark umbra, but further outwards a faintly luminous penumbra. 

 Under favourable circumstances other bright lines appear in the 

 inner part of the penumbra, their origin being explained by 

 repeated internal reflexions. 



We have finally to consider 

 the rays which penetrate the 

 lumen without internal re- s 

 flexion, and which undergo, 

 in consequence, only a four- 

 fold refraction. It is clear 

 from what has already been 

 said, that they will illuminate 

 the central portion of the 

 hollow cylinder, but disappear 

 towards the edges. The points 

 where the penumbra begins, 

 and where it passes into the 

 umbra, alter their position in 

 accordance with the magnitude FIG. 115. 



N' 



