PHENOMENA OF ALL MICROSCOPES 21 



the same amplitude as in Fig. II. 4, provided that the numerical value 

 of A remains unchanged, but would be found to lead the S wave by 

 3^ wavelength. These conclusions about the phase difference between 

 the undeviated and deviated waves are so important to phase microscopy 

 that they are worth stating as the following theorem (Theorem 1): 



When the light transmissions of the particle and the surround are equal 

 and when the optical path of the particle differs from the optical path of the 

 surround by a sinall fraction of a wavelength, the deviated wave is retarded 

 or advanced in phase by }4 wavelength with respect to the undeviated wave 

 according as the optical path of the particle exceeds or is less than the optical 

 path of an equal thickness of the surround. 



Since the deviated wave is focused upon the neighborhood of the 

 geometrical image of the particle and since the undeviated wave is 



The statement that the sum of the D and S waves shall be equal to the P wave 

 is now equivalent to writing 



Vp = Vs + yd 



or 



s sin {z + A) = s sin 2 + d sin (z + 4>). (4) 



If A is so small that sin A — ► A, Eq. 4 reduces to the equation 



' .5 A cos z = d cos 4> sin z + d sin 4> cos z. (5) 



If the waves D + S are to be equal to the P wave for all values of z, Eq. 5 must be 

 true for all values of z. But this situation holds if and only if 



d cos 4> sin 2=0; 



sA cos z = d sin cos z. (6) 



Hence cos ^ = so that 



*=±^ (7) 



with 



sA = d sin </>. (8) 



When A > so that the optical path of the particle exceeds that of the surround, 

 we must choose the alternative 4> = +7r/2, since neither s nor d can be negative. 

 When A < so that the optical path of the particle is less than that of the surround, 

 we must choose the alternative (/> = — 7r/2. For either alternative the amplitude d 

 of the deviated wave is given from Eq. 8 by 



d=s\A\, (9) 



in which [a] denotes the absolute value of A. We see, therefore, that the amplitude 

 of the deviated wave is pr(jportional to both s and |a| when A is small and that the 

 deviated wave is retarded l)y ±34 wavelength with respect to the undeviated wave 

 according as A is greater than or less than zero, i.e., according as the optical path of 

 the particle is greater than or less than the optical path of an equal thickness of 

 the surround. 



