214 



THEORY OF MICROSCOPIC OBSERVATION. 



of the angles of aperture a> and S, the thickness of the wall, and the 

 refractive power of the different media. For the determination 

 of the latter by a formula which will be generally valid, the 

 following consideration may be utilized. If the ray $ T 

 (Fig. 115), incident from below, were, as in the .previous case, 

 reflected at the inner wall, the angle of deviation of the ray 

 emerging towards R' would be given by the equation 



2p = 4 a" + 2a - 2 (a" + a). 



As there is no reflexion, the path of the rays undergoes the 

 same modification as if, in the above figure, the line C N' and 

 the reflected ray corresponding with it were turned, like the. hand 

 of a watch, round the point 6', but to the left, till C N' coincides 

 with C N. If we mentally follow this movement it will be seen 

 that the angle of deviation 2 p is increased by the angle N' C N = 

 180 2 a". Adding this value to the one above obtained, we 

 get the angle of deviation 2 p for the ray which has been refracted 

 four times, and hence 



p = 90 + a" + a - (a" + a). 



The direction of the emergent ray is determined from this for any 

 given inclination of the incident ray. If, therefore, the angles of 

 aperture of the objective (o>) and of the diaphragm (8) are given, 

 the limits of the umbra and penumbra in the plane of adjustment 

 can be calculated. For the distances of these limiting lines from 



the centre, if R = 1, n = , ^ , <w = 60, and B = 12, we 



1'ooOb 



obtain the values given in the following table : 



In reality the limits of the shadow always lie somewhat 

 further inwards, if the above suppositions are approximately 

 accurate. It is evidently in consequence of this, that the inten- 

 sity of the marginal rays, for which a"', for instance, may be 80 

 and upwards, is considerably weakened through the repeated 

 reflexions at the refracting surface, and will therefore become nil 

 for the observing eye before it should do so theoretically. Hence 



