WILSON. — RECTILINEAR OSCILLATOR THEORY. 113 



which is nearly equivalent to it, since x and d 2 x/dt 2 are nearly equal, 

 the amplitude of the motion is seen to suffer a reduction in the ratio 



during n complete oscillations. We may now use experiments on the 

 distinctness of interference fringes for different path-lengths to 

 obtain an estimate for k, and we find that 2 1CH is an entirely 

 reasonable value. See Abraham, loc. cit.) 



The complete solution of (5') will be of the form 



x= d e aT + c~ 0T (C 2 cosyr -1- C 3 shm), 



where a,—/? ±71 are the roots of the cubic 



r°- + r - k r 3 = 0. 



The root a is positive and of great magnitude, approximately 1/k, 

 say, | 10 8 . As Planck says, this root has no physical significance. 6 

 This means that for every determination of the constants C\, C, 2 C 3 , 

 the value of C\ must vanish and the solution of the equation reduce 

 to 



x = p- /3r (C 2 cosyt + C 3 shvyr). 



It is certainly unfortunate that we are forced to discard so arbi- 

 trarily one root of an equation which we have labored so carefully to 

 establish. Moreover, as the solution of the equation now depends 

 only on two constants, the elimination of these constants by differ- 

 entiation will give an equation of the second order which accounts 

 satisfactorily for the phenomenon as far as its mathematical side is 

 concerned. This equation, which is almost identical with (7), con- 

 tains a friction term, it is true, and is unsatisfactory physically be- 

 cause of this fact. But there is no real reason for considering it as 

 less satisfactory than (5') which has a physically meaningless root and 

 which moreover, while ostensibly allowing for radiation on a physical 

 basis, actually requires the rate of radiation to be maximum where 

 there is a general agreement that it should be minimum, and minimum 

 where it should be maximum. 



3. Discussion of (6). We turn next to the equation (6') and, 

 for the sake of brevity, write it as 



Kp+ vf+ vx= 0, (8) 



/ and v designating respectively acceleration and velocity when the 



6 If we make clear the assumption of the quasi-stationary state as a prelimi- 

 nary condition in our approximations which lead to the equation, the large 

 root is sure to be inadmissible by virtue of our assumption. 



